Characters and chromatic symmetric functions

Let $P$ be a poset, $inc(P)$ its incomparability graph, and $X_{inc(P)}$ the corresponding chromatic symmetric function, as defined by Stanley in {\em Adv. Math.}, {\bf 111} (1995) pp.~166--194. Certain conditions on $P$ imply that the expansions of $X_{inc(P)}$ in standard symmetric function bases yield coefficients which have simple combinatorial interpretations. By expressing these coefficients as character evaluations, we extend several of these interpretations to {\em all} posets $P$. Consequences include new combinatorial interpretations of the permanent and other immanants of totally nonnegative matrices, and of the sum of elementary coefficients in the Shareshian-Wachs chromatic quasisymmetric function $X_{inc(P),q}$ when $P$ is a unit interval order.


Introduction
The Frobenius isomorphism from the space T n of symmetric group traces to the space Λ n of homogeneous degree-n symmetric functions, where ctype(w) is the cycle type of w, allows one to translate statements about the representation theory of the symmetric group S n to the language of symmetric functions. Conversely, one may use the inverse of the Frobenius isomorphism to study symmetric functions, such as Stanley's chromatic symmetric functions X G [30], in terms of S n -class functions. In particular, for G the incomparability graph inc(P ) of a poset P , we will expand X G in the standard symmetric function bases, and we will use the inverse Frobenius isomorphism to interpret the resulting coefficients. Our main tool is reminiscent of the Cauchy and dual Cauchy identities [24, §I.4] for symmetric functions in two sets of variables, (1.2) i,j≥1  In Section 2 we present standard bases of the trace space of the Hecke algebra H n (q), the trace space of the symmetric group algebra Z[S n ], and the space Λ n of homogeneous degree-n symmetric functions. We show that the expansion of any homogeneous degree-n symmetric function in any standard basis of Λ n yields coefficients which are trace evaluations. In Section 3 we apply this result to the standard expansions of chromatic symmetric functions of the form X inc(P ) and the related symmetric functions ωX inc(P ) . We obtain combinatorial interpretations for resulting coefficients of X inc(P ) and ωX inc(P ) for all posets P , thus extending previous results which hold only for special classes of posets. In particular, we interpret the monomial and power sum coefficients of ωX inc(P ) in Theorems 3.7 and 3.10, respectively. We also obtain a new proof in Proposition 3.5 of Kaliszewski's interpretation of hook-Schur coefficients. In each case, the trace evaluations allow for very simple proofs of our results. In Section 4 we apply our interpretations of trace evaluations to functions of totally nonnegative matrices, obtaining new interpretations of these. In particular, we interpret "hook" irreducible character immanants in Theorem 4.11, induced trivial character immanants in Theorem 4.15, and power sum immanants in Theorem 4.16. These results include two new interpretations (4.10) -(4.11) of the permanent of a totally nonnegative matrix and play an important role in the evaluation of hyperoctahedral group characters at elements of the type-BC Kazhdan-Lusztig basis [29]. These results also lead to a new expression in Section 5 for the sum of elementary coefficients of the Shareshian-Wachs chromatic quasisymmetric function X inc(P ),q when P is a unit interval order.
Let H n (q) be the (type A) Hecke algebra, generated over Z[q 1 2 , q¯1 2 ] by T s 1 , . . . , T s n−1 subject to relations T 2 s i = (q − 1)T s i + q for i = 1, . . . , n − 1, For each w ∈ S n and w = s i 1 · · · s i ℓ a reduced expression, define the natural basis element T w = T s i 1 · · · T s i ℓ (which does not depend upon the choice of a reduced expression). (See, e.g., [3].) The (modified) Kazhdan-Lusztig basis of H n (q) as a Z[q 1 2 , q¯1 2 ]-module consists of elements { C w (q) | w ∈ S n } related to the natural basis by where ≤ is the Bruhat order on S n , and where {P v,w (q) | v, w ∈ S n } are the recursively defined Kazhdan-Lusztig polynomials. (Our basis element C w (q) is q ℓ(w) 2 times the basis element C ′ w in [20].) When w avoids the patterns 3412 and 4231 (the one-line notation w 1 · · · w n contains no subword w i 1 w i 2 w i 3 w i 4 whose letters have values appearing in the same relative order as 4231 or 3412), each polynomial P v,w (q) is identically 1.
