Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements

For irreducible characters $\{ \chi_q^\lambda \,|\, \lambda \vdash n \}$, induced sign characters $\{ \epsilon_q^\lambda \,|\, \lambda \vdash n \}$, and induced trivial characters $\{ \eta_q^\lambda \,|\, \lambda \vdash n \}$ of the Hecke algebra $H_n(q)$, and Kazhdan-Lusztig basis elements $C'_w(q)$ with $w$ avoiding the patterns 3412 and 4231, we combinatorially interpret the polynomials $\chi_q^\lambda(q^{l(w)/2}C'_w(q))$, $\epsilon_q^\lambda(q^{l(w)/2} C'_w(q))$, and $\smash{\eta_q^\lambda(q^{l(w)/2} C'_w(q))}$. This gives a new algebraic interpretation of chromatic quasisymmetric functions of Shareshian and Wachs, and a new combinatorial interpretation of special cases of results of Haiman. We prove similar results for other $H_n(q)$-traces, and confirm a formula conjectured by Haiman.


Introduction
The symmetric group algebra ZS n and the (Iwahori-) Hecke algebra H n (q) have similar presentations as algebras over Z and Z[q 1 2 , q¯1 2 ] respectively, with multiplicative identity elements e and T e , generators s 1 , . . . , s n−1 and T s 1 , . . . , T s n−1 , and relations s 2 i = e T 2 s i = (q − 1)T s i + qT e for i = 1, . . . , n − 1, s i s j s i = s j s i s j T s i T s j T s i = T s j T s i T s j for |i − j| = 1, s i s j = s j s i T s i T s j = T s j T s i for |i − j| 2.
Analogous to the natural basis S n of ZS n is the natural basis {T w | w ∈ S n } of H n (q), where we define T w = T s i 1 · · · T s i whenever s i 1 · · · s i is a reduced (short as possible) expression for w in S n . We call the length of w and write = (w). It is known that (w) is equal to inv(w), the number of inversions in the one-line notation w 1 · · · w n of w, i.e., the number of pairs (i, j) with i < j and w i > w j . The specialization of H n (q) at q 1 2 = 1 is isomorphic to ZS n .
In addition to the natural bases of ZS n and H n (q), we have the (signless) Kazhdan-Lusztig bases [13] {C w (1) | w ∈ S n }, {C w (q) | w ∈ S n }, defined in terms of certain where denotes the Bruhat order on S n and we define q v,w = q (w)− (v) 2 . We have the identity P v,w (q) = 1 when w avoids the patterns 3412 and 4231, i.e., when no subword w i 1 w i 2 w i 3 w i 4 of w 1 · · · w n consists of letters which appear in the same relative order as 3412 or 4231 [17]. These particular permutations are of interest to algebraic geometers because they correspond to smooth Schubert varieties. (See [3,Ch. 4,Ch. 13].) Representations of ZS n and H n (q) are often studied in terms of linear maps called characters. (See [22,Ch. 1] for definitions.) The span of the S n -characters is called the space of S n -class functions, and has dimension equal to the number of integer partitions of n. Three well-studied bases are the irreducible characters {χ λ | λ n}, induced sign characters { λ | λ n}, and induced trivial characters {η λ | λ n}, where λ n denotes that λ is a partition of n. Letting S λ denote the Young subgroup of S n of type λ, we have λ := sgn ↑ Sn S λ , η λ := triv ↑ Sn S λ . The span of the H n (q)-characters, called the space of H n (q)-traces, has the same dimension and analogous character bases {χ λ q | λ n}, { λ q | λ n}, {η λ q | λ n}, specializing at q 1 2 = 1 to the S n -character bases. Each of the two spaces has a fourth basis consisting of monomial class functions {φ λ | λ n} or traces {φ λ q | λ n}, and a fifth basis consisting of power sum class functions {ψ λ | λ n} or traces {ψ λ q | λ n}. These are defined via the inverse Kostka numbers {K −1 λ,µ | λ, µ n} and the numbers {L λ,µ | λ, µ n} of row-constant Young tableaux of shape λ and content µ by Each of these functions is not a character, but is a difference of two characters. In each space, the five bases are related to one another by the same transition matrices which the electronic journal of combinatorics 23(2) (2016), #P2.7 relate the Schur {s λ | λ n}, elementary {e λ | λ n}, complete homogeneous {h λ | λ n}, monomial {m λ | λ n}, and power sum {p λ | λ n} bases of the space Λ n of homogeneous degree n symmetric functions. It follows from the theory of symmetric functions that basis elements in each space are integer linear combinations of irreducible characters in that space. (For more information on the transition matrices and symmetric functions, see [2,Sec. 2], [29,Ch. 7], respectively.) A correspondence between these class functions and symmetric functions is given by the Frobenius characteristic map ch(θ) := 1 n! w∈Sn θ(w)p ctype(w) , where ctype(w) is the cycle type of w. In particular, we have ch(χ λ ) = s λ , ch( λ ) = e λ , ch(η λ ) = h λ , ch(φ λ ) = m λ , ch(ψ λ ) = p λ , which explains our names for the fourth and fifth bases. We naturally use the Greek ancestors , η of e, h in our class function notation, but we prefer not to use the ancestors σ, µ, π of s, m, p. Instead we follow the standard practices of using χ for irreducible characters, while reserving µ for integer partitions, and π for path families in planar networks. We follow [11], [32] in using φ, and we use ψ because psi begins with p.
For any S n -class function θ belonging to the bases above, and for any element z of the natural or Kazhdan-Lusztig basis of ZS n , we have θ(z) ∈ Z. This follows from the linearity of θ and the fact that χ λ (w) can be expressed as the trace of an integer matrix. (See, e.g., [22,Sec. 2.3].) On the other hand, we do not in general have an elementary formula for the integer θ(z). This incomplete understanding of the S n -class functions is unfortunate, since the functions encode much important information about S n . For some class functions and basis elements, we may associate sets R, S to the pair (θ, z) to combinatorially interpret the integer θ(z) as (−1) |S| |R|, or simply as |R| if θ(z) ∈ N. We summarize results and open problems in the following table.
Our understanding of H n (q)-traces is even less complete. We know that irreducible H n (q)-characters {χ λ q | λ n} satisfy χ λ q (T w ) ∈ Z[q] for all w ∈ S n , since χ λ q (T w ) can be expressed as the trace of a Z[q] matrix [13]. Thus for any element θ q of the mentioned H n (q)-trace bases and any element z ∈ span Z[q] {T w | w ∈ S n }, we have θ q (z) ∈ Z[q] as well. For instance, elements of a modified Kazhdan-Lusztig basis {q e,w C w (q) | w ∈ S n } belong to this span. On the other hand, we do not have a general elementary formula for the polynomial θ q (z). This is unfortunate, since H n (q)-characters are important in the study of H n (q) and quantum groups. In some cases, we may associate sequences (S k ) k 0 , (R k ) 0 of sets to the pair (θ q , z) to combinatorially interpret θ q (z) as k (−1) |S k | |R k |q k , or simply as k |R k |q k if θ q (z) ∈ N[q]. We summarize results and open problems in the following table.
interpretation of θ q (T w ) as interpretation of θ q (q e,w C w (q)) as k |R k |q k for w avoiding 3412 and 4231? The polynomial χ λ q (T w ), and therefore all polynomials θ q (T w ) above, may be computed via a q-extension of the Murnaghan-Nakayama algorithm, developed in [14], [15], [21], [33]. However, none of these has a conjectured expression of the type stated above. Interpretations of θ q (q e,w C w (q)) are not known for general w ∈ S n , but results concerning containment in N[q] follow principally from work of Haiman [11]. In all cases above, coefficients of θ q (q e,w C w (q)) are symmetric about q e,w . They are also unimodal for θ q ∈ {η λ q , λ q , χ λ q } and conjectured to be so for θ q ∈ {ψ λ q , φ λ q } [11, Lem. 1.1, Conj. 2.1]. For w avoiding the patterns 3412 and 4231, formulas for η λ q (q e,w C w (q)), λ q (q e,w C w (q)), χ λ q (q e,w C w (q)), and ψ λ q (q e,w C w (q)) follow from work of Athanasiadis [1], Gasharov [7], Shareshian-Wachs [24], and the authors. There is no conjectured combinatorial interpretation of φ λ q (q e,w C w (q)), the electronic journal of combinatorics 23(2) (2016), #P2.7 even for w avoiding the patterns 3412, 4231, although for particular partitions λ interpretations are given by the authors in Section 10. The problem of combinatorially interpreting θ q (q e,w C w (q)) when w does not avoid the patterns 3412 and 4231 is open. Another way to understand the evaluations θ(w) is to define a generating function Imm θ (x) for {θ(w) | w ∈ S n } in the polynomial ring Z[x 1,1 , . . . , x n,n ]. Similarly, we may define a generating function Imm θq (x) for {θ q (T w ) | w ∈ S n } in a certain noncommutative ring A n (q). In some cases these generating functions have simple forms. We summarize known results in the following tables. Simple expressions for Imm η λ (x) and Imm λ (x) are due to Littlewood [19] and Merris-Watkins [20], and a simple expression for Imm ψ λ (x) follows immediately from a standard definition of ψ. An expression for Imm χ λ (x) as a coefficient of a generating function in two sets of variables was given by Goulden-Jackson [9]. There is no conjectured simple formula for Imm φ λ (x), although a simple formula for particular partitions λ was stated by Stembridge [32]. Simple expressions for Imm η λ q (x) and Imm λ q (x) are due to the fourth author and Konvalinka [16], as is a (less simple) expression for Imm χ λ q (x) as a coefficient in a generating function in two sets of variables.
In Section 2 we discuss generating functions Imm θ (x) for S n -class functions θ and generating functions Imm θq (x) for H n (q)-traces θ q . These generating functions belong to the ring Z[x 1,1 , . . . , x n,n ] and to a certain q-analog A n (q) of Z[x 1,1 , . . . , x n,n ] known as the quantum matrix bialgebra. In Section 3 we relate these generating functions to structures called zig-zag networks [25], which serve as combinatorial interpretations for Kazhdan-Lusztig basis elements indexed by permutations avoiding the patterns 3412 and 4231. In Section 4 we introduce a partial order on paths in zig-zag networks, and show how this poset and results in the literature lead to combinatorial interpretations for the evaluations of S n -class functions at the above Kazhdan-Lusztig basis elements of ZS n . For the remainder of the article, we concentrate on combinatorial interpretations of trace evaulations of the form θ q (q e,w C w (q)) where w avoids the patterns 3412 and 4231. In Sections 5 -6 we interpret η λ q (q e,w C w (q)) and λ q (q e,w C w (q)). In Section 7 we recall the relationship between S n -class functions and Stanley's chromatic symmetric functions, and prove that H n (q)-traces are similarly related to the Shareshian-Wachs chromatic quasisymmetric functions. In Sections 8 -9 we use results of Shareshian-Wachs and Athanasiadis to interpret χ λ q (q e,w C w (q)) and ψ λ q (q e,w C w (q)). One of our interpretations proves a formula conjectured by Haiman [11,Conj. 4.1]. Finally, in Section 10 we state several results concerning φ λ q (q e,w C w (q)).
the electronic journal of combinatorics 23(2) (2016), #P2.7 2 Generating functions for θ(w) and θ q (T w ) when θ (θ q ) is fixed For a fixed S n -class function θ, we create a generating function for {θ(w) | w ∈ S n } by writing a matrix of variables x = (x i,j ) i,j∈[n] and defining where w 1 · · · w n is the one-line notation of w. We call this polynomial the θ-immanant.
A simple formula for the ψ λ -immanant involves a sum over all permutations of cycle type λ. Specifically, we have where z λ is the product 1 α 1 2 α 2 · · · n αn α 1 ! · · · α n !, and λ has α i components (or parts) equal to i for i = 1, . . . , n. No such simple formulas are known for the χ λ -immanants or φ λimmanants in general, although Stembridge gave a formula [32,Thm. 2.8] for Imm φ λ (x) when λ is the rectangular partition (k) r = (k, . . . , k). In this case we have where the sum is over all ordered set partitions of [n] = [kr] of type (r) k . (See [9] for an expression for Imm χ λ (x) as a coefficient of a generating function in two sets of variables.) For a fixed H n (q)-trace θ q , we create a generating function for {θ q (T w ) | w ∈ S n } as before, except that we interpret polynomials in x = (x i,j ) as elements of the quantum the electronic journal of combinatorics 23(2) (2016), #P2.7 matrix bialgebra A n (q), the noncommutative Z[q 1 2 , q¯1 2 ]-algebra generated by n 2 variables x 1,1 , . . . , x n,n , subject to the relations for all indices 1 i < j n and 1 k < n. As a Z[q 1 2 , q¯1 2 ]-module, A n (q) has a natural basis of monomials x 1 ,m 1 · · · x r ,mr in which index pairs appear in lexicographic order. The relations (2.4) allow one to express other monomials in terms of this natural basis.
