Cyclic descents, matchings and Schur-positivity

A new descent set statistic on involutions, defined geometrically via their interpretation as matchings, is introduced in this paper, and shown to be equi-distributed with the standard one. This concept is then applied to construct explicit cyclic descent extensions on involutions, standard Young tableaux and Motzkin paths. Schur-positivity of the associated quasisymmetric functions follows.

Cyclic descent extensions were introduced in the study of Lie algebras [15] and descent algebras [7]. Surprising connections of cyclic descent extensions to a variety of mathematical areas were found later. For connections of cyclic descents to Kazhdan-Lusztig theory see [20]; for topological aspects and connections to the Steinberg torus see [8]; for twisted Schützenberger promotion see [20,13]; for cyclic quasisymmetric functions and Schur-positivity see [1,2,5]; for higher Lie characters see [2]; and for Postnikov's toric Schur functions and Gromov-Witten invariants see [3].
The question addressed in [2] was: which conjugacy classes in S n carry a cyclic descent extension? Cellini's cyclic descent map does not provide a cyclic descent extension on most conjugacy classes. However, it turns out that most conjugacy classes carry such an extension. yields a pair (cDes, p) which is a cyclic descent extension for this conjugacy class.
A full characterization of the conjugacy classes in S n which carry a cyclic descent extension was given. The proof of Theorem 4, presented in [2], is algebraic and not constructive. In this paper we present a solution of this problem for the conjugacy classes of involutions. For n k 0 with n − k even, let I n,k be the conjugacy class of involutions with k fixed points in the symmetric group S n . We present a purely combinatorial constructive proof of the following result.
Theorem 6. For every n > k > 0 with n − k even, I n,k carries a cyclic descent extension. For k = 0 and k = n there is only an Escherian cyclic extension.
In order to construct an explicit cyclic descent extension for conjugacy classes of involutions with fixed points, we have to consider first the conjugacy class of fixed-pointfree involutions. It will be shown that a certain geometrically-defined set-valued function on perfect matchings is equidistributed with the standard descent set on fixed-point-free involutions, leading to an Escherian cyclic descent extension for this conjugacy class of involutions (k = 0) and an ordinary (non-Escherian) extension for 0 < k < n.
For n k 0 with n − k even, let M n,k be the set of partial matchings on n points, labeled by [n] := {1, . . . , n}, with exactly k unmatched points.
Definition 8. The standard descent set of a matching m ∈ M n,k , denoted Des(m), is defined via the one-line notation of the corresponding involution in I n,k . Definition 9. The geometric descent set of a matching m ∈ M n,k , denoted GDes(m), consists of the geometric descents of m, defined as follows. Draw the n points on a horizontal line in the real plane and label them by 1, . . . , n from left to right. Indicate a matched pair {i, j}, with i < j, by drawing an arc in the upper half plane from the point labeled i to the point labeled j. An index i ∈ [n − 1] is a geometric descent of the matching m ∈ M n,k if one of the following conditions holds: 2. The arc containing i intersects the arc containing i + 1.