Let T n,q be the Z[q 1 2 , q¯1 2 ]-module of H n (q)-traces, linear functionals θ q : H n (q) → Z[q 1 2 , q¯1 2 ] satisfying θ q (gh) = θ q (hg) for all g, h ∈ H n (q). For any trace θ q : T w → a(q) in T n,q , the q 1 2 = 1 specialization θ : w → a(1) belongs to the space T n := T n,1 of Z[S n ]-traces from Z[S n ] → Z (S n -class functions). Like the Z-module Λ n , the trace spaces T n,q and T n have dimension equal to the number of integer partitions of n. The Frobenius Z-module isomorphism (1.1) and its q-extension, Frob q : T n,q → Λ n , θ q → Frob(θ), define bijections between standard bases of Λ, T n , and T n,q . Schur functions correspond to irreducible characters, while elementary and homogeneous symmetric functions correspond to induced sign and trivial characters, where S λ is the Young subgroup of S n indexed by λ and H λ (q) is the corresponding parabolic subalgebra of H n (q). The power sum, monomial, and forgotten bases of Λ n correspond to bases of T n (T n,q ) which are not characters. We call these the power sum , and forgotten {γ λ | λ ⊢ n} ({γ λ q | λ ⊢ n}) traces, respectively. These are the bases related to the irreducible character bases by the same matrices of character evaluations and inverse Koskta numbers that relate power sum, monomial, and forgotten symmetric functions to Schur functions, where χ µ (λ) := χ µ (w) for any w ∈ S n having ctype(w) = λ. Just as the power sum symmetric functions form a Q-basis of Λ n , the power sum traces form Q-bases of T n and T n,q . The power sum traces of T n also have the natural definition where z λ = λ 1 · · · λ ℓ α 1 ! · · · α n ! and α i is the number of parts of λ equal to i. It can be useful to record trace evaluations in a symmetric generating function. In particular, for g ∈ Q(q) ⊗ H n (q), we record induced sign character evaluations by defining This symmetric generating function in fact gives us all of the standard trace evaluations.
equivalently, ωY q (g) is equal to While this follows from (1.2) -(1.3), we provide a short proof.
Proof. Consider the second and fourth sums in (2.5), in which the symmetric functions and traces satisfy Using (2.6) to expand the fourth sum in the monomial symmetric function basis, we have i.e., it is equal to the second sum. Similarly, for each of the remaining sums λ θ λ q (g)t λ in (2.5), there is a matrix (M λ,µ ) λ,µ⊢n and equations relating it to the fourth sum. In particular, Since each symmetric function is a quasisymmetric function, researchers sometimes express elements of Λ n in terms of bases of the Z-module QSym n of degree-n quasisymmetric functions. (See [32, §7.19] for definitions). The coefficients arising in such expansions also can be viewed as trace evaluations. In particular, let {F n,S | S ⊆ [n − 1]} be the fundamental quasisymmetric function basis of QSym n . For any Young tableau U of shape λ = (λ 1 , . . . , λ r ), let U 1 , . . . , U r denote its rows, and let • denote concatenation of rows. Define the inverse descent set of U by Now we have the following fundamental quasisymmetric expansion of Y q (g).
Corollary 2.2. For λ = (λ 1 , . . . , λ r ) ⊢ n and S ⊆ [n − 1], define b(λ, S) to be the number of standard Young tableaux U of shape λ with ides(U) = S. Then we have of a symmetric function satisfy The result now follows from Proposition 2.1.
To say that the functions {Y q (g) | g ∈ H n (q)} arise often in the study of symmetric functions would be an understatement; essentially every element of Z[q] ⊗ Λ n has this form. Proposition 2.3. Every symmetric function in Z[q] ⊗ Λ n has the form Y q (g) for some element g ∈ Q(q) ⊗ H n (q).
Proof. Fix a symmetric function in Z[q]⊗Λ n , express it in the elementary basis as λ⊢n a λ e λ , and define the H n (q) element and w µ is the maximal element of the Young subgroup S µ of S n . By [16,Prop. 4.1], we have Of course, for g ∈ Q[S n ], the q 1 2 = 1 specialization Y (g) := Y 1 (g) of (2.4) satisfies the q 1 2 = 1 specializations of Proposition 2.1, Corollary 2.2, and Proposition 2.3.

Chromatic symmetric functions
Closely related to symmetric generating functions for Z[S n ]-traces are symmetric generating functions for graph colorings. Define a proper coloring of a (simple undirected) graph G = (V, E) to be an assignment κ : V → {1, 2, . . . , } of colors (positive integers) to V such that adjacent vertices have different colors. For G on |V | = n vertices and any composition α = (α 1 , . . . , α ℓ ) n, say that a coloring κ of G has type α if α i vertices have color i for i = 1, . . . , ℓ. Let c(G, α) be the number of proper colorings of G of type α. Stanley [30] defined the chromatic symmetric function of G to be where the first sum is over all proper colorings of G. By Proposition 2.3 we see that for each graph G on n vertices, there exists an element g ∈ Q[S n ] such that X G = Y (g). Such an element g is not uniquely determined by G, and is not in general easily described in terms of the structure of G. On the other hand, the evaluations of traces at such elements are easily described in terms of G.
Observation 3.1. Let G be a graph on n vertices and let g ∈ Q[S n ] satisfy Y (g) = X G . Then for each trace θ = λ⊢n a λ ǫ λ ∈ T n , we have θ(g) = λ⊢n a λ c(G, λ).
For every trace θ ∈ T n , Proposition 2.3 and Observation 3.1 allow us to define where g is any element in Q[S n ] satisfying Y (g) = X G . By Proposition 2.1, we have that Some conditions on graphs G and traces θ imply the numbers θ(G) to be positive, and sometimes the resulting positive numbers have nice combinatorial interpretations, particularly when G is the incomparability graph of a poset. (See, e.g., [8], [27], [30].) Given a poset P , define its incomparability graph inc(P ) to be the graph having a vertex for each element of P and an edge {i, j} for each incomparable pair of elements of P . For positive integers a, b, call a poset (a + b)-free if it has no induced subposet isomorphic to a disjoint sum of an a-element chain and a b-element chain. For example, the following poset P is (3 + 1)-free and (2 + 2)-free, and has incomparability graph inc(P ) = G. .