As a generating function for {θ q (T w ) | w ∈ S n }, we define in A n (q), and call this the θ q -immanant. The H n (q) sign character (T w → (−1) (w) ) immanant and trivial character (T w → q (w) ) immanant are called the quantum determinant and quantum permanent, Specializing A n (q) at q 1 2 = 1 gives the commutative polynomial ring Z[x], with elements det q (x) and per q (x) specializing to the classical determinant det(x) and permanent per(x). Simple formulas for the λ q -immanants and η λ q -immanants employ quantum determinants and quantum permanents of submatrices of x. In particular, the fourth author and Konvalinka [16,Thm. 5.4] proved quantum analogs of the Littlewood-Merris-Watkins identities (2.1), Imm λ q (x) = (I 1 ,...,Ir) det q (x I 1 ,I 1 ) · · · det q (x Ir,Ir ), Imm η λ q (x) = (I 1 ,...,Ir) per q (x I 1 ,I 1 ) · · · per q (x Ir,Ir ), (2.5) where the sums are as in (2.1). See [16] for an expression for the χ λ q -immanant as a coefficient of a generating function in two sets of variables. No formulas are known for the ψ λ q -or φ λ q -immanants. It would be interesting to state a q-analog of (2.3) for rectangular partitions λ = (k) r .

Planar networks and path matrices
Call a directed planar graph G a planar network of order n if it is acyclic and may be embedded in a disc with 2n boundary vertices labeled clockwise as source 1, . . . , source n (with indegrees of 0) and sink n, . . . , sink 1 (with outdegrees of 0). In figures, we will draw sources on the left and sinks on the right, implicitly labeled 1, . . . , n from bottom the electronic journal of combinatorics 23(2) (2016), #P2.7 to top. Edges will be implicitly oriented from left to right. Given a planar network G, define the path matrix B = B(G) = (b i,j ) of G by b i,j = number of paths in G from source i to sink j. (3.1) The path matrix of any planar network is totally nonnegative, i.e., each square submatrix has nonnegative determinant. Specifically, for sets I = {i 1 , . . . , i k }, J = {j 1 , . . . , j k } ⊂ [n], and the corresponding submatrix B I,J := (b i,j ) i∈I,j∈J , we have that det(B I,J ) is equal to the number of families (π 1 , . . . , π k ) of mutually nonintersecting paths from sources i 1 , . . . , i k (respectively) to sinks j 1 , . . . , j k (respectively). This fact is known as Lindström's Lemma [18]. (See also [12].) We will call two planar networks G 1 , G 2 isomorphic and will write For example, consider two isomorphic planar networks, their common path matrix B and its submatrix B 23,13 : We can interpret det(B 23,13 ) = 2 as counting the two path families An easy fact about planar networks is the following.
Observation 3.1. Let G be a planar network of order n and assume that for some indices i, i , j, j ∈ [n], G contains a path π i from source i to sink j and a path π i from source i to sink j . If these two paths intersect, then G also contains a path from source i to sink j and a path from source i to sink j. If i < i and j < j then the paths π i and π i cross.
Given planar networks G, H of order n, in which all sources have outdegree 1 and all sinks have indegree 1, define G • H to be the concatenation of G and H, where for the electronic journal of combinatorics 23(2) (2016), #P2.7 i = 1, . . . , n, sink i of G is dropped, source i of H is dropped, and the unique edge in G from vertex x to sink i and the unique edge in H from source i to vertex y are merged to form a single edge from x to y in G•H. Note that for star networks We will be interested in concatenations consists of t distinct, pairwise nonnesting intervals, We define a relation ≺· on the set of intervals appearing in the concatenation (3.2) by This is the covering relation of a partial order . For each planar network G of the form (3.2) satisfying the conditions following (3.2), we define a related planar network F by considering each covering pair and deleting all but one of the k paths from the central vertex of G [c i ,d i ] to the central vertex of G [c j ,d j ] . Following [25], we call the resulting network F a zig-zag network. In the special case that we have c 1 > · · · > c t , we call F a descending star network. The descending star networks (up to isomorphism) of order 4 are The zig-zag networks of order 4 which are not descending star networks are . (3.4) It is easy to see that if F is a zig-zag network of order n, then there is at most one interval in the concatenation (3.2) containing n, and this interval must be maximal or minimal (or both) in the partial order .
It was shown in [25,Thm. 3.5,Lem. 5.3] that zig-zag networks of order n correspond bijectively to 3412-avoiding, 4231-avoiding permutations in S n . To summarize this bijection, we let F be the zig-zag network obtained from the concatenation G in (3.2), and the electronic journal of combinatorics 23(2) (2016), #P2.7 construct another concatenation of star networks as follows.
.) Call the resulting augmented network G . Now visually follow paths from sources to sinks in G , passing "straight" through each star, to create a 3412-avoiding, 4231-avoiding permutation w ∈ S n . See [25,Sec. 3] for a description of the inverse of this bijection, which maps w to G . We will let F w and G w denote the zig-zag network and augmented star network corresponding to a fixed 3412-avoiding, 4231-avoiding permutation w, and we will let w(F ) denote the 3412-avoiding, 4231-avoiding permutation corresponding to a fixed zig-zag network F .
For example, suppose that F is the zig-zag network obtained from the concatenation G = G [3,7] • G [5,8] • G [8,9] • G [1,2] • G [2,4] of star networks of order 9. Drawing the poset on these intervals from left to right, we have Call a sequence π = (π 1 , . . . , π n ) of source-to-sink paths in a planar network G of order n a path family. We will always assume that path π i begins at source i. If for some w ∈ S n with one-line notation w 1 · · · w n , each component path π i terminates at sink w i , we will say that π has type w and we will write type(π) = w. If the union of the paths of π is equal to G, we will say that π covers G. For example, the planar network G [2,4] • G [1,3] can be covered by many different path families, including two of type e, and two of type 2341: . (3.5) The result [25,Lem. 5.3] states that the path families covering a zig-zag network F w correspond bijectively to elements of a principal order ideal in the Bruhat order: Thus w is the unique Bruhat-maximal permutation for which some path family of type w covers F w . It follows also that there is exactly one path in F w from any source i to the corresponding sink i, and at most one path from source i to sink j = i. Whether or not such a path exists may be determined by the intervals in the corresponding concatenation of star networks and the partial order .
Observation 3.4. Let F be a zig-zag network which corresponds to the concatenation (3.2). There exists a path in F from source i to sink j if and only if we have one of the following.
It is easy to see that if F is a descending star network with sources i and j belonging to the same connected component, then the second condition in Observation 3.4 is equivalent to the inequality i > j. This fact allows us to refine Observation 3.1 slightly.
Lemma 3.5. Let π 1 , π 2 be paths in a descending star network F from sources 1 < 2 to sinks m 1 , m 2 , respectively. Then the paths π 1 and π 2 intersect if and only if there exists a path in F from source 1 to sink m 2 .
Proof. Assume that F has order n and corresponds to the concatenation (3.2) of star networks. Let x 1 , . . . , x t be the vertices in F corresponding to the central vertices of the t star networks in (3.2), and for any index j ∈ [n], let f (j) and g(j) denote the indices of the first and last intervals, respectively, in [c 1 , d 1 ], . . . , [c t , d t ] to contain j. Then for any indices , m ∈ [n], the unique path in F from source to sink m contains the vertices x f ( ) , . . . , x g(m) . It is easy to see that if the intersection of π 1 and π 2 is nonempty, then there is a path in F from source 1 to sink m 2 . Now suppose that there is a path in F from source 1 to sink m 2 . By Observation 3.4, we have m 2 < 1 or we have an interval [c k , d k ] containing both 1 and m 2 . Either case implies that we have f ( 2 ) f ( 1 ) g(m 2 ), and the paths π 1 and π 2 share the vertex x f ( 1 ) .
The subset of zig-zag networks which are descending star networks can be characterized using pattern avoidance. Proof. Let G be the concatenation (3.2) of star networks which leads to F v , and define G as in the example above.
Suppose first that F v is not a descending star network. Then in the concatenation (3.2) there exists an index i such that the interval [c i , d i ] is minimal in the poset , and an index j > i such that Considering the relationship between G and v, one sees that we have v d i < c j and that for some index Now suppose that F v is a descending star network. We claim that if F v arises from the concatenation (3.2), then v avoids the pattern 312 and satisfies v c 1 > · · · > v d 1 . By inspection of (3.3), this is true for descending star networks of orders 1 -4. Now assume this to be true for each descending star network F w of order 1, . . . , n − 1, and let F v be a descending star network of order n. If in the first interval of (3.2) we have d 1 < n, then v n = n and F v 1 ···v n−1 is a descending star network of order n − 1. Thus F v has the claimed properties. If in the first interval of (3.2) we have d 1 = n, then consider the descending star network F w arising from the concatenation b] is the unique Bruhat-maximal permutation for which a path family of this type covers the star network G [a,b] . By the above argument, w avoids the pattern 312, and satisfies w c 2 > · · · > w d 2 , and w i = i for i = d 2 + 1, . . . , n. It follows that the letters in positions c 1 , . . . , n of s [c 1 ,d 2 ] w form an increasing sequence, and that v satisfies v c 1 > · · · > v n . Thus the subword v 1 · · · v c 1 −1 = w 1 · · · w c 1 −1 avoids the pattern 312, as does the subword v c 1 · · · v n . If v contains any subword z 3 z 1 z 2 that matches the pattern 312, then z 3 z 1 must be a subword of v 1 · · · v c 1 −1 while z 2 is a letter of the decreasing word v c 1 · · · v n . But in this case, z 3 z 1 is a subword of w 1 · · · w c 1 −1 while z 2 is a letter of w c 1 · · · w n , which contradicts our assumption that w avoids the pattern 312.
It follows that there are 1 n+1 2n n descending star networks of order n, since there are this many 312-avoiding permutations in S n , all of which avoid the patterns 3412 and 4231. Also related to pattern avoidance are the sizes of the stars in the concatenation (3.2).
the electronic journal of combinatorics 23(2) (2016), #P2.7 Theorem 3.7. Let w ∈ S n avoid the patterns 3412 and 4231. Then w 1 · · · w n contains a decreasing subsequence of size k if and only if some interval in the concatenation (3.2) corresponding to F w has cardinality at least k.
Proof. Let w i 1 , . . . , w i k be a decreasing subsequence of w. Then there is a path in F w from source i j to sink w i j for j = 1, . . . , k. These paths pairwise intersect, since i r < i s if and only if w ir > w is . Since F w is acyclic, these paths must all intersect at a single vertex. Such a vertex must correspond to the central vertex of a star network indexed by an interval of cardinality at least k in the concatenation (3.2).
The converse is clearly true if we have t = 1 in (3.2). Suppose that the converse holds for each zig-zag network corresponding to a concatenation of t − 1 star networks. Let F w correspond to a concatenation (3.2) of t star networks, and let F v correspond to the concatenation i t, has cardinality at least k. If i t − 1, then v contains a decreasing subsequence of size k. By [25,Cor. 3.7], there is a reduced expression for w which consists of a reduced expression for v, followed by some reduced expression s i 1 · · · s i k for the permutation v −1 w. It is well known that each permutation in the sequence (v, vs i 1 , vs i 1 s i 2 , . . . , w) preserves all inversions of the previous permutation and introduces one more. It follows that w also has a decreasing subsequence of length k. If i = t, then apply the above argument to w −1 , which corresponds to the concatenation G [ct,dt] • · · · • G [c 1 ,d 1 ] . It is well known that w has a decreasing subsequence of length k if and only if w −1 does.
We may use path matrices of zig-zag networks to evaluate S n -class functions at Kazhdan-Lusztig basis elements {C w (1) | w avoids the patterns 3412 and 4231}. Specifically, if B is the path matrix of F w , then by [25,Sec. 4,Thm. 5.4] we have θ(C w (1)) = Imm θ (B). (3.6) This fact is a crucial ingredient in the proofs of Theorems 4.6, 4.7, which interpret the evaluations of certain S n -class functions at Kazhdan-Lusztig basis elements of ZS n . In order to interpret the evaluations of H n (q)-traces at Kazhdan-Lusztig basis elements of H n (q), we will prove a q-extension of Equation (3.6) in Proposition 3.8. Namely, we will show that path matrices can also be used to evaluate H n (q)-traces at the (modified) Kazhdan-Lusztig basis elements {q e,w C w (q) | w avoids the patterns 3412 and 4231}. A bit of care is required though: the evaluation Imm θq (B) does not make sense because the substitution x i,j → b i,j does not respect the relations (2.4) and therefore does not give a well-defined map from A n (q) to Z[q 1 2 , q¯1 2 ]. Thus we define a Z[q 1 2 , q¯1 2 ]-linear map for each n × n integer matrix B by Proposition 3.8. Let θ q be an H n (q)-trace and let w ∈ S n avoid the patterns 3412 and 4231. Then the path matrix B of F w satisfies θ q (q e,w C w (q)) = σ B (Imm θq (x)). Proof. The right-hand side is equal to By Theorem 3.3 ([25, Lem. 5.3]), the product b 1,v 1 · · · b n,vn is 1 when v w and is 0 otherwise. Thus the above expression is equal to Since w avoids the patterns 3412 and 4231, [17] implies that the Kazhdan-Lusztig polynomials {P v,w (q) | v w} are identically 1. (See also [3,Ch. 6].) Comparing to (1.1), we see that the parenthesized sum is equal to q e,w C w (q).