i is unmatched and i + 1 is matched.
See Figure 1 for an example. x Des(m) y GDes(m) = m∈M 2n,0 x GDes(m) y Des(m) .
For a matching m ∈ M n,k let cr(m) and ne(m) be the crossing number and nesting number of m, respectively; see Definition 29 below. Using Lemma 10 we will prove the following.
Theorem 11. For every n k 0 with n − k even Bijective proofs of Lemma 10 and Theorem 11 will be described in Section 3. Let M n := n k=0 M n,k be the set of all matchings on n points, labeled by 1, . . . , n. Let um(m) be the number of unmatched points in a matching m ∈ M n . For a partition λ let ht(λ) be the number of parts in λ, let oc(λ) be the number of odd parts in the conjugate partition, and let s λ be the corresponding Schur function. For D ⊆ [n − 1] let F n,D be the corresponding fundamental quasisymmetric function. For definitions and more details see Subsection 4.1. The following Schur-positivity phenomenon follows from the proof of Theorem 11.
The existence of cyclic descent extensions, on conjugacy classes of involutions with fixed points and other combinatorial sets, follows. To verify this observe, first, that there is a very natural cyclic extension of GDes on M n,k .
Definition 13. Draw n points on a circle and label them by 1, . . . , n clockwise. Indicate a matched pair by drawing a chord between the corresponding points. A point i ∈ [n] is a cyclic geometric descent of a matching m ∈ M n,k if one of the following conditions holds (where addition is modulo n): 2. The chord containing i intersects the chord containing i + 1.
3. i is unmatched and i + 1 is matched.
The cyclic geometric descent set of m is denoted by cGDes(m).
Let r : M n,k → M n,k correspond to clockwise rotation by 2π/n. Here is an explicit version of Theorem 6.
Proposition 14. Assume that n k 0 with n − k even.
(a) If 0 < k < n then the pair (cDes,ι −1 • r •ι) is a (non-Escherian) cyclic extension of Des on I n,k .  The cyclic descent extension from Proposition 14 can be further refined to certain subsets of I n,k , yielding a combinatorial cyclic descent extension for sets of standard Young tableaux of bounded height with a given number of odd columns. Letting the height be at most 2 with all columns even, or height at most 3 with no further restrictions, give explicit cyclic descent extensions for the sets of Dyck paths and Motzkin paths of fixed length, respectively. These cyclic extensions coincide with those determined by Dennis White (as described in [17]) and Bin Han [12].
The structure of this paper is as follows. Section 2 contains some preliminary background. Section 3 contains bijective proofs of the equidistribution results, Lemma 10 and Theorem 11. Section 4 contains a proof of the Schur-positivity result, Theorem 12. Section 5 deals with cyclic descent extensions and proves Proposition 14, thus Theorem 6. Finally, Section 6 presents a non-constructive proof of a refinement of Theorem 11, based on a very recent unpublished result of Gessel.
Let S n denote the symmetric group consisting of all permutations of [n]. A permutation π ∈ S n will be represented by the one-row notation π = [π 1 , . . . , π n ] ∈ S n , where π i := π(i) (i ∈ [n]); denote also Fix(π) := {i ∈ [n] : π(i) = i}, the set of fixed points of π. Recall that the descent set of a permutation π ∈ S n is Des(π) := {i ∈ [n − 1] : π(i) > π(i + 1)}. Another important family of combinatorial objects for which there is a well-studied notion of descent set is the set of standard Young tableaux (SYT). Let SYT(λ) denote the set of standard Young tableaux of shape λ, where λ is a partition of n. We draw tableaux in English notation, as in Figure 3. The descent set of T ∈ SYT(λ) is For example, the descent set of the SYT in Figure 3 is {1, 3, 5, 6}.  The Robinson-Schensted (RS) correspondence is a bijection π → (P π , Q π ) from permutations in S n to pairs of standard Young tableaux (SYT) of the same shape and size n. The common shape λ of the insertion tableau P π and the recording tableau Q π is called the shape of the permutation π. We recall basic properties of the RS correspondence that will be used in the paper. For more details see, e.g., [22].
The height ht(λ) of a shape λ is the number of rows in λ.

Proposition 15. [23]
For every permutation π ∈ S n , the height of the shape of π is equal to the maximal length of a decreasing subsequence in the one-line notation of π. 1. P π = Q π −1 , thus Q π = P π −1 and P π = Q π if and only if π ∈ S n is an involution.

