Schur-Positivity of Short Chords in Matchings

We prove that the set of matchings with a fixed number of unmatched vertices is Schur-positive with respect to the set of short chords. Two proofs are presented. The first proof applies a new combinatorial criterion for Schur-positivity, while the second is bijective. The coefficients in the Schur expansion are derived, and interpreted in terms of Bessel polynomials. We present a Knuth-like equivalence relation on matchings, and show that every equivalence class corresponds to an irreducible representation. We proceed to find various refined Schur-positive sets, including the set of matchings with a prescribed crossing number and the set of matchings with a given number of pairs of intersecting chords. Finally, we characterize all the matchings $m$ such that the set of matchings avoiding $m$ is Schur-positive.


Introduction
Given N ∈ N and a set A, together with a set-valued function D : A → 2 [N −1] which is sometimes called a statistics function, we define its quasisymmetric generating function  Note that cyclic short chords are not counted. That is, for every m ∈ M N we have N / ∈ Short(m). This holds even where (1, N ) ∈ m.
Enumerative properties of short chords in perfect matchings were studied extensively. The distribution of the number of short chords in perfect matchings appears in Entry A079267 of the OEIS [26], and was analyzed for example by Killpatrick and Cameron [17]. McSorley and Feinsilver [21] discovered that short chords of matchings are closely related to the Bessel polynomials, as will be discussed in Section 4.3.
The study of short chords of matchings is also motivated by the involutive length and its corresponding poset studied by Adin, Postnikov and Roichman [1]. Related posets derived from the Bruhat order were studied by Richardson and Springer [22], Hultman [15], Deodhar and Srinivasan [8] and others, motivated by the topology of linear algebraic groups (see [15] for details). Further discussion on the algebraic motivations can be found in Section 6.3.
We prove the following: Theorem 1.6. Let n ∈ N and f ∈ N 0 , and denote N = 2n + f . Then the set M N,f is Schur-positive with respect to Short. Furthermore, its Schur expansion is given by the following formula: |{m ∈ M N −2k,f | Short(m) = ∅}|s N −k,k .
It turns out that the Schur coefficients of Q Short (M N,f ) may be explicitly interpreted in terms of Bessel polynomials, see Corollary 4.13 below.
We give two proofs of Theorem 1.6. The first proof relies on a new criterion for Schur-positivity of sparse statistics: Definition 1.7. We say that a set J ⊆ [N − 1] is sparse if {j, j + 1} S for every 1 ≤ j ≤ N − 2.
For a set A and a function D : A → 2 [N −1] , we say that D is sparse if D(a) is sparse for all a ∈ A. Theorem 1.8. Let A be a finite set with a statistic D : A → 2 [N −1] , and denote n = ⌊ N 2 ⌋. Then the following statements are equivalent: • D is sparse, and for every sparse J ⊆ [N − 1], the cardinality of the set {a ∈ A | D(a) ⊇ J} depends on the size of J only. • A is symmetric with respect to D, with a Schur expansion of the form Q(A) = n k=0 c k s N −k,k for some c k ∈ Z.
Furthermore, if these statements hold then A is Schur-positive, and its Schur expansion is This new criterion implies Theorem 1.6, as shown at the beginning of Section 4, and can be applied to many other sets as well. This criterion will be applied elsewhere to prove Schur-positivity of various graph colorings.
The second proof of Theorem 1.6 provides a bijection from the set of matchings to a multiset of SYTs, and applies the bijective criterion of Adin and Roichman to Schur-positivity [2,Prop. 9.1], presented in Theorem 2.5.
Following the bijective proof, we define the following notions, analogous to the elementary Knuth transformations and the Knuth equivalence of permutations: Definition 1.9 (Definition 5.1 below). We say that two matchings m 1 , m 2 ∈ M N are Knuth-like equivalent if one can be obtained from the other by a sequence of elementary Knuth-like transformations: • Replace the chords (i, i + 1), (i + 2) with the chords (i), (i + 1, i + 2) or vice versa (i.e., interchange a short chord with an adjacent unmatched vertex). • Replace the chords (i, i + 1), (i + 2, j) (regardless of whether i + 2 < j or i + 2 > j) with the chords (i, j), (i + 1, i + 2) or vice versa (i.e., interchange a short chord with an adjacent endpoint of another chord).
Definition 1.10 (Definition 4.3 below). The core of a given matching m ∈ M N , denoted core(m), is obtained by repeatedly removing short chords from the matching until no short chords remain. The remaining vertices are then re-indexed with natural numbers starting from 1 while preserving their relative order.
Analogously to the Knuth equivalence, we prove the following result: We show that Knuth-like classes of matchings, similarly to Knuth classes of permutations, have the Schur functions as their generating functions: Theorem 1.12 (Corollary 5.10 below). Every set M ⊆ M N that is closed under the Knuth-like equivalence is Schur-positive with respect to Short. Moreover, if M is a Knuth-like equivalence class, then its generating function is Q(A) = s N −k,k , where N − 2k is the number of vertices of core(m) for some arbitrary matching m ∈ M.
Furthermore, utilizing the Knuth-like classes, we explore in Section 5 various refined Schurpositive sets, including: • The set of k-crossing matchings (i.e., with k the maximal number of a set of pairwise intersecting chords). • The set of matchings with exactly k pairs of intersecting chords. Lastly, we define M N,f (m) as the set of matchings in M N,f that avoid the pattern m. In Proposition 5.15, we provide a characterization of the matchings m for which M N,f (m) is Schurpositive for all N, f . The remainder of this paper is organized as follows: Section 2 provides the necessary background information. In Section 3, we prove Theorem 1.8. In Section 4, we focus on matchings. We first derive Theorem 1.6 from Theorem 1.8. Then, we present a bijective proof, followed by a description of the Schur coefficients of matchings in terms of the Bessel polynomials. In Section 5, we utilize the bijection and refine the Schur-positivity property of Theorem 1.6. Finally, in Section 6, we conclude with further research.
We say that two compositions α, β N are equivalent (denoted α ∼ β), if β is a rearrangement of the entries of α.
Denote the ring of symmetric functions by Sym and the ring of quasisymmetric functions by QSym (see, for example, [14, Section 1] for details). The space of homogeneous symmetric functions of degree N is denoted by Sym N , and the space of homogeneous quasisymmetric functions of degree N is denoted by QSym N .
The space QSym N has several important bases. In this work, we focus on the fundamental basis, consisting of the functions The space Sym N has several standard bases as well. In this work, we focus on the Schur basis, which consists of the Schur functions s λ , where λ is a partition of N . The definition and properties of Schur functions can be found in [23,Section 4.4]. Here, we adopt the combinatorial approach for the Schur functions, as described in Theorem 2.4 and Theorem 2.5 below.