Stated from another point of view, Proposition 3.8 asserts that for w avoiding the patterns 3412 and 4231, the zig-zag network F w combinatorially encodes the modified Kazhdan-Lusztig basis element q e,w C w (q) in the sense that where the sum is over all path families which cover F w . In a planar network G of order n, the source-to-sink paths have a natural partial order Q = Q(G). Given paths π i , ρ j , originating at sources i, j, respectively, we define π i < Q ρ j if i < j and π i and ρ j do not intersect. Let P (G) be the subposet of Q(G) induced by paths whose source and sink indices are equal. For each zig-zag network F w , the poset P (F w ) has exactly n elements: there is exactly one path from source i to sink i, for i = 1, . . . , n. The posets P (G) corresponding to the descending star networks G in (3.3) are . (4.1) These are precisely the (3+1)-free, (2+2)-free posets on four elements, where (a+b)-free means that no induced subposet is a disjoint union of an a-element chain and a b-element chain. Such posets are also called unit interval orders. It is known that unit interval orders may be naturally labeled by the numbers [n] so that i < P j implies i < j as integers and so that the conditions i < j < k and i incomparable to k in P imply that {i, j, k} is an antichain in P Proof. Let P and U be the two sets of posets in the theorem. We define a map ζ : U → P as follows. By the above fact on incomparability, we may naturally label P ∈ U by 1, . . . , n so that its maximal antichains are [a 1 , b 1 ], . . . , [a t , b t ], with a 1 > · · · > a t . Now define F to be the descending star network corresponding to the concatenation G [a 1 ,b 1 ] • · · · • G [at,bt] , and let (π 1 , . . . , π n ) be the unique path family of type e that covers F . Let ζ(P ) be the poset P (F ).
We claim that for each poset P ∈ U, the map i → π i is an isomorphism of P and ζ(P ). To see this, note that i < P j if and only if i < j as integers and i, j belong to no common antichain in P , i.e., if and only if i < j and i, j belong to no common interval While a star network is covered by a unique path family of type e, a concatenation of star networks need not be. Nevertheless, such a concatenation is covered by a unique noncrossing path family of type e: the concatenation of the unique path families of type e that cover the component star networks. Given a concatenation G of star networks, let P (G) be the n-element subposet of P (G) induced by the paths in the unique noncrossing path family of type e which covers G. For example, the first two figures in (3.5) show the two path families of type e which cover G [2,4] • G [1,3] . Call these (π 1 , π 2 , π 3 , π 4 ) and (π 1 , π 2 , π 3 , π 4 ), respectively. Ordering all six of these source-i-to-sink-i paths, or only the mutually noncrossing paths {π 1 , π 2 , π 3 , π 4 }, we form the posets P (G [2,4] • G [1,3] ) = π 2 π 4 1 π π 3 2 π π 3 ' ' ,P (G [2,4] • G [1,3] respectively. It is easy to show that for a concatenation G of star networks, the posetP (G) does not depend upon the ordering of the factors of G.
It is also easy to show that the path poset of a zig-zag network F may be obtained directly from the concatenation of star networks which lead to F as in Section 3. Proposition 4.3. Let F be a zig-zag network constructed from a concatenation G of star networks as after (3.2). Then P (F ) is isomorphic toP (G).
Proof. Let (ρ 1 , · · · , ρ n ) and (τ 1 , . . . , τ n ) be the unique noncrossing path families of type e in G and F , respectively, and let x 1 , . . . , x t be the vertices of For example, it is easy to see the equivalence of the four permutations corresponding to the seventh descending star network in (3.3) and the fourth, seventh, and eighth zig-zag networks in (3.4): in each case the path poset is isomorphic to the seventh unit interval order in (4.1). It is also easy to see that we have w ∼ w −1 for w avoiding the patterns 3412 and 4231: the networks F w and F w −1 differ only by reflection in a vertical line. Proof. Fix w ∈ S n avoiding the patterns 3412 and 4231, let G = G [c 1 ,d 1 ] •· · · · · ·•G [ct,dt] be the concatenation of star networks which leads to the zig-zag network F w as after (3.4), and let u ∈ S t be the unique permutation satisfying c u 1 > · · · > c ut . Then the concatenation G [cu 1 ,du 1 ] • · · · · · · • G [cu t ,du t ] leads to a descending star network F v . By Theorem 3.6, v avoids the pattern 312, and by Propositions 4.2 and 4.3 we have P (F v ) ∼ = P (F w ). By Theorem 4.1, v is the only 312-avoiding permutation in its equivalence class.

Planar network tableaux
To combinatorially interpret evaluations of S n -class functions and H n (q)-traces, we will repeatedly fill a (French) Young diagram with a path family π = (π 1 , . . . , π n ) covering a zig-zag network F w , and will call the resulting structures F w -tableaux, or more specifically π-tableaux. (See, e.g., [2, Sec. 2] for French notation.) If π has type v, then we will also say that each π-tableau has type v. Since π induces a subposet Q π of the poset the electronic journal of combinatorics 23(2) (2016), #P2.7 Q(F w ), π-tableaux form a special case of Gessel and Viennot's P -tableaux [8]: they are Q π -tableaux. Several properties which π-tableaux may posess can be defined for P -tableaux where P is an arbitrary poset. We say that a P -tableau U has shape λ for some partition λ = (λ 1 , . . . , λ r ) if it has λ i cells in row i for all i. If U has λ i cells in column i for all i, we say that U has shape λ , where we define λ to be the partition whose ith part is equal to the number of cells in row i of U . Call an element x ∈ P a nontrivial record in a row of U if it is greater in P than all entries appearing to its left in the same row, and if it is not the leftmost entry of its row.
• Call U column-strict (row-strict ) if whenever elements x, y appear consecutively from bottom to top in a column (left to right in a row), then we have x < P y.
• Call U row-semistrict if whenever elements x, y appear consecutively from left to right in a row, then we have x < P y or x incomparable to y in P .
• Call U cyclically row-semistrict if it is row-semistrict and the above condition also holds when x, y are the rightmost and leftmost (respectively) entries in a row.
• Call U standard if it is both column-strict and row-semistrict.
• Call U record-free if no row contains a nontrivial record.
Another property of π-tableaux depends upon each path being labeled by its source vertex. Call a row of U left anchored (right anchored ) if its leftmost (rightmost) entry has the least source vertex of all paths in the row.
More properties of π-tableaux depend upon the fact that each element of the poset Q π has a source vertex and a (potentially different) sink vertex. Given a π-tableau U , let L(U ) and R(U ) denote the Young tableaux of integers obtained from U by replacing paths π 1 , . . . , π n with their corresponding source and sink indices, respectively.
• Call U row-closed if each row of R(U ) is a permutation of the corresponding row of L(U ).
• Call U cylindrical if for each row of L(U ) containing indices i 1 , . . . , i k from left to right, the corresponding row of R(U ) contains i 2 , . . . , i k , i 1 from left to right.
It will be convenient to let T (F w , λ) be the set of F w -tableaux of shape λ. Proof. Observe that for a row-closed, left row-strict tableau Similarly, for a row-semistrict tableau V ∈ T (F v , λ) of type e, the tableau V k is itself a row-semistrict tableau of type e in T (F v | I k , λ k ). Letting r = λ k , we have that F v | I k is a descending star network of the form F u for some u ∈ S r avoiding the pattern 312. We therefore construct a bijection which preserves the set of path indices appearing in each row of a tableau, and we state it as a product of bijections on one-rowed tableaux.
Map each left row-strict tableau U ∈ T (F u , (r)) to a row-semistrict tableau U of type e in T (F u , (r)) as follows. Let ρ = (ρ 1 , . . . , ρ r ) be the unique path family of type e covering F u . Let U contain the path family π = (π 1 , . . . , π r ) from left to right, and define w ∈ S r to be the word of right indices of these paths. Write w −1 in cycle notation, with each cycle starting with its greatest element, and cycles ordered by increasing greatest elements. (See Cycle Structure subsection in [28,Sec. 1.3]). Drop the parentheses, and interpret the resulting string x = x 1 · · · x r of letters as the one-line notation of an element of S r . Then write the paths ρ x 1 , . . . , ρ xr from left to right in U .
To see that U is row-semistrict, assume that we have ρ To see that the map U → U is a bijection, we construct its inverse. Let V be a row-semistrict tableau of type e in T (F u , (r)), containing paths ρ x 1 , . . . , ρ xr from left to right. Define w ∈ S n to be the permutation whose cycle notation is given by . . , x i k are the records of the word x 1 · · · x r , i.e., x i j = max{x 1 , . . . , x i j }. Then write w −1 = w −1 1 · · · w −1 r in one-line notation and define V to be the tableau in T (F u , (r)) whose ith entry is the unique path in F u from source i to sink w −1 i . It is clear that this map, if well defined, is inverse to the map U → U , and therefore that the two are bijections. (See [28,Sec. 1.3]).
To see that the necessary paths exist in F u , consider a cycle (x j , . . . , x j+ ) of w and the pairs (i, , any integer inequality x j+a > x j+a+1 implies that there are paths in F u from sources x j+a and x j+a+1 to (both) sinks x j+a and x j+a+1 . In particular, there are paths in F u from sources x j and x j+1 to sinks x j and x j+1 . Now assume that there are paths from source x j+s to sink x j+s−1 and from source x j to sink x j+s , and consider the pair (x j+s+1 , x j+s ). If x j+s+1 < x j+s , then by the above argument there is a path from source x j+s+1 to sink x j+s . Since there are paths from sources x j and x j+s+1 to sink x j+s , the two sources must belong to the same connected component of F u . By the comment following Observation 3.4, we also have a path from source x j to sink x j+s+1 . If on the other hand x j+s+1 < x j+s , then since x j is the maximum index in its cycle we have x j > x j+s+1 > x j+s . Since there are paths in F u from source x j to sink x j+s and the electronic journal of combinatorics 23(2) (2016), #P2.7 from source x j+s+1 to sink x j+s+1 , Observation 3.1 implies that there are also a paths from source x j to sink x j+s+1 and from source x j+s+1 to x j+s . By induction, we have that for a = 1, . . . , , there is a path from source x j+a to sink x j+a−1 , and that there is also a path from source x j to sink x j+ .

Interpretation of S n -class function evaluations
The equivalence relation in (4.2) has applications in the enumeration of certain F -tableaux and in the evaluation of S n -class functions.
Theorem 4.6. Let v, w ∈ S n avoid the patterns 3412 and 4231 and satisfy v ∼ w, and let X be a property of F -tableaux which depends only upon the poset P (F ) (rather than on Q(F )). Then F v -tableaux and F w -tableaux having property X are in bijective correspondence. Moreover, for any S n -class function θ we have θ(C v (1)) = θ(C w (1)).
Proof. Since the property X depends only upon the poset P (F v ) ∼ = P (F w ), we have a bijection between the sets of F v -tableaux and F w -tableaux having property X. Now apply Lindström's Lemma and (3.6) to the first Littlewood-Merris-Watkins identity in (2.1) to see that for all λ n, the evaluations λ (C v (1)) and λ (C w (1)) are equal to the numbers of column-strict F v -tableaux and F w -tableaux, respectively, of type e and shape λ. Since column-strictness of these tableaux depends only upon P (F v ) ∼ = P (F w ), the above bijection implies that we have λ (C v (1)) = λ (C w (1)) for all λ n. Since { λ | λ n} is a basis for the space of S n -class functions, each class function θ satisfies For some S n -class functions θ, and all 3412-avoiding, 4231-avoiding permutations w, we may combinatorially interpret θ(C w (1)) in terms of a zig-zag network F w as follows.
(ii-b) Since row-semistrictness in F w -tableaux of type e is a property of the poset P (F w ), we may apply Theorem 4.6 and Lemma 4.5 to the interpretation in (ii-a).
(iii) Applying (i) and Theorem 4.1 to Gasharov's [7, Thm. 2], we obtain the claimed interpretation for 312-avoiding permutations. (See also Section 7.) Since standardness of F w -tableaux depends only upon P (F w ), we may apply Theorem 4.6 to extend the result to 3412-avoiding, 4231-avoiding permutations as well.
(iv-a) Apply the definition (3.1) of path matrix to the identity (2.2).
(iv-b) Let v ∼ w avoid the pattern 312. We define a map from cylindrical tableaux in T (F v , λ) to cyclically row-semistrict tableaux in T (F v , λ) having type e as follows. For U ∈ T (F v , λ) cylindrical with rows k of L(U ) and R(U ) containing indices i 1 , . . . , i λ k and i 2 , . . . , i λ k , i 1 (respectively) from left to right, and ρ = (ρ 1 , . . . , ρ n ) the unique path family of type e covering F v , create a cyclically row-semistrict tableau U ∈ T (F v , λ) by inserting ρ i 1 , . . . , ρ i λ k into row k, from right to left. This map is bijective since in the descending star network F v , there exists a (unique) path from source i j to sink i m if and only if i j > i m or ρ i j and ρ im intersect. Thus the claimed interpretation holds for 312-avoiding permutations. Since cyclical row-semistrictness in F w -tableaux of type e depends only upon P (F w ), we may apply Theorem 4.6 to extend the result to 3412-avoiding, 4231avoiding permutations as well.