Proposition 17. [24]
The number of columns of odd length in the shape of an involution with k fixed points is equal to k.
Let SYT n denote the set of all SYT of size n, and let SYT n,k denote the set of standard Young tableaux of size n having k columns of odd length. Consider the map Q : S n → SYT n defined by mapping π ∈ S n to the corresponding Robinson-Schensted recording tableau Q π .
Corollary 18. The map Q, restricted to involutions with a fixed number of fixed points, is a descent-set-preserving bijection from the set I n,k of involutions in S n with k fixed points to the set SYT n,k of standard Young tableaux of size n with k odd columns.
Proof. By Proposition 16.1, the map Q determines a bijection from the set of involutions I n to the set of all SYT of size n. By Proposition 16.2, this map is descent-set-preserving and, by Proposition 17, the pre-image of SYT n,k is I n,k .
Let X be a totally-ordered set of letters, and let U and V be disjoint finite subsets of X. Let S U and S V be the groups of permutations on U and V , respectively.
For σ ∈ S U and τ ∈ S V , the set of shuffles of σ and τ , denoted by σ ¡ τ , is the set of all permutations of the disjoint union U V in which the letters of U appear in the same order as in σ and the letters of V appear in the same order as in τ . For sets A and B of permutations on disjoint finite totally-ordered sets of letters U and Observation 19. By the definition of the RS correspondence, the smallest n − k letters in P π form a sub-tableau which depends only on their relative positions in π.
In particular, letting σ be a permutation on [k] and τ a permutation on [n] \ [k], all π ∈ σ ¡ τ have a common sub-tableau of P π consisting of the smallest k letters. Proposition 20. For every σ ∈ I n−k,0 and π ∈ σ ¡ [n − k + 1, . . . , n], the number of odd columns in the RS shape of π is equal to k.
Proof. Since σ ∈ I n−k,0 , by Proposition 17 all the columns of its shape have even length. By Observation 19, the shape of the sub-tableau consisting of the smallest n − k letters in P π , which are the letters of σ, is the shape of σ. On the other hand, for every shuffle π ∈ σ ¡ [n − k + 1, . . . , n] and every n − k < i < n, i ∈ Des(π −1 ). By Proposition 16 this implies that i ∈ Des(P π ) for all such i, so that the largest k letters in P π belong to distinct columns and increase from left to right. They are therefore in the bottom cells of the odd columns of P π , and the result follows.
The proof of Proposition 20 implies the following.
Corollary 21. For every σ ∈ I n−k,0 and π ∈ σ ¡ [n − k + 1, . . . , n], the largest k letters in P π appear in the bottom cells of the k odd columns of P π , and they are increasing from left to right.

Involutions and oscillating tableaux
Consider the Young lattice whose elements are all partitions, ordered by inclusion of the corresponding Young diagrams. A standard Young tableau of shape λ may be viewed as a maximal chain, in the Young lattice, from the empty partition to λ; see, e.g., [4, §14.2.5.1]. A variation of this description yields oscillating tableaux, which correspond to general paths in the Hasse diagram of the Young lattice, from the empty diagram to a diagram of shape λ. The size of the oscillating tableau is the length of the path, and its shape is λ. We focus on closed paths of length 2n from the empty diagram to itself; in other words, on oscillating tableaux of size 2n with an empty shape. The set of all such oscillating tableaux will be denoted by O 2n . A key tool in this paper is Sundaram's bijection s : I 2n,0 → O 2n , from the set I 2n,0 of fixed-point-free involutions in S 2n to the set O 2n of oscillating tableaux of size 2n and empty shape; see [28]. We hereby describe this bijection. [28]) Let π ∈ I 2n,0 . We start with λ 0 = ∅. For 1 d 2n, define a standard Young tableau of shape λ d , with letters forming a subset of [2n], from a presumably-defined standard Young tableau of shape λ d−1 , as follows. Let t d = (i, j), i < j, be the unique transposition which affects d in the factorization of π into a product of n disjoint transpositions. If d = i, insert j into the tableau of shape λ d−1 using Robinson-Schensted insertion and get a tableau of shape λ d . If d = j, delete j from the tableau of shape λ d−1 and apply jeu-de-taquin to get a tableau of shape λ d . We get a sequence of 2n + 1 tableaux of shapes λ d , 0 d 2n. Ignoring the letters in the tableaux yields a sequence of shapes, which is the oscillating tableau corresponding to π.

Definition 22. (Sundaram's bijection
Thus, the oscillating tableau corresponding to π is A special case of [28,Theorem 5.3] is the following.
Theorem 24. The map s : I 2n,0 → O 2n defined above is a bijection.
A characterization of the descents of π in the language of oscillating tableaux follows. In all other cases, i ∈ Des(π).

Definition 26. For an oscillating tableau
Another bijection t : I 2n,0 → O 2n , using growth diagrams, was described by Roby [21, §4.2]. This bijection applies a growth diagram algorithm to (a half of) the permutation matrix corresponding to a fixed-point-free involution π ∈ I 2n,0 , with empty boundary conditions, and reads an oscillating tableau t(π) from the main diagonal. It relates to Sundaram's bijection via conjugation.
Proof. Conjugating a permutation π ∈ S 2n by w 0 corresponds to reflecting its permutation matrix about its vertical midline as well as about its horizontal midline. This is equivalent to a 180-degree rotation of the permutation matrix. An inspection of the algorithm shows that for π ∈ I 2n,0 this simply reverses the oscillating tableau t(π) on the main diagonal. Combining this with Proposition 27 completes the proof.