Then define w = w(P ) = w 1 · · · w n by For example, elements of the poset P in (3.5) are already labeled to satsify (3.6), Thus we compute w(P ) = 34521. The labeling (3.6) of P is natural in the sense that elements labeled a 1 , a 2 satisfy (3.8) a 1 < P a 2 =⇒ a 1 < a 2 (as integers).
The map P → w(P ) is a bijection from n-element unit interval orders to the 1 n+1 2n n 312-avoiding permutations in S n , and gives us the following result [8,Cor. 7.5].
Proposition 3.2. Let P be an n-element unit interval order and w = w(P ) the corresponding 312-avoiding permutation in S n . Then we have X inc(P ) = Y ( C w (1)).
Combinatorial interpretations of numbers θ(inc(P )) often involve structures called Ptableaux and statistics on these. Define a P -tableau of shape λ ⊢ |P | to be a filling of a (French) Young diagram of shape λ with the elements of P , one per box. Given such a Ptableau U, let U i be the ith row (from the bottom) of U, and let U i,j be the jth entry in row i. If P -tableau U consists of a single row, we will call it a P -permutation. In particular, each concatenation U i 1 • · · · • U i k of the rows (in any order) of a P -tableau U is a P -permutation. If the elements of a poset are [n] := {1, . . . , n}, we will sometimes write a P -permutation as an ordinary permutation v 1 · · · v n ∈ S n . For example, (3.9) shows a poset P , a P -tableau U of shape 32, the second row U 2 and entry U 2,1 of U, and the P -permutation U 2 • U 1 which may be viewed as an element of S 5 . The statistics we apply to P -tableaux are P -analogs of traditional permutation statistics. Call a position (i, j) in U a P -descent if U i,j > P U i,j+1 , and define des P (U) to be the number of P -descents in U. Define U to be the P -tableau obtained from U by ordering the elements in each row from least to greatest labels. That is, U i,j is the entry of U i whose label is jth-smallest, as an integer. Call a position (i, j) in U a P -excedance if U i,j > P U i,j , and define exc P (U) to be the number of P -excedances in U. Call a position (i, j) in U a P -record if U i,1 , . . . , U i,j−1 < P U i,j , and call the record nontrivial if j = 1. Define rec P (U) to be the number of nontrivial P -records in U. For example, we look again at the poset P in (3.9), the P -tableaux there, and three more, We have des P (T ) = des P (U) = des P (V ) = 0, while des P (W ) = des P (U 2 • U 1 ) = 1, because 3 > P 1 and 4 > P 1. We have that rec P (U) = rec P (U 2 • U 1 ) = rec P (W ) = 0 since the first entry in each row of these tableaux is greater than or incomparable to the remaining entries in the same row. On the other hand, rec P (T ) = 2 since 1 < P 3 and 2 < P 4, and rec P (V ) = 1 since 1 < P 3. Reordering the entries of each row in the tableaux above we obtain Comparing T , U, V with T , U , V , we have exc P (T ) = exc P (U) = exc P (V ) = 0. Comparing W and W , we see that position (1, 1) is the only P -excedance: 3 > P 1. Thus exc P (W ) = 1. Comparing U 2 • U 1 to U 2 • U 1 , we see that positions (1, 1) and (1, 2) are both P -excedances: 5 > P 1, 4 > P 2. Thus exc P (U 2 • U 1 ) = 2. Using comparability in P and the above statistics, we define six classes of P -tableaux. Call a P -tableau U of shape λ (1) P -descent-free or row-semistrict if des P (U) = 0, (2) column-strict if the entries of each column satisfy U i,j < P U i+1,j , (3) standard if it is column-strict and row-semistrict, P -record-free if it has no nontrivial P -records. For example, we may examine the tableaux in (3.10) for these properties to obtain the table where the row-semistrict tableaux T and U fail to be cyclically row-semistrict because their first rows begin with 1 and end with 3 > P 1.

Induced sign characters / monomial coefficients of X inc(P )
. By definition, the induced sign characters satisfy ǫ λ (inc(P )) = # column-strict P -tableaux of shape λ ⊤ for all graphs G = (V, E) and posets P . Since ǫ n (G) = 1 if G has no edges and is 0 otherwise, we can easily express ǫ λ (G) in terms of subgraphs of G. For J ⊆ V = [n], let J = [n] J, and define G J = subgraph of G induced by vertices J, P J = subposet of P induced by elements J.
Given α = (α 1 , . . . , α r ) n, call a sequence (I 1 , . . . , I r ) of subsets of [n] an ordered set Using (3.11) and the language of ordered set partitions, we can decompose some trace evaluations θ(G) as follows. (1) If λ = (λ 1 , . . . , λ r ) ⊢ n is the weakly decreasing rearrangement of the parts of µ ⊢ k and ν ⊢ n − k, then we have where the first sum is over ordered set partitions of [n] of type λ. Proof.
(2) Express t 1 , t 2 in the elementary bases of Λ k , Λ n−k as (3.14) and let λ(µ, ν) ⊢ n be the weakly decreasing rearrangement of the parts of µ and ν. Then we have By (3.12) and (3.14), θ(G) equals We will use this fact to prove similar formulas for induced trivial characters and power sum traces.