(iv-c) Shareshian  (iv-d) The number of tableaux in (iv-b) is equal to the cardinality of the subset that are right-anchored, times λ 1 · · · λ r . This subset is precisely the right-anchored row-semistrict F w -tableaux of type e and shape λ. Alternatively, we may use the Shareshian-Wachs argument of (iv-c).
(v-a) This was first stated in [4], and will be proved in Theorem 10.3. A different interpretation was given in [34, Thm. 2.5.1].
Conspicuously absent from Theorem 4.7 is an interpretation of monomial class function evaluations of the form φ λ (C w (1)) which holds for all λ n. As we have mentioned in the first  Problem 4.8. For (special cases of) w ∈ S n and λ n, find a combinatorial proof that φ λ (C w (1)) is nonnegative.

Inversions in path tableaux
For λ n and w ∈ S n avoiding the patterns 3412 and 4231, parts (i) -(iv-d) of Theorem 4.7 interpret λ (C w (1)), η λ (C w (1)), χ λ (C w (1)), and ψ λ (C w (1)) as cardinalities of certain sets of F w -tableaux. Using these same sets of F w -tableaux and variations of the permutation statistic inv, we show in Sections 5, 6, 8, 9 that λ q (q e,w C w (q)), η λ q (q e,w C w (q)), χ λ q (q e,w C w (q)), and ψ λ q (q e,w C w (q)) are generating functions for tableaux on which the statistics take the values k = 0, 1, . . . . Specifically, we adapt the permutation statistic inv for use on path tableaux as follows. Let π = (π 1 , . . . , π n ) be a path family of type v in some zig-zag network F , and let U be a π-tableau. Let (π i , π j ) be a pair of intersecting paths in F such that π i appears in a column of U to the left of the column containing π j . Call ( Sometimes we will compute inversions in a one-rowed tableau formed by concatenating all of the rows of a path tableau U . Let U i be the ith row of U , and let U 1 • · · · • U r and U r • · · · • U 1 be the F -tableaux of shape n consisting of the rows of U concatenated in increasing and decreasing order, respectively. We will also compute inversions in the transpose U of a path tableau U , whose rows are the columns of U . It is easy to see that inversions in these one-rowed and transposed tableaux are related by the identities 5 Interpretation of η λ q (q e,w C w (q)) Let w ∈ S n avoid the patterns 3412 and 4231, and let B be the path matrix of F w . Using (2.5) and Proposition 3.8, we have where the sum is over all ordered set partitions (I 1 , . . . , I r ) of [n] of type λ = (λ 1 , . . . , λ r ). Let S λ denote the Young subgroup of S n generated by and let S − λ be the set of Bruhat-minimal representatives of cosets of the form S λ u, i.e., the elements u ∈ S n for which each of the subwords the electronic journal of combinatorics 23(2) (2016), #P2.7 is strictly increasing. It is clear that such elements correspond bijectively to the ordered set partitions (I 1 , . . . , I r ) in (5.1). Expanding the product of permanents, we obtain monomials of the form q u,v times where v is the concatenation, in order, of rearrangements of the r words (5.2). Thus v may be written as yu with y ∈ S λ , or as uy with y ∈ u −1 S λ u. Now the sum in (5.1) becomes Let us therefore consider evaluations of the form To combinatorially interpret these evaluations, let π = (π 1 , . . . , π n ) be a path family (of arbitrary type) which covers a zig-zag network F , and define U (u, π) to be the πtableau of shape (n) containing π in the order π u 1 , . . . , π un . Clearly the left tableau of U (u, π) is u 1 · · · u n . If the right tableau is v 1 · · · v n then π has type u −1 v. If s i is a left descent for u, then right inversions in U (u, π) and U (s i u, π) are related as follows.
Proposition 5.1. Fix u, v ∈ S n , let F be a zig-zag network, and let π = (π 1 , . . . , π n ) be a path family of type u −1 v which covers F . If s i u < u then we have Proof. The tableaux U (u, π) and U (s i u, π) are identical except that π u i appears before π u i+1 in U (u, π). Thus we have but not in U (u, π), rinv(U (s i u, π)) + 1 if (π u i , π u i+1 ) is a right inversion in U (u, π) but not in U (s i u, π), rinv(U (s i u, π)) if (π u i , π u i+1 ) is not a right inversion in U (s i u, π) or U (u, π).
Since s i u < u, we have u i > u i+1 . If s i v > v, then we have v i < v i+1 , and Observation 3.1 implies that the paths π u i and π u i+1 intersect, forming a right inversion in U (s i u, π) and not in U (u, π).
If s i v < v, then we have v i > v i+1 , and the paths π u i and π u i+1 do not form a right inversion in U (s i u, π). Suppose that some path family π = (π 1 , . . . , π n ) of type u −1 s i v covers F . Then the tableau U (u, π ) satisfies L(U (u, π )) = (u 1 , . . . , u n ), the electronic journal of combinatorics 23(2) (2016), #P2.7 By the uniqueness of source-to-sink paths in zig-zag networks, this tableau is identical to the tableau U (u, π) except for the paths π u i and π u i+1 in positions i and i + 1, which terminate at sinks v i+1 < v i , respectively. By Observation 3.1, the paths π u i , π u i+1 cross and the paths π u i , π u i+1 intersect, forming a right inversion in the tableau U (u, π). On the other hand, suppose that no path family of type u −1 s i v covers F . Since π has type u −1 v, we can deduce that either there is no path in F from source u i to sink v i+1 or there is no path from source u i+1 to sink v i . By Observation 3.1, the paths π u i , π u i+1 do not intersect and therefore do not form a right inversion in U (u, π).
Now we evaluate σ B (q u,v x u,v ), first in the case that u = e.
Proposition 5.2. Let w in S n avoid the patterns 3412 and 4231, let B be the path matrix of F w , and fix v in S n . Then we have ,π)) if there exists a path family π of type v which covers F w , 0 otherwise.
Proof. By definition we have First assume that there exists a (unique) path family π of type v that covers F w . Then we have b e,v = 1. In the tableau U (e, π), paths appear in the order (π 1 , . . . , π n ). Now observe that for each inversion in v, i.e., each pair (v i , v j ) with i < j and v i > v j , the paths π i (from source i to sink v i ) and π j (from source j to sink v j ) cross in F w and therefore form a right inversion in U (e, π). Conversely, for each noninversion (v i , v j ) in v, the paths π i and π j do not form a right inversion in U (e, π). Thus we have (v) = rinv(U (e, π)), and the expression in (5.4) is equal to q rinv(U (e,π)) . Now assume that there is no path family of type v which covers F w . Then we have b e,v = 0 and the expressions in (5.4) are equal to 0.
More generally, we evaluate σ B (q u,v x u,v ) as follows.
Proposition 5.3. Let w in S n avoid the patterns 3412 and 4231, let B be the path matrix of F w , and fix u, v in S n . Then we have σ B (q u,v x u,v ) = q rinv(U (u,π)) if there exists a path family π of type u −1 v covering F w , 0 otherwise.
Proof. We use induction on the length of u. By Proposition 5.2, the claimed formula holds when u has length 0. Now assume that the formula holds when u has length 1, . . . , k − 1, and consider u of length k. Choosing a left descent s i of u, we may write Suppose first that we have s i v > v. Then by induction we have s i u,π)) if there exists a path family π of type u −1 v covering F w , 0 otherwise, and Proposition 5.1 implies that the claim is true in this case. Now suppose that we have s i v < v and consider path families of types u −1 v and u −1 s i v which cover F w . If there are no path families of types u −1 v and u −1 s i v which cover F w , then by induction we have If there exists a path family π = (π 1 , . . . , π n ) of type u −1 v which covers F w , but no path family of type u −1 s i v which covers F w , then by induction and Proposition 5.1 we have If there exists a path family π = (π 1 , . . . , π n ) of type u −1 s i v which covers F w , then the paths π u i , π u i+1 from sources u i > u i+1 to sinks v i+1 < v i (respectively) cross. It follows that there exists a path family π = (π 1 , . . . , π n ) of type u −1 v which covers F w , and which agrees with π except that the paths π u i , π u i+1 from sources u i > u i+1 to sinks v i > v i+1 (respectively) intersect but do not cross. By induction and the existence of π and π we have The tableaux U (s i u, π) and U (s i u, π ) agree except in positions i and i + 1, where paths π u i+1 and π u i form a right inversion, but π u i+1 and π u i do not. This fact and Proposition 5.1 imply that we have rinv(U (s i u, π )) = rinv(U (s i u, π)) + 1 = rinv(U (u, π)), and σ B (q s i u,s i v x s i u,s i v ) = q rinv(U (u,π))−1 , Thus by Equations (5.5), (5.6), and (5.7) the claim is true when s i v < v.
Now we have the following q-analog of Theorem 4.7 (ii-a).
Theorem 5.4. Let w ∈ S n avoid the patterns 3412 and 4231. Then for λ = (λ 1 , . . . , λ r ) n we have where the sum is over all row-closed, left row-strict F w -tableaux of shape λ.
By Proposition 5.3, this is equal to (y,π) q rinv(U (u,π)) , where the sum is over pairs (y, π) such that y ∈ u −1 S λ u and π is a path family of type y which covers F . If such a path family π exists for a given permutation y, it is necessarily unique. Thus as y varies over u −1 S λ u we have that U (u, π) varies over all bijective path tableaux of shape (n) which satisfy 1. For j = 1, . . . , r, the paths in positions λ 1 + · · · + λ j−1 + 1, . . . , λ 1 + · · · + λ j are indexed by I j , in increasing order.
2. The sequence of sink indices of these same paths are a rearrangement of I j .
Thus the expression in (5.8) may be rewritten as where this last sum is over all row-closed, left row-strict F w -tableaux U of shape λ for which path indices of U j are I j for j = 1, . . . , r. Summing over ordered set partitions and using (5.1), we have the desired result.
For example, consider the network , where i, j represents the unique path from source i to sink j. These tableaux U of shape 31 yield tableaux U 1 • U 2 of shape 4, which have 1, 2, 3, and 4 right inversions, respectively. Together, the tableaux contribute q + q 2 + q 3 + q 4 to η 31 q (q e,3421 C 3421 (q)) = 1 + 3q + 6q 2 + 6q 3 + 3q 4 + q 5 . Expanding η λ q in terms of irreducible characters and Kostka numbers, η λ q = K µ,λ χ µ q , and using Haiman's result [11, Lem 1.1], we have that the coefficients of η λ q (q e,w C w (q)) are symmetric and unimodal about q e,w for all w ∈ S n . In the case that w avoids the patterns 3412 and 4231, it would be interesting to explain this phenomenon combinatorially in terms of Theorem 5.4.
It would also be interesting to extend Theorem 5.4 to include a q-analog of Theorem 4.7 (ii-b). In particular, the identity suggests that an answer to Problem 4.8 and its q-analog are related to a set partition of tableaux counted by η (n) . It is not clear whether such a partition is more easily expressed in terms of left row-strict tableaux of shape (n), or row-semistrict tableaux of type e and shape (n).
Problem 5.5. Find a statistic stat on F -tableaux such that we have where the sum is over all row-semistrict F w -tableaux of type e and shape λ.
As a consequence of Theorem 5.4, we obtain the following q-analog of Theorem 4.6.
Theorem 5.6. Let v, w ∈ S n avoid the patterns 3412 and 4231 and satisfy v ∼ w, and let (X, stat) be a property of F -tableaux and a statistic on F -tableaux which depend only upon the poset P (F ). Then F v -tableaux and F w -tableaux having property X and satisfying stat(U ) = k are in bijective correspondence. Moreover, for each H n (q)-trace θ q we have θ q (q e,v C v (q)) = θ q (q e,w C w (q)).
Proof. Since the pair (X, stat) depends only upon the poset P (F v ) ∼ = P (F w ), we have a bijection between the sets of F v -tableaux and F w -tableaux having property X and satisfying stat(U ) = k. In particular, we have η λ q (q e,v C v (q)) = η λ q (q e,w C w (q)), and since {η λ q | λ n} is a basis of the space of H n (q)-traces, we have θ q (q e,v C v (q)) = θ q (q e,w C w (q)) for all H n (q)-traces θ q as well.
Let θ q be an H n (q)-trace. If the posets P (F v ), P (F w ) of two zig-zag networks are dual, rather than isomorphic, we still have θ q (q e,v C v (q)) = θ q (q e,w C w (q)). In this case v and w satisfy v ∼ w 0 ww 0 and (v) = (w), where w 0 is the longest element of S n . Since natural basis elements of H n (q) satisfy T w 0 T w T −1 w 0 = T w 0 ww 0 , and since any H n (q)-trace θ q satisfies θ q (gh) = θ q (hg) for all h, g ∈ H n (q), we have that the equations θ q (q e,w 0 ww 0 C w 0 ww 0 (q)) = θ q (q e,w T w 0 C w (q)T −1 w 0 ) = θ q (q e,w C w (q)) hold for all w ∈ S n .