Matchings
Chen et al. [6] generalized Sundaram's bijection, described in Subsection 2.2 above, and applied it to the enumeration of crossings and nestings in perfect matchings and partitions.
Definition 29. Let m ∈ M n be a matching.
1. The crossing number cr(m) of m is the maximal r such that there exist matched pairs 2. The nesting number ne(m) of m is the maximal r such that there exist matched pairs {i 1 , j 1 }, {i 2 , j 2 }, . . . , {i r , j r } in m with 1 i 1 < · · · < i r < j r < · · · < j 1 n.
Theorem 11 follows. The bijectionι : M n,k → M n,k is used to prove Theorem 12 in Subsection 4.2, and to determine cyclic descents on involutions in Section 5.
2. there exist a < i and b > i + 1, such that (a, i) and (i + 1, b) are chords in π.
3. there exist a < i and b > i + 1, such that (i, b) and (a, i + 1) are chords in π.
4. there exist i + 1 < a < b such that (i, a) and (i + 1, b) are chords in π.
5. there exist i + 1 < a < b such that (i + 1, a) and (i, b) are chords in π.
6. there exist a < b < i such that (a, i) and (b, i + 1) are chords in π.
7. there exist a < b < i such that (a, i + 1) and (b, i) are chords in π.
By Definition 22 of an oscillating tableau and basic properties of the insertion algorithm, what we do in the i th and (i + 1) st steps of the first 5 cases above is 1. add a box and then delete a box.

delete a box and then add a box.
3. add a box and then delete a box. 4. add a box and then add another box in a weakly higher row.
5. add a box and then add another box in a strictly lower row.
(7) delete a box and then delete another box in a strictly higher row. This is the description for π. Forπ = ι(π) we have a conjugate oscillating tableau. The description for cases (1)-(3) remains the same, whereas case (4) is switched with case (5) and case (6) is switched with case (7). By Observation 25, this translates to i ∈ Des(π) in cases (1), (3), (4) and (6), but not in the other cases. This completes the proof.
Remark 35. Arguments, similar to those used in the proof of Lemma 10, were used by Kim [14] to prove the symmetry of the Eulerian and Mahonian distributions on I 2n,0 .
The following refinement of Lemma 10 follows.

Proof of Theorem 11
In this subsection we describe a map ι : M n,k → M n,k , for any 0 k n, which generalizes the bijection ι : M 2n,0 → M 2n,0 used in the previous subsection. It will be shown thatι is a bijection which maps the descent set to the geometric descent set and the crossing number to the nesting number, implying Theorem 11.
Recall that M n,k is naturally identified with I n,k (Remark 7). In the rest of this section it will be more convenient to consider involutions in I n,k , rather than matchings in M n,k , since the shuffle operation and the RS correspondence used here are defined in terms of permutations (in particular, involutions). Definition 38. Fix n k 0, with n − k even.

For
where Q τ is the recording tableau of τ : where SYT n denotes the set of standard Young tableaux of size n and I n is the set of involutions in S n . Note that q(τ ) is an involution by Proposition 16.1.