Kaliszewski [18,Prop. 4.3] extended this result to all posets P when λ is a hook shape. We give an alternate proof of this fact using (3.11) and the inverse Kostka numbers, which satisfy For partitions λ, µ with |µ| ≤ |λ| = n and µ i ≤ λ i for all i, define a (skew) Young diagram of shape λ/µ to be the diagram obtained from a Young diagram of shape λ by removing the µ i leftmost boxes in row i for all i. Call a Young diagram a border strip if it contains no 2 × 2 subdiagram of boxes. Define a special ribbon diagram of shape µ ⊢ n and type λ = (λ 1 , . . . , λ ℓ ) ⊢ n to be a Young diagram of shape µ subdivided into border strips (ribbons) of sizes λ 1 , . . . , λ ℓ , each of which contains a cell from the first row of µ. Given a special ribbon diagram Q, define sgn(Q) to be −1 to the number of pairs of boxes in Q which are horizontally adjacent and which belong to the same ribbon. It is known that we have where the sum is over all special ribbon diagrams Q of shape µ and type λ. Special ribbon diagrams also relate column-strict P -tableaux of hook shape µ = k1 n−k ⊢ n to column-strict P -tableaux of shape λ ⊢ n majorizing µ ⊤ . To state this relationship precisely, we first observe that subsets of [k − 1] correspond bijectively to special ribbon diagrams of shape µ. Let D be a Young diagram of shape µ. For each subset S ⊆ [k − 1], define Q(µ, S) to be the special ribbon diagram of shape µ whose ribbons are the equivalence classes defined by for all j ∈ S, and define λ(µ, S) to be the type of Q(µ, S). For example, when µ = 31 3 = (3, 1, 1, 1) the ribbon diagrams in (3.18) correspond to the subsets ∅, {2}, {1} {1, 2}, respectively. This bijection leads to another. Lemma 3.4. Fix hook partition µ = k1 n−k and subset S ⊆ [k − 1], and define Q = Q(µ, S) and λ = λ(µ, S) as above. There is a bijection between column-strict P -tableaux U of shape µ satisfying U 1,j > P U 1,j+1 for all j ∈ S, and column-strict P -tableaux of shape λ ⊤ .
Proof. Let ϕ be the claimed bijection. Given a column-strict P -tableau U of shape µ satisfying U 1,j > P U 1,j+1 for all j ∈ S, create P -tableau ϕ(U) as follows.
(1) Let D be a Young diagram of shape λ ⊤ .
(2) Label the ribbons of Q from left to right as 1, . . . , r, and let q i be the number of boxes in ribbon i.
(4) For i = 1, . . . , r, place the elements of U under ribbon i into the leftmost unused column of D which contains exactly q i boxes, so that elements strictly increase in the column from bottom to top. The resulting P -tableau is clearly column-strict of shape λ ⊤ . To invert ϕ, suppose we are given a column-strict P -tableau T of shape λ ⊤ .
(1) Superimpose Q onto an empty tableau U of shape µ.
(2) For i = 1, . . . , r, fill U by placing the elements from column i of T onto the leftmost available ribbon of Q of length λ ⊤ i boxes, so that elements decrease from top to bottom, and from left to right. By the definition of Q, the resulting P -tableau U has entries which satisfy the required inequalities, and it is easy to see that ϕ(U) = T .
As an example of the above bijection, fix µ = 61, S = {1, 5} ⊆ {1, 2, 3, 4, 5}, and consider the pair where U is a column-strict P -tableau of shape µ which satisfies U 1,i > P U 1,i+1 , for i = 1, 5. Corresponding to S is the special ribbon diagram Proposition 3.5. For any n-element poset P and hook shape k1 n−k ⊢ n, the evaluation χ k1 n−k (inc(P )) equals the number of standard P -tableaux of shape k1 n−k .
Proof. Fix µ = k1 n−k , let a(µ) be the number of standard P -tableaux of shape µ, and for each subset S ⊆ [k − 1], let b(µ, S) be the number of column-strict P -tableaux U of shape µ which satisfy By the principle of inclusion-exclusion, these are related by Each subset S ⊆ [k − 1] corresponds to a special ribbon diagram Q = Q(µ, S) of shape µ as described before Lemma 3.4. The partition λ(µ, S) satisfies |S| = k − ℓ(λ(µ, S)). By Lemma 3.4, b(µ, S) is also the number of pairs (Q, T ) with Q a special ribbon diagram of shape µ and type λ(µ, S), and T a column-strict P -tableau of shape λ ⊤ . Thus we may rewrite (3.23) by summing over pairs (S, U) and (Q, T ) satisfying the above conditions, Now collect terms in the last sum which correspond to special ribbon diagrams sharing the same partition type(Q) ⊢ n. Summing first over λ ⊢ n and then over special ribbon diagrams Q of shape µ and type λ, we have that a(µ) is By (3.17) and (3.16), this is λ⊢n K −1 λ,µ ⊤ ǫ λ (inc(P )) = χ µ (inc(P )).

3.3.
Induced trivial characters / monomial coefficients of ωX inc(P ) . Given graph O 2 is acyclic; O 1 is not. By [30] we have for all graphs G that (3.24) η n (G) = # acyclic orientations of G, and as a consequence (or by Proposition 3.5) we have for all posets P that (3.25) η n (inc(P )) = #P -descent-free P -permutations.