Proposition 6.1. Let w ∈ S n avoid the patterns 3412 and 4231, and let B be the path matrix of F w . Fix λ n, u ∈ S − λ , and y ∈ u −1 S λ u. Then we have σ B (q −1 u,uy x u,uy ) = q rinv(U (u,π,λ) ) if there exists a path family π of type y covering F w , 0 otherwise.
We claim that this expression reduces further to rinv(V ). To see this, recall that the tableaux L(V 1 ) • · · · • L(V r ) and R(V 1 ) • · · · • R(V r ) contain the one-line notations of u the electronic journal of combinatorics 23(2) (2016), #P2.7 and uy, respectively. Since u belongs to S − λ and each tableau R(V j ) is a permutation of the corresponding tableau L(V j ), we have (u) = #{(k, k ) | k > k , k in an earlier row of L(V ) than k } = #{(k, k ) | k > k , k in an earlier row of R(V ) than k }.
Now fix a row V j of V and consider right inversions in V j . Let π i , π i be two paths in this row, with π i appearing first. Since u belongs to S − λ , we have i < i . Let k and k be the corresponding sink indices. If we have k > k , then the paths cross and form a right inversion in V j . On the other hand, if we have k < k , then the paths do not form a right inversion in V j , even if they intersect. Thus we have rinv(V j ) = inv(R(V j )) for all j and rinv(V 1 ) + · · · + rinv(V r ) = (uy) − (u), (6.2) as desired.
While the final sum in (6.1) has signs, we will use a sign-reversing involution to cancel some of the the terms there, and to obtain a signless sum more amenable to combinatorial interpretation. Fix a partition λ = (λ 1 , . . . , λ r ) n, an ordered set partition I of [n] of type λ, and a zig-zag network F , and let T I = T I (F ) be the set of all row-closed, left rowstrict F -tableaux U of shape λ such that L(U ) j = I j (as sets) for j = 1, . . . , r. Observe that all row-strict F -tableaux of type e satisfying L(U ) j = I j (as sets) for j = 1, . . . , r belong to T I . Let us define an involution ζ : T I → T I as follows.
1. If U is a row-strict tableau of type e, then define ζ(U ) = U .

Otherwise,
(a) Let i be the least index such that U i is not row-strict.
(b) Let (j, j ) be the lexicographically least pair of indices in L(U ) i such that π j and π j intersect.
(c) Let (k, k ) be the sink indices of paths π j and π j , respectively.
(d) Define ζ(U ) to be the tableau obtained from U by replacing π j and π j by the unique paths in F from source j to sink k and source j to sink k. Proof. For each F -tableau U ∈ T I satisfying ζ(U ) = U , the claimed equality is obvious. Now let U be an F -tableau not fixed by ζ. Define the indices i, j, j , k, k and paths π j , π j as in the definition of ζ, and let π j , π j be the two new paths created in the final step of the definition of ζ. Since the tableaux U and ζ(U ) agree everywhere except in the two cells in column i containing the paths π j , π j , π j , π j , (6.3) it is clear that the right inversions of these tableaux are equal except possibly for inversions involving a path π h in a column other than i and one of the paths (6.3). We claim that these remaining right inversions in U and ζ(U ) correspond bijectively. In particular, we have (a) π h forms a right inversion with π j in U if and only if it forms a right inversion with π j in ζ(U ) .
(b) π h forms a right inversion with π j in U if and only if it forms a right inversion with π j in ζ(U ) .
To see this, observe that j < j , and consider the intersection of π h with π j and π j . By Lemma 3.5 we have four cases: 1. π h intersects both π j and π j .
3. π h intersects only the path whose sink is min(k, k ). 4. π h intersects neither π j nor π j . Now considering the intersections of π h with π j and π j , we see that the above cases imply respectively that π h intersects both π j and π j , only π j (not π j ), only the path in {π j , π j } whose sink index is min(k, k ), and neither π j nor π j . In all cases, the equivalences (a) and (b) are true.
Since the tableaux U and ζ(U ) agree except in one row, we have the following. Proposition 6.3. Let U ∈ T I be a tableau not fixed by ζ and let i be the index satisfying Proof. Obvious. Now we have the following q-analog of Theorem 4.7 (i).
the electronic journal of combinatorics 23(2) (2016), #P2.7 Theorem 6.4. Let w ∈ S n avoid the patterns 3412 and 4231. Then for λ n we have λ q (q e,w C w (q)) = where the sum is over all column-strict F w -tableaux of type e and shape λ .
Proof. Let B be the path matrix of F w and let (I 1 , . . . , I r ) be a set partition of [n] of type λ. By (5.2) -(5.3), there is a permutation u ∈ S − λ corresponding to (I 1 , . . . , I r ) such that we have u,uy x u,uy ).
By Proposition 6.1 this is equal to where the sum is over pairs (y, π) such that y ∈ u −1 S λ u and π is a path family of type y which covers F w . If such a path family exists, it is necessarily unique. Thus as y varies over u −1 S λ u we have that U (u, π, λ) varies over all tableaux in T I . Thus by (6.2) this sum is equal to Now consider a tableau V ∈ T I which satisfies ζ(V ) = V . By Propositions 6.2 -6.3, the term of the above sum corresponding to the tableau ζ(V ) is Thus all terms corresponding to tableaux V and ζ(V ) = V cancel one another in the sum (6.5), leaving terms only for the tableaux V ∈ T I which are fixed by ζ, i.e., the row-strict tableaux. For these tableaux we have rinv(V 1 ) = · · · = rinv(V r ) = 0, rinv(V ) = inv(V ).
Furthermore, V is row-strict of shape λ if and only if V is column-strict of shape λ . Thus we may again rewrite (6.4) as where the sum is over all column-strict F w -tableaux of shape λ satisfying U j = I j (as sets). Summing over ordered set partitions and using (6.1), we have the desired result.
For example, consider the descending star network F 3421 in (5.10). It is easy to verify that there are exactly two column-strict F 3421 -tableaux of type e and shape 31: where π i is the unique path from source i to sink i. These tableaux have 2 and 3 inversions, respectively. Since 31 = 211, the tableaux together give 211 q (q e,3421 C 3421 (q)) = q 2 +q 3 . It is clear that all twenty-four F 3421 -tableaux of type e and shape 4 are column-strict. Counting inversions in these tableaux gives 1111 q (q e,3421 C 3421 (q)) = 1 + 3q + 8q 2 + 8q 3 + 3q 4 + q 5 . It is also clear that there are no F 3421 -tableaux of type e and shapes 22, 211, or 1111. Thus we have 22 q (q e,3421 C 3421 (q)) = 31 q (q e,3421 C 3421 (q)) = 4 q (q e,3421 C 3421 (q)) = 0. Expanding λ q in terms of irreducible characters and Kostka numbers, λ q = K µ ,λ χ µ q , and using Haiman's result [11, Lem 1.1], we have that λ q (q e,w C w (q)) is symmetric and unimodal about q e,w for all w ∈ S n . In the case that w avoids the patterns 3412 and 4231, it would be interesting to explain this phenomenon combinatorially in terms of Theorem 6.4. Corollary 6.5. Fix λ = (λ 1 , . . . , λ r ) n and w ∈ S n avoiding the patterns 3412 and 4231. If w 1 · · · w n has a decreasing subsequence of length greater than r, then λ q (q e,w C w (q)) = 0. Proof. Let (π 1 , . . . , π n ) be the unique path family of type e which covers F w . By Theorem 3.7 there exist r + 1 paths π i , . . . , π i+r which share a vertex. No two of these can appear together in a column of a column-strict F w -tableau. Thus no such tableau has shape λ .
More generally, it is known that we have λ q (q e,w C w (q)) = 0 unless λ sh(w) in the majorization order, where sh(w) is the partition associated to w by the Robinson-Schensted row insertion algorithm. (See, e.g., [11,Prop. 4.1 (3)].) This implies Corollary 6.5, since a decreasing subsequence of length greater than r in w 1 . . . w n implies that we have λ sh(w).

Chromatic symmetric and quasisymmetric functions
The evaluations of S n -class functions and H n (q) traces at Kazhdan-Lusztig basis elements are closely related to certain symmetric and quasisymmetric functions defined by Stanley [27] and Shareshian and Wachs [23].
Let Λ n be the Z-module of homogeneous degree n symmetric functions. In [27] Stanley defined certain chromatic symmetric functions {X G | G a simple graph on n vertices } in Λ n and studied expansions of these in various bases of Λ n . Given G = (V, E) and defining P to be the set of positive integers, we call a function κ : V → P a proper coloring of G if κ(u) = κ(v) whenever (u, v) ∈ E. Then we have the definition where the sum is over all proper colorings of G. When G is the incomparability graph of a poset P , we will write X P = X inc(P  where c P,λ is the number of ordered set partitions of P whose blocks are chains of cardinalities λ 1 , λ 2 , . . . . These symmetric functions are related to S n -class function evaluations as follows. Theorem 7.1. Let P be an n-element unit interval order, let v ∈ S n be the 312-avoiding permutation satisfying ζ(F v ) = P as in the proof of Theorem 4.1, and let w ∈ S n be any 3412-avoiding, 4231-avoiding permutation satisfying w ∼ v as in (4.2). Then we have Proof. It is easy to see that c P,λ is equal to the number of column-strict P -tableaux of shape λ . By Theorems 4.1, 4.6, this is the number of column-strict F w -tableaux of shape λ . Thus by Theorem 4.7 (i) we have c P,λ = λ (C w (1)).
Expanding X P in other bases of Q ⊗ Λ n , including the forgotten basis {f λ | λ n}, we see that other class function evaluations appear as coefficients.
Corollary 7.2. Let P , v, w be as in Theorem 7.1. Then we have We remark that Theorem 7.1 and Corollary 7.2 do not hold for arbitrary w and P . Not all chromatic symmetric functions X P can be expressed as λ n λ (C w (1))m λ for appropriate w ∈ S n , nor can all symmetric functions of this form be expressed as X P for an appropriate labeled poset P .

Proof. The transition matrices relating the class function bases
Stanley and Stembridge [27,Conj. 5.1], [30,Conj. 5.5] conjectured that X P is elementary nonnegative when P is (3 + 1)-free, and Gasharov [7,Thm. 2] proved the weaker statement that X P is Schur nonnegative in this case. Guay-Paquet [10, Thm. 5.1] showed that the above conjecture and result are equivalent to the analogous statements in which P is assumed to be a unit interval order. Thus by Theorem 4.6 and Corollary 7.2 the nonnegativity statements are consequences of Haiman's conjecture and result that for all w ∈ S n , λ n we have φ λ q (q e,w C w (q)) ∈ N[q] and χ λ q (q e,w C w (q)) ∈ N[q], respectively [11, Conj. 2.1, Lem. 1.1]. Specifically, the nonnegativity statements are obtained from Haiman's by restricting to the case that w avoids the pattern 312 and by specializing at q = 1.
Let QSym n be the Z-module of homogeneous degree n quasisymmetric functions in the commuting indeterminates x 1 , x 2 , . . . , i.e., the generalization of homogeneous degree the electronic journal of combinatorics 23(2) (2016), #P2.7 n symmetric functions Λ n in which monomials of the forms x α 1 1 · · · x α k k and x α 1 i 1 · · · x α k i k are required to have the same coefficient only when i 1 < · · · < i k . In [23,Sec. 4], Shareshian and Wachs defined a q-analog X G,q of the chromatic symmetric function X G = X G,1 , with {X G,q | G a simple labeled graph on n vertices } belonging to Z[q] ⊗ QSym n , and studied expansions of these functions in various bases. Assume that V is labeled by [n] and let asc(κ) = #{(u, v) ∈ E | u < v and κ(u) < κ(v)}. Then we have the definition where the sum is over all proper colorings of G. In [24,Thm. 4.5] Shareshian and Wachs showed that this function is in fact symmetric with coefficients in Z[q] when G is the incomparability graph of an appropriately labeled n-element unit interval order P . Specifically, we require for each pair x, y ∈ P satisfying #{z ∈ P | z < x} − #{z ∈ P | z > x} < #{z ∈ P | z < y} − #{z ∈ P | z > y}, (7.3) that the label of x be less than that of y. (The equivalence of this requirement to that stated in [24,Thm. 4.5] 4].) When G is the incomparability graph of a labeled poset P , we will write X P,q = X inc(P ),q , and we may give an alternate definition of {X P,q | P an n-element poset } which is analogous to (7.1). (See also [24,Eq. (6.2)].) Proposition 7.3. Let P be a unit interval order, labeled as in (7.3). Then we have and the sum is over column-strict P -tableaux of shape λ .