Letι
The following proposition implies Theorem 33.
To prove Proposition 39 we first generalize the concepts of crossing and nesting numbers, from matchings (or, equivalently, involutions) to shuffles of fixed-point-free involutions with increasing sequences.
Proof. By Definition 38, ϕ is a bijection. To show that it maps GDes to Des, let 1 i n and consider the four possible cases.
This proves the claim regarding GDes and Des. The claim about crossing and nesting numbers follows from Definition 38, Definition 41 and Theorem 32.
To prove Proposition 39 we also need the following lemmas.
This completes the proof.
Observe that (τ ) − (σ) ∈ {0, 1}; hence, by Proposition 15, By Definition 41 and Remark 46 we deduce Recall the map Q : S n → SYT n sending each π ∈ S n to the corresponding RS recording tableau Q π . Recall also the notation SYT n,k for the set of all SYT of size n with k odd columns.
Lemma 48. For any n k 0 with n − k even, the map Q restricts to a descent-setpreserving bijection from the set of shuffles I n−k,0 ¡ [n − k + 1, . . . , n] to SYT n,k .
To prove that Q is a bijection, we will construct an inverse. Assuming that τ ∈ I n−k,0 ¡ [n − k + 1, . . . , n], it will be shown that τ can be reconstructed from its recording tableau Q τ , which can be an arbitrary element of SYT n,k . Assume that τ ∈ σ ¡[n−k+1,...,n], where σ ∈ I n−k,0 . By Corollary 21, the largest k letters in P τ appear in the bottom cells of the k odd columns of P τ , and they are increasing from left to right. These cells can be identified from Q τ , which has the same shape as P τ . We want to recover the positions of these k largest letters in τ , i.e., the values τ −1 (i) for n − k + 1 i n. Recall, from Proposition 16.1, that if τ corresponds (under RS) to the pair (P τ , Q τ ) then τ −1 corresponds to (P τ −1 , Q τ −1 ) = (Q τ , P τ ). Apply to Q τ , k times, the inverse of the RS insertion algorithm, as in [22, proof of Theorem 3.1.1]. Here is the first step: • Let T n := Q τ . Assume that the bottom cell of the the rightmost odd column is in the r th row. Let i r be the entry in this cell.
• Let i r−1 be the largest letter smaller than i r in the (r − 1) st row. Delete i r from the r th row and replace i r−1 by i r .
• Repeat this step until i 1 , the largest letter smaller than i 2 in the first row, is replaced by i 2 .
• The letter i 1 is the position of n in τ , namely τ −1 (n).
Apply the same procedure to the resulting tableau T n−1 (with τ −1 (n) removed) to find the position of n − 1, namely τ −1 (n − 1), and so on. After k steps, the positions of the increasing subsequence of τ consisting of the largest k letters have been determined. At this stage we get T n−k , which is the P tableau corresponding to the sequence τ −1 (1), . . . , τ −1 (n−k). This sequence has the same relative order of letters as the sequence σ −1 (1), . . . , σ −1 (n − k). Replacing the n − k letters in T n−k by 1, . . . , n − k with the same relative order therefore yields P σ −1 = Q σ . It is clear from the algorithm that all the columns of T n−k , and therefore of Q σ , have even lengths. By Corollary 18 (for k = 0) there is a unique fixed-point-free involution σ with this Q σ as a Q tableau. Since σ is an involution, Proposition 16.1 implies that P σ = Q σ , and therefore σ is the RS preimage of (Q σ , Q σ ). This completes the proof. . This tableau has n = 8 cells and k = 2 columns of odd length. Thus τ ∈ σ ¡ [7,8] for some σ ∈ I 6,0 .
Start with T 8 = Q τ . The bottom cell of the rightmost odd column in T 8 appears in the first row. Thus r = 1, the entry there is i r = i 1 = 6, and therefore τ −1 (8) = 6. The resulting tableau after deleting this letter is Recall the map q from Definition 38.4: for τ ∈ I n−k,0 ¡[n−k+1,...,n], the involution q(τ ) is the RS preimage of (Q τ , Q τ ), where Q τ is the RS recording tableau of τ .
Let A be a set of combinatorial objects, equipped with a set-valued function D : A → 2 [n −1] . We say that A is symmetric (Schur-positive) with respect to D if is a symmetric (respectively, Schur-positive) function. Determining whether a given symmetric (quasisymmetric) function is Schur-positive is a major problem in contemporary algebraic combinatorics [27, §3].
The following theorem is due to Gessel.
Theorem 51. [26,Theorem 7.19.7] For every partition λ n, Thus SYT(λ) is symmetric and Schur-positive with respect to the standard descent set.
We say that a statistic f : • Statistics on S n which are invariant under Knuth relations; e.g., the length of the longest increasing subsequence, the inverse descent number, and the inverse major index. This follows from Theorem 51 above together with Proposition 16.
• The inversion number on S n (reduced to the inverse major index by Foata's bijection). For a far reaching generalization see [25,Theorem 6.3].
Theorem 12, to be proved in the following subsection, implies that on the set M n of all matchings on n points, the pair (cr, um) of the crossing number and the number of unmatched points is Schur-positive with respect to the geometric descent set GDes.