We will extend this result to all posets in Theorem 3.7 and will include combinatorial interpretations related to P -excedance-free P -tableaux and acyclic orientations. To do so, we consider some straightforward extensions of permutation statistics to P -permutations. Let w be a P -permutation, and let exc P (w) and aexc P (w) be the numbers of P -excedances and P -antiexcedances in w, Let des P (w) and asc P (w) be the numbers of P -descents and P -ascents in w, Define the standard cycle notation of w ∈ S n to be the cycle notation in which cycles are listed in increasing order of their greatest elements, and these greatest elements are listed first in each cycle. Let σ : S n → S n be the bijection [31, §1.3] defined by setting σ(w) equal to the permutation whose one-line notation is obtained by erasing parentheses from the standard cycle notation of w. For example, to compute σ(5243761), we write 5243761 in standard cycle notation as (2)(4, 3)(6)(7, 1, 5), since the greatest elements of the cycles satisfy 2 < 4 < 6 < 7. Then we erase parentheses to obtain 2436715.
The following result is a strengthening of [31, Exercise 3.60c]. It first appeared with a different proof in [34,Thm. 4.6].
Proposition 3.6. For any poset P , the statistics des P , asc P , exc P , aexc P are equally distributed on the set of all P -permutations.
Proof. (des P ∼ asc P ) We have asc P (w) = des P (w n · · · w 1 ). (exc P ∼ aexc P ) We have aexc P (w) = exc P (w −1 ). (des P ∼ aexc P ) Assume first that P is naturally labeled (3.8). We claim that the map σ satisfies des P (σ(w)) = aexc P (w). To see this, write w in standard cycle notation and σ(w) in one-line notation as Suppose that j is a P -descent of σ(w). By the natural labeling of P , we have a j > a j+1 and therefore a j can not appear last in its cycle in the standard cycle notation for w. Thus we have a j > P a j+1 = w(a j ) and position a j is a P -antiexcedance of w. Thus position a j is a P -antiexcedance of w. Now suppose that j is not a P -descent of σ(w). Then in the standard cycle notation for w we have either that a j , a j+1 appear consecutively in a cycle and satisfy a j > P a j+1 = w(a j ), or that a j appears last in its cycle and satisfies a j ≤ w(a j ). By the natural labeling of P , this last inequality implies a j > P w(a j ). Thus in both cases position a j is not a P -antiexcedance of w. Now assume that P is nonnaturally labeled, and let P ′ be a naturally labeled copy of P . Then for some u ∈ S n , the poset isomorphism P → P ′ is given by i → u i , and the bijection w → uw from S n to itself satisfies aexc P (w) = aexc P ′ (uw).
Given a graph G on n vertices and an ordered set partition (I 1 , . . . , I r ) of [n] of type λ ⊢ n, call the sequence (G I 1 , . . . , G Ir ) an ordered induced subgraph partition of G of type λ. Let I λ (G) be the set of such sequences, and define an acyclic orientation of an element of I λ (G) to be a sequence (O 1 , . . . , O r ), where O j is an acyclic orientation of G I j . For example, consider the ordered set partition (234, 15) of type (3,2). A graph G, its ordered induced subgraph partition (G 234 , G 15 ), and one acyclic orientation (O 1 , O 2 ) of this are (1) # P -descent-free P -tableaux of shape λ, (2) # P -excedance-free P -tableaux of shape λ, (3) # acyclic orientations of sequences (inc(P I 1 ), . . . , inc(P Ir )) ∈ I λ (inc(P )).
For i = 1, . . . , r, do (1) Initialize U i to be the empty Young diagram of shape λ i . The special case of Theorem 3.7 (2) corresponding to P a unit interval order and λ = n has an interpretation in terms of the Bruhat order on S n . Corollary 3.8. Let P be a unit interval order on [n] labeled as in (3.6) and let w ∈ S n be the corresponding 312-avoiding permutation as in (3.7). Then we have Proof. Define the matrix A = (a i,j ) by By [28,Lem. 5.3 (3)], the product a 1,v 1 · · · a n,vn is 1 if v ≤ w and is 0 otherwise. But a 1,v 1 · · · a n,vn = 1 if and only if i < P v i for i = 1, . . . , n, i.e., if and only if v is P -excedance free.
3.4. Power sum traces / scaled power sum coefficients of ωX inc(P ) . It is known that we have ψ λ (inc(P )) ≥ 0 for all P [30], and (3.29) ψ λ (inc(P )) = # cyclically row-semistrict P -tableaux of shape λ = # P -record-free, row-semistrict P -tableaux of shape λ for all unit interval orders P labeled as in (3.6) [1, Thm. 4], [8,Thm. 4.7], [27, §7]. We will extend these results to all posets in Theorem 3.10, and will include more combinatorial interpretations involving inc(P ) and a related directed graph. Define ngr(P ) to be the directed graph whose vertices are the elements of P and whose edges are the ordered pairs {(i, j) ∈ P 2 | i > P j}, including loops (i, i) for all i ∈ P . For example, a poset P and related directed graph ngr(P ) are To prove our results, we will use the transition matrix which relates the elementary and power sum bases of Λ n . In particular, where c µ equals the number of subgraphs of the (labeled) cycle graph whose connected components are paths on µ 1 , . . . , µ k vertices. Clearly such subgraphs correspond bijectively to subsets S ⊆ [n], and we define µ(S) to be the weakly decreasing sequence of component cardinalities of C n,S . We will also use the set of P -permutations whose (cyclic) P -descent set contains S, Lemma 3.9. For any n-element poset P and subset S ⊆ [n], the permutations in B(S) correspond bijectively to column-strict P -tableaux of shape µ(S). In particular, we have |B(S)| = ǫ µ(S) (inc(P )).