Proof. Each proper coloring κ of inc(P ) may be viewed as an assignment of colors to elements of P so that each subset of elements having a given color forms a chain. By [24,Thm. 4.5], X P,q is symmetric. Thus for λ n, the coefficient in X P,q of m λ is welldefined: it is the coefficient of x λr 1 x λ r−1 2 · · · x λ 1 r , i.e., the sum of q asc(κ) , over all colorings κ that assign color 1 to a λ r -element chain, color 2 to a λ r−1 -element chain, . . . , color r to a λ 1 -element chain. Each such coloring corresponds to a column-strict P -tableau U of shape λ. Specifically, each λ r+1−i -element chain of color i corresponds to column r + 1 − i of U , for i = 1, . . . , r. Now observe that a pair (u, v) in P , with u < v as integers, forms an ascent of κ if and only if it forms an inversion in U . Specifically, (u, v) is an edge in inc(P ) if and only if u, v are incomparable in P , and we have κ(u) < κ(v) if and only if u appears in a column of U to the right of the column containing v. Just as Stanley's chromatic symmetric functions X P are related to S n -class function evaluations in Theorem 7.1, the Shareshian-Wachs chromatic quasisymmeric functions X P,q are related to H n (q)-trace evaluations.
Theorem 7.4. Let P be an n-element unit interval order labeled as in (7.3), let v ∈ S n be the corresponding 312-avoiding permutation as in Theorem 4.1, and let w ∈ S n be any 3412-avoiding, 4231-avoiding permutation satisfying w ∼ v as in (4.2). Then we have X P,q = λ n λ q (q e,w C w (q))m λ .
Proof. By Proposition 7.3, c P,λ (q) is equal to the sum of q inv(U ) over column-strict Ptableaux of shape λ . By Theorems 4.1, 5.6, we may sum over column-strict F w -tableaux. Now Theorem 6.4 gives c P,λ (q) = λ (q e,w C w (q)).
Expanding X P in other bases of Q[q] ⊗ Λ n and following the proof of Corollary 7.2, we see that other trace evaluations appear as coefficients.
Corollary 7.5. Let P , v, w be as in Theorem 7.4. Then we have As before, Theorem 7.4 and Corollary 7.5 do not hold for arbitrary w and P . Not all chromatic symmetric functions X P,q can be expressed as λ n λ q (q e,w C w (q))m λ for appropriate w ∈ S n , nor can all symmetric functions of this form be expressed as X P,q for an appropriate labeled poset P .
Shareshian and Wachs conjectured that X P,q belongs to span N[q] {e λ | λ n} when P is a unit interval order labeled as in (7.3) [23,Conj. 4.9], and proved that X P,q belongs to span N[q] {s λ | λ n} for such posets [24,Thm. 6.3]. These statements do not hold for the more general (3 + 1)-free posets, since the functions X P,q are not always symmetric in this case. Nevertheless, the result of Guay-Paquet [10,Thm. 5.1] shows that the statements generalize those of Stanley, Stembridge, and Gasharov mentioned after Corrolary 7.2. By Theorem 4.6 and Corollary 7.5, the statements are special cases (corresponding to w avoiding the pattern 312) of Haiman's conjecture and result [11, Conj. 2.1, Lem 1.1] that for all w ∈ S n , λ n we have φ λ q (q e,w C w (q)) ∈ N[q] and χ λ q (q e,w C w (q)) ∈ N[q]. Shareshian and Wachs also conjectured [24,Sec. 7], and Athanasiadis proved [1,Thm. 4] that X P,q belongs to span N[q] {(−1) n− (λ) z −1 λ p λ | λ n} when P is a unit interval order labeled as in (7.3). By Theorem 4.6 and Corollary 7.5 this is equivalent to the assertion that we have ψ λ q (q e,w C w (q)) ∈ N[q] for w avoiding the pattern 312. Thus this result is a special case of the (unpublished) conjecture that ψ λ q (q e,w C w (q)) ∈ N[q] for all w ∈ S n , λ n, which is a weakening of Haiman's conjecture [11,Conj. 2.1] since ψ λ q is a nonnegative linear combination of monomial traces (1.2).
Theorem 8.1. Let w ∈ S n avoid the patterns 3412 and 4231. For λ n we have where the sum is over all standard F w -tableaux of type e and shape λ.
Proof. Let P = P (F w ). By Corollary 7.5, χ λ q (q e,w C w (q)) is equal to the coefficient of s λ in X P,q . By [24,Thm. 6.3] and Theorems 4.1, 5.6, this is precisely the claimed sum.
For example, consider again the descending star network F 3421 in (5.10). It is easy to verify that there are exactly two standard F 3421 -tableaux of type e and shape 31: the two column-strict F 3421 -tableaux of type e and shape 31 in (6.6) are also row-semistrict. Thus we have χ 31 q (q e,3421 C 3421 (q)) = 211 q (q e,3421 C 3421 (q)) = q 2 + q 3 . On the other hand, not all twenty-four F 3421 -tableaux of type e and shape 4 are row-semistrict: the six tableaux with π 4 immediately preceding π 1 are not. It is easy to verify that the eighteen remaining tableaux give χ 4 q (q e,3421 C 3421 (q)) = 1 + 3q + 5q 2 + 5q 3 + 3q 4 + q 5 . Since there are no column-strict F 3421 -tableaux of shapes 22, 211, or 1111, there are no standard F 3421tableaux of these shapes either, and we have χ 22 q (q e,3421 C 3421 (q)) = χ 211 q (q e,3421 C 3421 (q)) = χ 1111 q (q e,3421 C 3421 (q)) = 0. Combining Theorems 3.7 and 8.1, we have the following analog of Corollary 6.5.
More generally, it is known that we have χ λ q (q e,w C w (q)) = 0 unless λ sh(w) in the majorization order. (See the comment following Corollary 6.5.) This implies Corollary 8.2, since a decreasing subsequence of length greater than λ 1 in w 1 . . . w n implies that we have λ sh(w) .
9 Interpretation of ψ λ q (q e,w C w (q)) Combining results in Sections 4, 5, 7 with those of Shareshian, Wachs, and Athanasiadis now leads to q-analogs of Theorem 4.7 (iv-c)-(iv-d). We will sometimes find it useful to reflect path tableaux in a vertical line, and will write U R for the reverse of tableau U . For instance, we have U = π π π π 1 2 3 4 , U R = π 1 π 4 π 2 π 3 , the electronic journal of combinatorics 23(2) (2016), #P2.7 where each path π i retains its original source, sink, and orientation. Thus U R i will denote the reverse of the ith row of U . Note that while U R may not be a tableau, because its cells are right-justified rather than left justified, the functions inv and rinv may still be applied to U R as at the end of Section 4. We will also use standard notation for the q-analogs of the nonnegative integers and factorial function. For a ∈ N we define [a] q = 1 + q + · · · + q a−1 for a 1, and [0] q = 0. We also define [a] q ! = [a] q [a − 1] q · · · [1] q for a 1, and [0] q ! = 1.
Theorem 9.1. Let w ∈ S n avoid the patterns 3412 and 4231. For λ = (λ 1 , . . . , λ r ) n, we have where the sum is over all record-free, row-semistrict F w -tableaux of type e and shape λ, where the sum is over all right-anchored, row-semistrict F w -tableaux of type e and shape λ, Proof. Let P = P (F w ). By Corollary 7.5, ψ λ q (q e,w C w (q)) is equal to (−1) n−r z λ times the coefficient of p λ in X P,q . By [1, Thm. 3.1] and Theorems 4.1, 5.6, this is equal to the right-hand side of (9.1). By [24,Lem. 7.7] and Theorems 4.1, 5.6, it is also equal to the right-hand side of (9.2).
Definition 9.1. Let F be a descending star network, and let be the concatenation of star networks which corresponds to F as in Section 3. Define the For example, the connected descending star network F 3421 in (5.10) corresponds to the concatenation G = G [2,4] • G [1,3] and two-element poset of intervals [2,4] ≺· [1,3]. Thus we have Note that for the identity element e ∈ S n we have O(F e ) = 1 if n = 1 and O(F e ) = 0 otherwise. Letting π = (π 1 , . . . , π n ) be the unique path family of type e covering F , define V (F, I) to be the unique (row-semistrict) π-tableau of shape λ for which L(V (F, I)) is a row-strict Young tableau containing indices I j in row j. For S a k-element subset of [n], let F | S denote the zigzag network of order k isomorphic to the subnetwork of F covered by paths {π i | i ∈ S}. Also essential to our proofs of Theorems 9.4 and 9.13 is a map ι : {w ∈ S n | w avoids the patterns 3412 and 4231, F w connected } → {w ∈ S n−1 | w avoids the patterns 3412 and 4231, F w connected }.
Let F w be a connected zig-zag network of order n 2 corresponding to the concatenation (9.4). By Observation 3.
We declare ι(w) to be the permutation whose descending star network F ι(w) of order n − 1 is obtained from F w by deleting the path from source n to sink n, and, in the case that d 1 = n and d 2 = n − 1 (d t = n and d t−2 = n − 1), by contracting one edge whose vertices correspond to the central vertices of G [c 1 ,n] and G [c 2 ,n−1] (G [c t−1 ,n−1] and G [ct,n] ). Equivalently, F ι(w) is the zig-zag network corresponding to the concatenation For example, let n = 6. F 256431 corresponds to the concatenation G [3,6] • G [2,5] • G [1,2] , with d 2 = 5 = n − 1. Removing the path from source 6 to sink 6, and contracting the edge whose endpoints correspond to the central vertices of G [3,6] and G [2,5] , we obtain F ι(256431) , which can be shown (as in the example preceding Theorem 3.3) to be F 25431 . Similarly, F 246531 corresponds to the concatenation G [3,6] •G [2,4] •G [1,2] , with d 2 = 4 < n−1.
Removing the path from source 6 to sink 6, we obtain F ι(246531) , which can be shown to be F 24531 . Inversions in right-anchored, row-semistrict path tableaux are closely related to the polynomials {O(F ) | F a zig-zag network of order 1, . . . , n} in Definition 9.1. In order to state this relationship precisely (Lemma 9.3) and state a third combinatorial formula for ψ λ q (q e,w C w (q)) (Theorem 9.4), we define a family of sets {Z(F ) | F a zig-zag network} of tableaux and maps between these. For F a zig-zag network of order m, let Z(F ) be the set of right-anchored, row-semistrict F -tableaux of type e and shape (m). Note that if F is not connected, then Z(F ) = ∅, since a right-anchored F -tableau cannot be row-semistrict when F is disconnected. Now let F w be a connected zig-zag network of order m which corresponds to the concatenation (9. by declaring U to be the tableau obtained from U by deleting π n , and by declaring k to be the number of indices in the interval [b, m − 1] appearing to the left of m in L(U ). For example, consider the network F 256431 in (9.7) and let U be the tableau π π π 4 π 3 5 6 2 1 π π .
Then the unique interval in G containing m = 6 is [3,6], and there are two indices in this interval appearing to the left of 6 in L(U ). Thus γ(U ) = (U , 2) where U is the tableau π π π 4 π 2 5 3 1 π .
Lemma 9.2. For each connected zig-zag network F w of order m, the map γ is a bijection.
To invert γ, find an entry i d of L(U ) which belongs to [b, m − 1] and has exactly k indices from the interval [b, m − 1] to its left. (This is possible, since 0 k m − b − 1.) Now create a new F w -tableau by inserting π m into U immediately before π i d . This map is well-defined, because π i d intersects π m . It is clear that the map inverts γ. Since U has type e, it is clear that the number of inversions in U R involving π m is equal to the number of indices in the interval [b, m − 1] appearing to the left of m in L(U ). It follows that inv(U R ) = inv(U R ) + k.
By the above lemma, we can interpret O(F ) as a generating function for inversions in tableaux belonging to Z(F ). Proof. When m = 1, both sides of (9.9) are 1. Now assume that (9.9) holds for all zig-zag networks corresponding to 3412-avoiding, 4231-avoiding permutations in S 1 , . . . , S m−1 , and consider w ∈ S m avoiding the patterns 3412 and 4231. Let F w correspond to the concatenation (9.4). If F w is disconnected, then Z(F w ) = ∅ and both sides of (9.9) are 0. If F w is connected, let [b, m] be the unique interval in (9.4) to contain m. By induction and Lemma 9.2 we have In the third case, we obtain an expression similar to (9.11). In all cases, the expression is equal to O(F w ).
Now we can state the precise relationship between the polynomials O(F ) and inversions in right-anchored, row-semistrict path tableaux.
Theorem 9.4. Let w ∈ S n avoid the patterns 3412 and 4231. For λ = (λ 1 , . . . , λ r ) n, we have Proof. By Theorem 9.1, ψ λ q (q e,w C w (q)) is equal to the right-hand side of (9.2). Grouping terms in the sum and using (4.3), we may rewrite this expression as (9.13) where U now varies over the subset of right-anchored, row-semistrict tableaux of type e and shape λ satisfying U j = I j for each component of the appropriate ordered set partition I = (I 1 , . . . , I r ). For fixed I, this inner sum can be rewritten as where W (j) varies over right-anchored, row-semistrict F w | I j -tableaux of shape λ j and type e ∈ S λ j , and U again varies as in (9.13). By Lemma 9.3, the first r sums are equal to O(F w | I 1 ), . . . , O(F w | Ir ), and it is easy to see that for any tableau U in the last sum, we have inv(U ) = inv(V (F w , I) 1 • · · · • V (F w , I) r ). Thus we obtain the right-hand side of (9.12).