Proof of Theorem 12
Recall from Section 1 the following notations: um(m) is the number of unmatched points in a matching m ∈ M n ; and, for a partition λ, ht(λ) is the number of parts in λ and oc(λ) is the number of odd parts in the conjugate partition.
The following proposition follows from Theorem 11. x Des(T ) .
The second equality follows from Theorem 11. The last equality is obtained from the interpretation of matchings as involutions, followed by the bijection to SYT via the Robinson-Schensted correspondence, using Corollary 18 and Lemma 45. The last equality follows from Theorem 51.

Cyclic descent extensions
The above setting is applied in this section to construct a cyclic descent extension for conjugacy classes of involutions and their refinements, that is, involutions with fixed cycle structure and nesting number. Let M n be the set of matchings on n points on the circle, labeled clockwise by 1, . . . , n. where addition is modulo n.
Proof of Proposition 14. Follows from Proposition 55.
Recall the map Q : S n → SYT n sending each π ∈ S n to the corresponding RS recording tableau Q π , and define h : I n,k → SYT n,k by h := Q •ι. A cyclic descent extension on the set SYT n,k,j := {T ∈ SYT n,k , 2j ht(T ) 2j + 1}.
is described in the following statement.
Proposition 56. Assume that n k 0 with n − k even, and 0 j (n − k)/2.
(a) If 0 < k < n, or k = 0 and j = n/2, then the pair is a (non-Escherian) cyclic extension of Des on SYT n,k,j .
(b) If k = n (and necessarily j = 0), or k = 0 and j = n/2, then the above pair is an Escherian cyclic extension of Des on SYT n,k,j .
Proof. Follows from Proposition 55, noting that, by Proposition 16 and Lemma 45, the restriction of Q to I n,k,j is a descent set preserving bijection onto SYT n,k,j .
Remark 57. Cyclic rotation of geometric configurations has been used before for the construction of cyclic descent extensions on standard Young tableaux of certain given shapes -rectangular shapes of height at most 3 [17] and flag shapes [16]. These results motivated our work, and some of them are indeed obtained as special cases: • Letting k = 0 and j = 1 in Proposition 56 yields a cyclic descent extension on standard Young tableaux of shape (n, n), since SYT 2n,0,1 = SYT(n, n). One can verify that this cyclic extension coincides with the one determined by Dennis White, as described in [17,Theorem 1.4].
• Recalling that the number of Motzkin paths of length n is equal to the number of standard Young tableaux of size n and at most three rows [19,9,4], consider Proposition 56 on the union of ∪ k SYT n,k,1 , namely j = 1 and k arbitrary. This determines a cyclic descent extension on Motzkin paths via Han's bijection [12], which coincides with Han's cyclic descent extension on Motzkin paths.

Equidistribution revisited
In an early version of this paper, the following conjecture was posed.
Conjecture 58. Let µ m and ν n be integer partitions with no common part. Let π and σ be permutations of cycle types µ and ν, respectively, with disjoint supports. Let A π,σ be the subset of the conjugacy class of cycle type µ ν m + n consisting of the permutations for which the relative order of the letters in the union of all cycles of µ is as in π, and the relative order of the letters in the union of all cycles of ν is as in σ. Then w∈Aπ,σ x Des(w) = τ ∈π¡σ x Des(τ ) .
The proof is partly algebraic and not bijective. Proposition 60 implies the following.
preserving descent set, nesting number and crossing number.
Proof. Take, in Proposition 60, σ ∈ I n−k,0 and π = [n − k + 1, . . . , n], the identity permutation in S which preserves the descent set and satisfies the following property: for every σ ∈ I n−k,0 and every τ ∈ σ ¡ [n − k + 1, . . . , n], the relative order of the letters in the union of all 2-cycles in φ(τ ) is equal to the relative order of the letters in σ. By Definition 41, the nesting and crossing numbers of τ are the same as those of σ. Thus φ preserves nesting and crossing numbers as well.
Remark 62. The explicit bijection q : I n−k,0 ¡ [n − k + 1, n − k + 2, . . . n] → I n,k from Lemma 50 preserves the descent set and the nesting number, but does not preserve the crossing number. The bijection φ : I n−k,0 ¡ [n − k + 1, n − k + 2, . . . n] → I n,k , whose existence is claimed in Corollary 61, preserves the crossing number as well.
The following refinement of Theorem 11 follows.
This completes the proof.
Problem 64. Find an explicit bijective proof of Theorem 63.