Proof. Fix w ∈ B(S). For each maximal interval [i, j] (mod n) with i, . . . , j ∈ S, we have the chain w i > P · · · > P w j+1 ; for i − 1, i ∈ S, we have the one-element chain w i . Let µ = µ(S) and insert the ℓ(µ) = n−|S| chains into the columns of a Young diagram of shape µ ⊤ to obtain a column-strict P -tableau. For chains of equal cardinalities, fill the leftmost available column of the tableau with the leftmost available chain in w (considering · · · > P w n > P w 1 > P · · · to be the leftmost chain of all, if it exists). It is easy to see that this map is invertible.
As an example of the above bijection, consider the poset, subset, and P -permutation Combining these P -descents to form chains (and collecting leftover 1-chains), we have w 7 > P w 1 > P w 2 = 753, w 3 = 1, w 4 > P w 5 > P w 6 = 642, which we can insert in order of weakly decreasing cardinality into a column-strict P -tableau of shape 331 ⊤ = 322 7 6 5 4 3 2 1 , where we have broken the tie between 3-element chains by inserting the leftmost chain (the one containing w 1 ) first. Now we may interpret ψ λ (inc(P )) as follows.
(1) We claim that for any n-element poset P , ψ n (inc(P )) is the number of cyclically rowsemistrict P -permutations (P -descent-free w 1 · · · w n with w n > P w 1 ). To see this, let a be the number of such P -permutations and define B(S) as in (3.32). By the principle of inclusion/exclusion and Lemma 3.9, the cardinalities a and |B(S)| are related by a =
(2) Stanley [30,Thm. 3.3] showed that φ n (P ) = ψ n (P ) equals the number of acyclic orientations of inc(P ) having exactly one sink. Reversing all edges in such an orientation and applying the bijection at the end of the proof of Theorem 3.7, we have that ψ n (P ) equals the number of P -record-free P -descent-free P -permutations. Then by (3.34) we have the desired result.
(3) Let U be a cyclically row-semistrict P -tableau of shape λ. Each pair of horizontally adjacent entries (U i,j , U i,j+1 ) (or (U i,λ i , U i,1 )) in U corresponds to an edge in ngr(P ), and the row U i corresponds to a cycle. Removing all other edges from ngr(P ) and listing the cycles in order of their corresponding rows of U, we obtain the desired disjoint cyclic vertex cover. (4) As described above, ψ n (P ) equals the number of acyclic orientations of inc(P ) having exactly one source. Now (3.34) gives the desired result.
Since φ 1 n = ǫ n and φ n = ψ n , it is tempting to conjecture a formula for φ λ (inc(P )) which combines column-strictness of Equation The author has found that for n ≤ 5, the sets of analogous tableaux for n-element posets have cardinalities no greater than the true values of φ λ (inc(P )). This suggests the following question.
Question 3.12. Do we have for all unit interval orders P and all partitions λ ⊢ |P |, that φ λ (inc(P )) is greater than or equal to (1) the number of standard, cyclically row-semistrict P -tableaux of shape λ? (2) the number of standard, P -record-free P -tableaux of shape λ?

Fundamental expansion of X inc(P ) .
We remark that for any graph G, there are known combinatorial interpretations for the coefficients arising in the fundamental expansions of X G and ωX G . These are easiest to express in the special case that G is the incomparability graph of an n-element poset P . Writing

Trace identities.
Symmetric function identities, Lemma 3.3, and the combinatorial interpretations stated in Equation (3.11) -Theorem 3.10 lead to some identities relating P -permutations to pairs of subposet permutations. For instance, the first identity in the following result implies that the number of ways to create a row-semistrict P -permutation and circle one element equals the number of P -permutations w 1 · · · w n with w 1 · · · w i a cyclically row-semistrict P Jpermutation for some i-element set J ⊆ [n] (0 < i ≤ n), and w i+1 · · · w n a row-semistrict P J -permutation.
Corollary 3.13. Let G be a graph on n vertices. We have Proof. Applying Lemma 3.3 (2) to the symmetric function identities we obtain the claimed graph identities.

Applications to total nonnegativity
Nonnegative expansions of chromatic symmetric functions in the standard bases are closely related to functions of totally nonnegative matrices. We will make this relationship precise in Corollary 4.7.