Proposition 9.5. Let w = w 1 · · · w n ∈ S n avoid the pattern 312 and define the sequence (f (1), . . . , f (n)) by f (j) = max{w 1 , . . . , w j }. Then we have (9.14) Proof. Setting λ = (n) in Theorem 9.4, we have ψ (n) q (q e,w C w (q)) = [n] q q 0 O(F w ). (9.15) If F w is not connected, then both sides of (9.15) are 0, and for some index k the prefix w 1 · · · w k of w belongs to S k . Thus f (k) = k and the right-hand side of (9.14) is 0 as well. Assume therefore that F w is connected. Since w avoids the pattern 312, F w is a descending star network and the intervals in the corresponding concatenation (9.4) form the chain [c 1 , d 1 ] ≺· · · · ≺· [c t , d t ] with d 1 = n, c t = 1. Thus the formula (9.5) for O(F w ) becomes the electronic journal of combinatorics 23(2) (2016), #P2.7 Defining g(j) = min{i | j ∈ [c i , d i ]} for j = 1, . . . , n, we may now rewrite (9.15) as Finally we claim that d g(j) = f (j) for j = 1, . . . , n − 1. We have f (j) d g(j) because there are no paths in F w from source j to sinks d g(j) + 1, . . . , n and therefore by Observation 3.4, no paths from sources 1, . . . , j − 1 to these sinks either. Similarly, we have f (j) d g(j) because j belongs to the interval [c g(j) , d g(j) ] and w c g(j) = d g(j) .
It is straightforward to show that the right-hand side of (9.16) coincides with the expression in [23,Thm. 7.1] for (−1) n−1 n = (−1) n− (n) z (n) times the coefficient of p n in X P,q .

Cylindrical F -tableaux and O(F )
In Theorem 9.13, we will prove an analog of Theorem 9.1 in which sums are taken over (left-anchored) cylindrical F -tableaux. To do so, we partition the set of cylindrical F -tableaux into equivalence classes as follows. Fix a permutation w ∈ S n avoiding the patterns 3412 and 4231, an integer partition λ n, and an ordered set partition I = (I 1 , . . . , I r ) of [n] of type λ. Let C(I, F w ) be the set of cylindrical F w -tableaux U such that for j = 1, . . . , r, the set of entries of L(U j ) is equal to I j . Let C L (I, F w ) be the subset of these tableaux which are left-anchored. Now the cylindrical analogs of the sums in (9.1) and (9.2) are where the left-hand sums are over cylindrical F w -tableaux of shape λ and left-anchored cylindrical F w -tableaux of shape λ, respectively. In both cases, it is easy to show that the inner right-hand sum factors as in Theorem 9.4. To state this factorization explicitly, we relate inv(U 1 • · · · • U r ) to intervals in the concatenation (9.4).
Lemma 9.6. Let w ∈ S n avoid the patterns 3412 and 4231, and let [c 1 , d 1 ], . . . , [c t , d t ] be the intervals appearing in the concatenation (9.4) of star networks that corresponds to F w . Let U be a cylindrical F w -tableau having r rows, and fix indices p 1 < p 2 in [r]. Then we have Proof. Let A and B denote the sets on the left-and right-hand sides of (9.18), respectively. Define a map ϕ : A → B as follows, assuming (π a , π b ) ∈ A. If a and b belong to a common interval [c i , d i ], then set ϕ((π a , π b )) = (a, b). Otherwise, read U p 1 cyclically from left to right starting at π b , and let π c be the first path which lies entirely above π a . Then set ϕ((π a , π b )) = (a, c). We claim that ϕ is a bijection. To see that ϕ is well defined, suppose that a and b belong to no common interval [c i , d i ], and let π f be the path in U p 1 terminating at sink b. If π f intersects π a , then there exists a path in F w from source a to sink b. Since a < b, Observation 3.4 and the comment following it imply that a and b belong to a common interval [c i , d i ], a contradiction. Thus the set of paths lying strictly above π a in U p 1 is nonempty, and the path π c is well defined. Suppose a and c belong to no common interval [c i , d i ], and let π g be the path in U p 1 cyclically preceding π c . Then π g terminates at sink c > a. By our choice of π c , the path π g must intersect π a . Thus there is a path in F w from source a to sink c. But this contradicts Observation 3.4.
The inverse ξ of ϕ may be described as follows, assuming (a, b) ∈ B. If π a intersects π b , then set ξ((a, b)) = (π a , π b ). Otherwise, read U p 1 cyclically from right to left starting at π b , and let π c be the first path such that a and c belong to no common interval [c i , d i ], and c > a. Then set ξ((a, b)) = (π a , π c ).
To see that ξ is well defined, suppose that π a does not intersect π b and that ξ((a, b)) = (π a , π c ). Let d be the index of the sink of π c , so that π d immediately follows π c . By our choice of π c , we have that a and d belong to a common interval [c i , d i ] or that d < a. Thus by Observation 3.4, there is a path in F w from source a to sink d, and by Lemma 3.5 paths π a and π c intersect. Now we claim that ϕ and ξ are in fact inverse to one another. This is clear when we restrict to pairs (a, b) belonging to a common interval in (9.4) such that the paths π a , π b intersect. Suppose therefore that π a intersects π b , and that a, b belong to no common interval in (9.4). Let ϕ((π a , π b )) = (a, c) and let ξ((a, c)) = (π a , π b ). Since π c lies entirely above π a , we have b = c. Suppose b = b and let π f be the path in U p 1 terminating at sink b . By the definitions of ϕ and ξ, and since a, b belong to no common interval in (9.4), the row U p 1 (up to cyclic rotation) has the form · · · π b · · · π b' π f · · · π c .
Since a, b belong to no common interval and a < b by the definition of ξ, we have by Observation 3.4 that there is no path in F w from source a to sink b . Thus π a and π f do not intersect, and π f must lie entirely above π a . It follows that ϕ((π a , π b )) = (a, f ) = (a, c), contradiction. Now suppose that π a does not intersect π b , and that a, b belong to some common interval [c i , d i ] in (9.4). Let ξ((a, b)) = (π a , π c ) and let ϕ((π a , π c )) = (a, b ). Since a and c belong to no common interval in (9.4), we have c = b . Suppose b = b and let g be the sink index of π b . By the definitions of ϕ and ξ, and since π b lies entirely above π a , the row U p 1 (up to cyclic rotation) has the form · · · π c · · · π b' π g · · · π b .
the electronic journal of combinatorics 23(2) (2016), #P2.7 By our choice of π b we have g > a, and there is no path in F w from source a to sink g. Thus by Observation 3.4 we have that a and g belong to no common interval in (9.4). But this implies that ξ((a, b)) = (π a , π g ) = (π a , π c ), a contradiction. Now we can factor the expressions in (9.17) as follows.
It is clear that a cylindrical tableau U is completely determined by the Young tableau L(U ). Under some conditions the insertion or deletion of a greatest letter in the left tableau of a cylindrical tableau yields a valid left tableau of another cylindrical tableau. In these cases, intersecting paths in the two cylindrical tableaux are closely related.
Lemma 9.8. Let F w be a connected zig-zag network of order n corresponding to the concatenation (9.4) of star networks (9.4), and let [b, n] be the unique interval containing n.
1. For any cylindrical F w -tableau U of shape (n), if T is the Young tableau obtained from L(U ) by deleting the entry n, then there exists a unique cylindrical F ι(w) -tableau U such that T = L(U ). 3. Let tableaux U and U in (1) contain the path families (π 1 , . . . , π n ) and (π 1 , . . . , π n−1 ), respectively. For all pairs (i, j), if π i , π i have the same sink index, and π j , π j have the same sink index, then π i and π j intersect if and only if π i and π j intersect.

For any cylindrical
Proof. (1) Let (i 1 , i 2 ) be a pair of cyclically consecutive entries in T . If these entries are also cyclically consecutive in L(U ), then there exists a (unique) path π i 1 in F w from source i 1 to sink i 2 . Since F ι(w) differs from F w by the removal of the unique path from source n to sink n and possibly the contraction of an edge to a single vertex, the image of π i 1 is again the unique path in F ι(w) from source i 1 to sink i 2 . If (i 1 , i 2 ) are not cyclically consecutive in L(U ), then i 1 cyclically precedes n and i 2 cyclically follows n in L(U ). Thus either i 1 or i 2 belongs to [b, n − 1]. Thus in G there is a path from source i 1 to the central vertex of G [b,n] and a path from this vertex to sink i 2 . It follows that there is a path in F w from source i 1 to sink i 2 . Uniqueness of U follows from uniqueness of source-to-sink paths in zig-zag networks. (See comment following Theorem 3.3.) (2) Let (i 1 , i 2 ) be a pair of cyclically consecutive entries in T . If these entries are also cyclically consecutive in L(V ), then there exists a (unique) path from source i 1 to sink i 2 in F ι(w) . Since each interval in the concatenation corresponding to F ι(w) is equal to or is contained in an interval in the concatenation corresponding to F w , we use Observation 3.4 and the fact that both F w and F ι(w) are connected to infer that there is a path in F w from source i 1 to sink i 2 . Again, uniqueness of V follows from uniqueness of source-to-sink paths in zig-zag networks.
(3) Let π i , π i , π j , π j satisfy the stated conditions. Then the source and sink indices of these paths are not equal to n. Suppose first that in the concatenation (9.4) corresponding to F w we have t = 1, or d 1 = n and d 2 < n − 1, or d t = n and d t−1 < n − 1. Then F ι(w) is the subgraph of F w obtained by deleting the unique path from source n to sink n. By the uniqueness of paths in descending star networks, we have π i = π i and π j = π j . Now suppose that in (9.4) we have d 1 = n and d 2 = n−1, or d t = n and d t−1 = n−1, and let x, y be the vertices in F w corresponding to the central vertices of the star networks G [b,n] and G [c 2 ,d 2 ] , respectively, or G [c t−1 ,d t−1 ] and G [b,n] , respectively. Then F ι(w) is obtained from F w by deleting the unique path from source n to sink n, and by contracting the edge (x, y) to a single vertex z. Thus if π i ∩ π j contains the edge (x, y) then π i ∩ π j contains the vertex z; if π i ∩ π j does not contain the edge (x, y), then π i ∩ π j = π i ∩ π j .
Given a zig-zag network F of order n, define Y n (F, i) to be the set of cylindrical Ftableaux U of shape (n) in which the first entry of L(U ) is i. In terms of our earlier notation, we have where we interpret [n] as the ordered set partition having one block. Let us examine the map U → U and the numbers inv(U ), inv(U ) defined by Lemma 9.8 (1) in the case that U ∈ Y (F w , i) and i n − 1. To be precise, for each pair (F w , i) where F w is a connected zig-zag network of order at least 2 with corresponding concatenation (9.4), poset of intervals, and [b, n] the unique interval containing n, the electronic journal of combinatorics 23(2) (2016), #P2.7 and 1 i n − 1, we define a map by declaring U to be the cylindrical tableau whose left tableau is obtained from L(U ) by deleting n, and by declaring k to be the number of paths following π n in U whose For example, let n = 6, recall (9.7), and consider the F 256431 -tableau and its left Young tableau U = with T = L(U ) and U ∈ Y 5 (F 25431 , 4). Since the only paths in U which follow π 6 and have source indices in [3,5] are π 5 , π 3 , we have δ 1 (U ) = (U , 2). Lemma 9.9. For each connected zig-zag network F w of order n, the map δ 1 is a bijection. Furthermore, if δ 1 (U ) = (U , k) then we have inv(U ) = inv(U ) + k.
Proof. Assume that F w corresponds to the concatenation (9.4) in which the unique interval containing n is [b, n], and let (U , k) be a pair in Y n−1 (F ι(w) , i) × {0, 1, . . . , n − b − 1}. To invert δ 1 , find an entry j ∈ [b, n − 1] in the tableau L(U ) with exactly k − 1 entries in [b, n − 1] to its right. (This is possible, since k n − 1 − b.) Now create a new Young tableau T by inserting the letter n into L(U ) immediately to the left of j if [b, n] is maximal in , or immediately to the right of j otherwise. By Lemma 9.8 (2) there is a unique F w -tableau U with L(U ) = T .