Symmetric and Schur-positive sets
As mentioned in Section 1, a set A is symmetric with respect to a statistic function D : A → 2 [N −1] if its generating function Q N,D (A) is a symmetric function. Moreover, it is Schur-positive if all Schur coefficients are nonnegative integers.
The entry in row i and column j of a tableau T ∈ SYT(λ) is denoted as T i,j . In addition, we define row i (T ) := {T i,j | 1 ≤ j ≤ λ i } as the set of entries in the i-th row of T . For example, if we consider the SYT shown in Figure 2, then T 3,2 = 8 and row 3 (T ) = {5, 8}.
In 2015, Adin and Roichman proved the following criterion. This criterion implies that proving the Schur-positivity of a set is achievable by establishing a statistic-preserving bijection between the set and SYTs of shapes corresponding to a specific multiset.
In this paper, we will also apply a recently formulated criterion for symmetry [19].
Definition 2.6. Let A be a finite set with a statistic D : ] . The set of elements that respect a given composition α N , denoted A D (α), consists of the elements a ∈ A such that D(a) ⊆ S α , where S α is the set corresponding to the composition α. When D is clear from the context, we may write A(α) instead.
Note that only sets of permutations are considered in [19]. However, Lemma 2.7 applies to other sets as well.
Another useful result about symmetric sets and symmetric functions is due to Bloom and Sagan:   We prove Lemma 3.1 in Section 3.1, and then show two proofs for Lemma 3.2. The first proof (presented in Section 3.2) is inductive. The second proof (presented in Section 3.3) is more involved, and it demonstrates the power of column superstandard tableaux, introduced by Hamaker, Pawlowski and Sagan [14], in proving Schur-positivity properties. We believe this approach can be applied in other cases as well.
Before proving these lemmas, let us prove a simple folklore lemma that will be useful for both lemmas: where λ ′ is the partition conjugate to λ.