Call a real n × n matrix A = (a i,j ) totally nonnegative if for each pair (I, J) of subsets of [n], the square submatrix A I,J := (a i,j ) i∈I,j∈J satisfies det(A I,J ) ≥ 0. Such matrices are closely related to directed graphs called planar networks. Define a (nonnegative weighted) planar network of order n to be a directed, planar, acyclic digraph D = (V, E) which can be embedded in a disc so that 2n distinguished vertices labeled clockwise as s 1 , . . . , s n , t n , . . . , t 1 lie on the boundary of the disc, with a nonnegative real weight c u,v assigned to each edge (u, v) ∈ E. We may assume that s 1 , . . . , s n , called sources, have indegree 0 and that t n , . . . , t 1 , called sinks, have outdegree 0. To every source-to-sink path, we associate a weight equal to the product of weights of its edges, and we define the path matrix A = A(D) = (a i,j ) i,j∈ [n] by setting a i,j equal to the sum of weights of all paths from s i to t j . For example, we have the following planar network D of order 3, in which unlabeled edges have weight 1, and its path matrix A. A result often attributed to Lindström [21] but proved earlier by Karlin and McGregor [19] asserts the total nonnegativity of such a matrix. where the sum is over all families π = (π 1 , . . . , π n ) of pairwise nonintersecting paths in D, with π i a path from s i to t i for i = 1, . . . , n, and where (4.2) wgt(π) := wgt(π 1 ) · · · wgt(π n ).
Theorem 4.2. For each n × n totally nonnegative matrix A, there exists a nonnegative weighted planar network D of order n whose path matrix is A.
In the language of Section 2, we have det(A) = Imm ǫ n (A). For some functions θ, the number Imm θ (A) is nonnegative for all totally nonnegative matrices A and has a nice combinatorial interpretation in terms of families of paths in D. We say that a path family π = (π 1 , . . . , π n ) in a planar network D has type w = w 1 · · · w n ∈ S n if for i = 1, . . . , n, the path π i begins at the source s i and terminates at the sink t w i . Define the sets Each path family π = (π 1 , . . . , π n ) ∈ P w (D) forms a poset P = P (π) defined by π i < P π j if i < j (as integers) and π i does not intersect π j . For example, the planar network D in (4.1) has two path families of type 132. These and their posets are Observe that if path families π, σ in D consist of the same multiset K of edges of D, then they satisfy wgt(π) = wgt(σ). Call such a multiset K a bijective skeleton, and define wgt(K) to be the product of its edge weights, with multiplicities. Define the sets Π(K) = {π ∈ P(D) | edge multiset of π is K}, where the sum is over all bijective skeletons K in D.
Proof. By the definition of path matrix, we can interpret each product of matrix entries appearing in Imm θ (A) as a 1,w 1 · · · a n,wn = Multiplying each product by θ(w), summing over w ∈ S n , and using the linearity of θ, we may thus express Imm θ (A) as Sometimes a combinatorial interpretation for Imm θ (A) comes from careful consideration of θ(z(K)); other times it comes from a simple expression for Imm θ (A), such as the Littlewood per(A I 1 ,I 1 ) · · · per(A Ir,Ir ), (4.5) where the sums are over ordered set partitions of [n] of type λ = (λ 1 , . . . , λ r ).
Proof. By Theorem 4.1 and the comment immediately following it, the term of (4.4) corresonding to a fixed ordered set partition (I 1 , . . . , I r ) is equal to the sum of weights of path families π = (π 1 , . . . , π n ) of type e in which for j = 1, . . . , r, paths indexed by I j are pairwise nonintersecting. This partitioned path family naturally forms a column-strict tableau U = U(π, I 1 , . . . , I r ) of shape λ ⊤ , if we place paths indexed by I j into column j. We may therefore write the right-hand-side of (4.4) as #{(I 1 , . . . , I r ) | U(π, I 1 , . . . , I r ) is column-strict of shape λ ⊤ } ǫ λ (inc(P (π))).
Proof. Weight the planar network D by algebraically independent real numbers and let A be its path matrix. By Proposition 4.3, we have where the sum is over all bijective skeletons K in D. Since the edge weights of D are algebraically independent, we may compare this expression to the right-hand-side of (4.6) to obtain ǫ µ (z(K)) = π∈Πe(K) ǫ µ (inc(P (π))).
Expanding θ in the induced sign character basis {ǫ µ | µ ⊢ n}, we obtain the desired result. Proposition 4.3 and Corollary 4.5 show that for θ ∈ T n we may compute Imm θ (A) by considering a planar network D having path matrix A, each path family π of type e in D, and the corresponding chromatic symmetric function X inc(P (π)) . Corollary 4.6. For D a planar network having path matrix A, we have where K varies over all bijective skeletons in D.
Thus if θ ∈ T n satisfies θ(inc(P )) ≥ 0 for all posets P , then it also satisfies Imm θ (A) ≥ 0 for all totally nonnegative matrices A. For the convenience of the reader we summarize this and other known implications as follows.

Irreducible character immanants.
While no combinatorial interpretation is known for Imm χ λ (A), Stembridge [35,Cor. 3.3] proved the following.  We do have a combinatorial interpretation in the special case that λ is a hook shape. wgt(π) (# standard P (π)-tableaux of shape k1 n−k ).
Proof. Let λ = k1 n−k . By Proposition 4.3, Corollary 4.5, and Proposition 3.5, we have where K varies over all bijective skeletons in D. This is equal to the claimed expression.
The case k = n (4.10) can also be deduced from Stanley's interpretation [30,Thm. 3.3] of χ n (inc(P )) as the number of acyclic orientations of inc(P ), using the bijection at the end of the proof of Theorem 3.7.