To compare inversions in U and U , write U = (π i 1 , . . . , π in ), U = (π i 1 , . . . , π i −1 , π i +1 , . . . , π in ), where π i = π n . Observe that all paths except π i −1 , π i in U have the same sources and sinks as the corresponding paths in U . Thus by Lemma 9.8 (3), two such paths form an inversion in U if and only if the corresponding paths form an inversion in U . Consider therefore inversions in U which involve one of the paths π i −1 , π i , and inversions in U which involve the path π i −1 . Suppose first that [b, n] is maximal in . By definition, there are k inversions in U of the form (π i , π c ), where π c terminates at a sink having index in [b, n − 1]. These in fact are the only inversions in U involving π i : since there are no paths in F w from source n to the electronic journal of combinatorics 23(2) (2016), #P2.7 sinks 1, . . . , b − 1, Observation 3.1 implies that π i cannot intersect any path terminating at one of these sinks. Now observe that path π i −1 terminates at sink n, while paths π i −1 and π n terminate at sink i +1 . By the maximality of [b, n], we have that i +1 belongs to the interval [b, n]. Thus the paths π i −1 and π i −1 are identical from their sources up to the vertex of F w (F ι(w) ) corresponding to the central vertex of G [b,n] . It follows that any path in U intersects π i −1 if and only if the corresponding path in U intersects π i −1 . Therefore we have inv(U ) = inv(U ) + k. Now suppose that [b, n] is not maximal in . Then it must be minimal, and since π i −1 terminates at sink n, we have i −1 b. Consider paths π c with c b. By the minimality of [b, n] in , we have that π c intersects π i −1 and π i at the vertex of F w (F ι(w) ) corresponding to the central vertex of G [b,n] . Thus (π c , π i −1 ) or (π i −1 , π c ) is an inversion in U if and only if the corresponding pair is an inversion in U . By definition, there are k inversions in U of the form (π i , π c ) with c b. Now consider paths π c with c < b. By Observation 3.1, no pair (π i −1 , π c ) is an inversion in U , since there is no path from source c to sink n in F w On the other hand, the paths π i and π i −1 are identical from the vertex of F w (or F ι(w) ) corresonding to the central vertex of G [b,n] to sink i +1 . Thus each pair (π i , π c ) is an inversion in U if and only if (π i −1 , π c ) is an inversion in U . It follows again that inv(U ) = inv(U ) + k.
Note that in the example preceding Lemma 9.9 we have inv(U ) = 6 = inv(U ) + 2, and δ 1 (U ) = (U , 2). Now let us examine a map U → U and the numbers inv(U ), inv(U ) closely related to those defined by Lemma 9.8 (1) in the case that U ∈ Y (F w , n). To be precise, for each connected zig-zag network F w of order at least 2 with corresponding concatenation (9.4), poset of intervals, and [b, n] the unique interval containing n, we define a map The tableau U exists and is unique by Lemma 9.8 (1). If [b, n] is maximal in then any path beginning at source n must terminate at a sink in this interval. Thus we have is not maximal in it must be minimal, and any path terminating at sink n must begin at a source in this interval. Thus we have i n ∈ [b, n − 1]. It follows that U belongs to the union in (9.21).
For example, let n = 6, recall (9.8), and consider the F 246531 -tableau and its left Young tableau U = with U ∈ Y 6 (246531, 6), and T having rightmost entry 4. Removing 6 from T and moving 4 to the leftmost position, we have the tableaux T = 4 1 2 5 3 , U = 4,1 1,2 2,3 3,5 5,4 , with T = L(U ) and U ∈ Y 5 (F 24531 , 4). Lemma 9.10. For each connected zig-zag network F w of order n, the map δ 2 is a bijection. Furthermore, if δ 2 (U ) = U ∈ Y n−1 (F ι(w) , j) and b is as in (9.21) then we have Proof. To invert δ 2 , let U be an element of Y (F ι(w) , j) for some 1 j n − 1. Create a Young tableau T from L(U ) by inserting n into the leftmost position and if [b, n] is not maximal in by moving j to the rightmost position. By Lemma 9.8 (2) there is a unique F w -tableau U = (π i 1 , . . . , π in ) satisfying L(U ) = T . To compare inversions in U and U , observe first that the paths π i 2 , . . . , π i n−1 in U have the same sources and sinks as the corresponding paths in is maximal in , (π in = π j , π i 2 , . . . , π i n−1 ) otherwise.
Thus by Lemma 9.8 (3), two such paths form an inversion in U if and only if the corresponding paths form an inversion in U . Consider therefore inversions in U which involve the paths π i 1 = π n or π in , and inversions in U which involve the path π in .
If [b, n] is maximal in , the path π i 1 = π n in U terminates at sink i 2 ∈ [b, n − 1]. Thus the path, which precedes all others in U , intersects only those n−b other paths in U which terminate at sinks [b, n] {i 2 }, and thus also pass through the vertex of F w corresponding to the central vertex of G [b,n] . Now observe that π in terminates at sink n of F w , while π in terminates at sink i 2 of F ι(w) . Since i 2 b, the paths π in and π in are identical up to the vertex of F w (F ι(w) ) which corresponds to the central vertex of G [b,n] . Thus any path π i k in U intersects π in if and only if the corresponding path π i k in U intersects π in . It follows that inv(U ) = inv(U ) + b − n in this case.
If [b, n] is not maximal in , then it is minimal. Since π in terminates at sink n, we have that i n b. Thus π in intersects and follows the n − i n + 1 paths π in+1 , . . . , π n which all intersect at least at the vertex of F w (or F ι(w) ) corresponding to the central vertex of G [b,n] . Now observe that the paths π n and π in both terminate at sink i 2 . Thus, since i n b, the paths are identical from the vertex of F w (F ι(w) ) which corresponds to the central vertex of G [b,n] until sink i 2 . Thus any path π k in U intersects π n if and only if the corresponding path π k in U intersects π in . It follows that inv(U ) = inv(U ) + n − j + 1 in this case.
Now we return to the problem of factoring the inner sums in (9.17).
Proposition 9.11. Fix w ∈ S m avoiding the patterns 3412 and 4231 and an index j ∈ [m]. Then we have U ∈Ym(Fw,j) q inv(U ) = q j−1 O(F w ). (9.22) Proof. We prove (9.22) by induction on m. The only zig-zag network of order m = 1 is F e , e ∈ S 1 . Thus the set Y 1 (F e , 1) consists of one tableau of shape (1) having no inversions. The left-and right-hand sides of (9.22) are therefore q 0 = 1 and q 0 O(F e ) = [0] q ! = 1, respectively. Now assume (9.22) to hold for zig-zag networks corresponding to 3412-avoiding, 4231avoiding permutations in S m−1 and consider F w with w ∈ S m avoiding the patterns 3412 and 4231. Let F w correspond to the concatenation (9.4) of star networks with [b, m] the unique interval containing m, let u = ι(w), and fix an integer j ∈ [m]. If F w is disconnected, then the set Y m (F w , j) is empty and both sides of (9.22) are 0. Assume therefore that F w is connected. If j < m, then by Lemma 9.9 and induction we have Now applying (9.11) to (9.23), (9.24), we have for j = 1, . . . , m that U ∈Ym(Fw,j) Applying (9.20) to the previous result, we have the following. Now we have the following q-analogs of Theorem 4.7 (iv-a).
By Theorem 9.4, this is ψ λ q (q e,w C w (q)). For example, consider the descending star network F 3421 in (5.10) and the sum in (9.25). It is easy to verify that there are eighteen cylindrical F 3421 -tableaux of shape 31. Four of these are which have 5, 2, 1, and 2 inversions, respectively. Together, they contribute q + 2q 2 + q 5 to ψ 31 q (q e,3421 C 3421 (q)) = 1 + 3q + 5q 2 + 5q 3 + 3q 4 + q 5 . Now consider the sum in (9.26). It is easy to verify that there are six left-anchored cylindrical F 3421 -tableaux of shape 31: the second and third tableaux in (9.27) and the four tableaux These tableaux U of shape 31 yield six tableaux U 1 • U 2 of shape 4: the second and third tableaux in (9.28) and the four tableaux which have 2, 1, 2, 3, 0, and 1 inversions, respectively. Together, the six tableaux contribute 1 + 2q + 2q 2 + q 3 to [3] q [1] q (1 + 2q + 2q 2 + q 3 ) = ψ 31 q (q e,3421 C 3421 (q)). As a consequence of Theorem 9.13, we now have the following analog of Corollary 6.5. Say that a partition λ of n is a refinement of another partition (or composition) µ of n if λ can be obtained from µ by replacing each part µ i by an integer partition of µ i and sorting the results into weakly decreasing order.
if and only if λ is not a refinement of µ.
Proof. It is clear that if λ is not a refinement of µ, then there is no cylindrical F w -tableau of shape λ. Therefore by Theorem 9.13 we have that ψ λ q (q e,w C w (q)) = 0. Suppose on the other hand that λ = (λ 1 , . . . , λ r ) is a refinement of µ, and let J 1 , . . . , J s be the subintervals of [n] corresponding to the connected components of F w . Then there exists an ordered set partition I = (I 1 , . . . , I r ) [n], whose type is a rearrangement of λ, such that each block of J is a union of several consecutive blocks of I. Let π = (π 1 , . . . , π n ) be the unique path family of type e covering F w . It is clear now that we can construct at least one record-free, row-semistrict F w tableau of type e and shape λ, by creating a row containing the paths π a , . . . , π b (in order) for each block [a, b] of I. By Theorem 9.1, we therefore have ψ λ q (q e,w C w (q)) = 0. It would be interesting to extend Theorem 9.13 to include a q-analog of Theorem 4.7 (iv-b). In particular the fourth identity in Equation (1.2) suggests that an answer to Problem 4.8 and its q-analog are related to a set partition of tableaux counted by ψ λ . It is not clear whether such a partition is more easily expressed in terms of record-free, rowsemistrict F w -tableaux of type e, right-anchored, row-semistrict F w -tableaux of type e, cylindrical F w -tableaux, left-anchored cylindrical F w -tableaux, or cyclically row-semistrict F w -tableaux of type e. Problem 9.15. Find a statistic stat on F -tableaux such that we have ψ λ q (q e,w C w (q)) = U q stat(U ) , where the sum is over all cyclically row-semistrict F w -tableaux of type e and shape λ.
the electronic journal of combinatorics 23(2) (2016), #P2.7 10 Results concerning φ λ q (q e,w C w (q)) Recall that the component statements of Theorem 4.7 pertaining to monomial traces are weaker than those pertaining to other traces. To state a q-analog of Theorem 4.7 (v-a), we will use several partial orders, including majorization and refinement of integer partitions. We will use the symbol to denote majorization and R to denote refinement, as defined before Corollary 9.14. We begin by stating an analog of Corollaries 6.5, 8.2.
Proof. Let w ∈ S n have a decreasing subsequence of length greater than λ 1 and recall that there exist integers {a λ,µ | λ, µ n} such that If φ λ q (q e,w C w (q)) = 0 then some partition µ in the above sum satisfies µ q (q e,w C w (q)) = 0. But the number of parts of µ is necessarily less than or equal to λ 1 . This contradicts Corollary 6.5.
Similarly, we have a partial analog of Corollary 9.14.
Proposition 10.2. Let w ∈ S n avoid the patterns 3412 and 4231 and let the component sizes of F w (in weakly decreasing order) be µ = (µ 1 , . . . , µ r ). Then for each partition λ n not refining µ we have φ λ q (q e,w C w (q)) = 0.
Proof. Observe that we may rewrite the last equation in (1.2) as since no row-constant Young tableau of shape λ has content ν unless λ refines ν. Inverting the matrix (L λ,ν ) λ,ν n and evaluating traces at q e,w C w (q) we have φ λ q (q e,w C w (q)) = ν R λ L −1 λ,ν ψ λ q (q e,w C w (q)). (10.1) Now suppose that we have λ R µ. Then each partition ν in (10.1) satisfies ν R µ. By Corollary 9.14, each term on the right-hand side of (10.1) is zero.
We remark that Propositions 10.1, 10.2 are not new: they follow from [11,Prop. 4.1]. For more facts about these evaluations, see [11,Sec 4]. Now we complete the proof of Theorem 4.7 (v-a) and provide a q-analog of this result. Let T C (F w , µ) denote the set of column-strict F w -tableaux of shape µ and type e. If j < k, then we have by induction and (10.4) that φ 2 i 1 n−2i q (q e,w C w (q)) = 0 for i < k, i = j. Now (10.5) implies that φ 2 k 1 n−2k q (q e,w C w (q)) = 0, and the claim is true. If j = k, then we have by (10.3) -(10.4) that (n−k,k) q (q e,w C w (q)) = φ 2 k 1 n−2k q (q e,w C w (q)) and by (10.5) the claim again is true.
We remark that the obvious q-analogs of Theorem 4.7 (v-b) are false. Consider the permutation w = 3142 and the evaluation φ 22 q (q e,w C w (q)) = q + q 2 . F w is the penultimate zig-zag network in (3.4), and there are two column-strict cylindrical F w -tableaux of shape 22, Unfortunately, as U varies over these tableaux we have Perhaps a correct q-analog of Theorem 4.7 (v-b) would help with the formulation of an interpretation of φ λ q (q e,w C w (q)) when w avoids the patterns 3412 and 4231. Given Theorems 5.4, 6.4, 8. 1, 9.1, 9.4, and 9.13, it seems reasonable to hope that F w -tableaux can play an important role in such an interpretation.
Problem 10.4. Find a property X of F w -tableaux and a statistic stat such that for λ n and w avoiding the patterns 3412 and 4231 we have where the sum is over all F w -tableaux U of shape λ having property X.