The permanent and induced trivial characters.
An obvious consequence of Proposition 4.3 is a combinatorial interpretation of the permanent of a totally nonnegative matrix. where wgt(π) is defined as in (4.2).

Power sum immanants.
Like the induced trivial character immanants {Imm η λ (A) | λ ⊢ n}, the power sum immanants {Imm ψ λ (A) | λ ⊢ n} have some combinatorial interpretations which are closely related to chromatic symmetric function coefficients, and others which are related to path families in a planar network. Call a π-tableau U of shape λ cylindrical if in each row U i = U i,1 · · · U i,λ i , we have R(U i,1 · · · U i,λ i ) = L(U i,2 · · · U i,λ i U i,1 ), i.e., each path begins where the preceding path in its row terminates. For example, let π be the path family in (4.12).
The interpretation (4.18) now follows from the definition of path matrix.

Monomial immanants.
By Corollary 4.7, the fact that we do not know φ λ (inc(P )) to be nonnegative for unit interval orders P ([33, Conj. 5.5]) implies that we do not know monomial immanants to evaluate nonnegatively on totally nonnegative matrices. We have the following conjecture of Stembridge [   (1) If λ = 21 n−2 then Imm φ λ (A) ≥ 0.
(2) If λ is the rectangular shape r k then # column-strict, cylindrical π-tableaux of shape r k .
The author has found that for n ≤ 5, the sets of analogous tableaux for certain planar networks have cardinalities no greater than the true values of Imm φ λ (A), where A is the path matrix of the planar network. Specifically, these planar networks, called descending star networks in [8, §3], correspond bijectively to unit interval orders. This suggests the following question.  Proof. Assume that φ λ q ( C w (q)) ∈ N[q] for all w. Consider θ ∈ T n and expand it in the monomial trace basis as θ = λ a λ φ λ . If a λ ≥ 0 for all λ, then by our assumption we have θ( C w (1)) = λ a λ φ λ ( C w (1)) ≥ 0, and statement (1) of Corollary 4.7 is true. Now suppose that a µ < 0 for some µ ⊢ n and let w µ be the maximal element of the Young subgroup S µ . Then by (2.7) [16,Prop. 4.1] and our assumption we have that θ( C wµ (1)) < 0. Since w µ avoids the patterns 3412 and 4231, statement (8) of the Corollary is false.

Immanant identities.
Analogous to Corollary 3.13 are three identities which follow from Corollary 4.8. The third of these is known as Muir's identity. Proof. Similar to proof of Corollary 3.13.
(1) Given an acyclic orientation O, define inv(O) to be the number of oriented edges (j, i) with j > i. We have where the sum on the right is over all acyclic orientations O of inc(P ) [26,Thm. 5.3]. (2) Given a P -permutation U, define inv P (U) = #{(i, j) | j > i (as integers), j > P i, and j appears to the left of i in U }.
We have (5.5) λ⊢n φ λ q (inc(P )) = U q inv P (U ) , where the sum is over all P -descent-free P -permutations U [26, Thm. 6.3]. It is possible to extend Theorem 3.7 (2) to obtain a similar formula for excedance-free Ppermutations as well. Given a P -tableau U, define inv(U) = #{(i, j) | j > i (as integers), and j appears to the left of i in U }.
Proposition 5.2. Let P be an n-element unit interval order, labeled as in (3.6) and corresponding by (3.7) to the 312-avoiding permutation w ∈ S n . Then we have where U in the final sum varies over all P -excedance-free P -permutations.
Since η n q (T v ) = q ℓ(v) , we have the first equality in (5.6). To see the second equality, let U = v 1 · · · v n be a P -permutation. Then U appears in the third sum of (5.6) if and only if exc P (U) = 0. By Corollary 3.8, this condition is equivalent to v ≤ w, and clearly we have ℓ(v) = inv(U).
Combining Proposition 5.2 with (5.4) and (5.5), we obtain equidistribution results for the three variations of inversion statistics. Unfortunately, the map w → σ(w −1 ) from Subsection 3.3, which satisfies exc P (w) = des P (σ(w −1 )) does not satisfy inv(w) = inv P (σ(w −1 )). (Neither does the map in [34,Rmk. 4.7] mentioned before Proposition 3.6.) Furthermore, the statistic pairs (exc P , inv) and (des P , inv P ) cannot be equidistributed on S n , since inv and inv P are not equidistributed on S n . This suggests the following problem for unit interval orders P , labeled as in (3.6).
Problem 5.3. Find a bijection ϕ from descent-free P -permutations to excedance-free Ppermutations which satisfies inv P (U) = inv(ϕ(U)). where the second and third sum are over descent-free P -permutations U having k P -records, and acyclic orientations O of inc(P ) having k sources.
It would be interesting to similarly refine Proposition 5.2.
Problem 5.5. Let P be an n-element unit interval order labeled as in (3.6), and let w(P ) be the corresponding 312-avoiding permutation as in (3.7). Find functions δ 1 , δ 2 so that λ⊢n ℓ(λ)=k φ λ q (inc(P )) = v≤w δ 1 (v,w)=k where U in the final sum is a P -permutation.

Acknowledgements
The author is grateful to an anonymous referee for numerous corrections and suggestions which helped improve the article significantly.