Proof of Lemma 3.1
The set A is symmetric, and its Schur expansion is Q N,D (A) = n k=0 c k s N −k,k . Therefore, by Theorem 2.5, we have For some values c λ . First of all, notice that the set Des(T ) is sparse for all T ∈ SYT(N − k, k). Therefore, by Equation (1), the set D(a) is sparse for all a ∈ A. It remains to show that for every sparse set J ⊆ [N − 1] of size k, we have |A(D ⊇ J)| = |A(D ⊇ odds(k))|. Denote α = (2 k , 1 N −2k ) N . Additionally, let i 1 < · · · < i N −k−1 ∈ [N − 1] \ J denote the elements not in J. The composition associated to the set [N − 1] \ J is denoted by Lemma 3.3, A is symmetric with respect to the complementary statisticD. Notably, the compositions α and β are equivalent, as they both have k occurrences of 1 and N − 2k occurrences of 2. By Lemma 2.7, we obtain that |AD(β)| = |AD(α)|. By Definition 2.6, we obtain that AD(β) = A(D ⊇ J) and AD(α) = A(D ⊇ odds(k)). Consequently, |A(D ⊇ J)| = |A(D ⊇ odds(k))|, as required.

First proof of Lemma 3.2
We start by introducing an observation, which will serve as a crucial step in our inductive argument.
where n = ⌊ N 2 ⌋. We prove it by induction on N ∈ N 0 : For N ≤ 1, the statement holds trivially. Next, we assume that the statement holds for every value smaller than N and proceed to prove it for N . Our goal is to establish the equation: for every set J ⊆ [N − 1], where c k = |A(D = odds(k))|. We begin by considering the case J = ∅. In this case, the only SYT T with Des(T ) = ∅ is the unique tableau of one row. Consequently, Equation (3) simplifies to |A(D = ∅)| = |A(D = odds(0))|, which obviously holds.
It now remains to prove Equation (3) for all J = ∅. By the inclusion-exclusion principle, it suffices to prove that the following holds for all J = ∅: The function D is assumed to be sparse, and the set Des(T ) is sparse for all T ∈ SYT(N − k, k). Therefore, if J is not sparse then Equation (4) holds.
Next, let J = ∅ be a sparse set, and denote |J| = i. The set odds(i) is sparse too, and has i elements. Therefore, by the assumptions of the lemma, we obtain that |A(D ⊇ J)| = |A(D ⊇ odds(i))|. In addition, by Theorem 2.4, the set SYT(N − k, k) is Schur-positive and has the Schur expansion Q(SYT(N − k, k)) = s N −k,k . Therefore, we can apply Lemma 3.1 and deduce that |SYT(N − k, k)(Des ⊇ J)| = |SYT(N − k, k)(Des ⊇ odds(i))| . Therefore, Equation 4 reduces to where i > 0. Thus, the proof will be completed when we establish Equation (5).
Notice that A 1 satisfies the requirements of Lemma 3.2 with respect to D 1 , where N is replaced by N − 2. Therefore, we may assume, by the induction hypothesis, that Lemma 3.2 holds for A 1 . Thus, we can apply Equation (4) for A 1 . If we substitute J = odds(i − 1) into this equation we obtain with Des(T ) ⊇ odds(i). Therefore, we can reformulate the equation and obtain that (5) is the omission of the summand corresponding to k = 0. However, this summand evaluates to zero, as c 0 |SYT(N, 0)(Des ⊇ odds(i))| = 0, due to the fact that every SYT of shape (N ) has no descents. Thus, it does not affect the overall expression.

Second proof of Lemma 3.2
As a first step of this proof, we prove that a set satisfying the conditions of Lemma 3.2 is symmetric.
Proof. Following Lemma 3.3, it suffices to prove that A is symmetric with respect to the complementary statisticD. As we proceed to prove it by Lemma 2.7, let us find |AD(α)| for all If a composition α has max i (α i ) > 2, then there exists j ∈ [N − 1] such that j, j + 1 / ∈ S α , where S α is the set corresponding to α. By Definition 2.6, for every element a ∈ AD(α) we havē D(a) ⊆ S α , so j, j + 1 / ∈D(a). Consequently, j, j + 1 ∈ D(a), and the set D(a) is not sparse. However, D is assumed to be a sparse function, so AD(α) = ∅. Now assume that max i (α i ) ≤ 2, and denote by k = |{i | α i = 2}| the number of occurrences of 2 in α. Thus, we have ℓ = N − k. Consider the set J = [N − 1] \ S α . Notably, the set J is sparse and consists of k elements. Let a ∈ A. We have a ∈ AD(α) if and only ifD(a) ⊆ S α , or equivalently, To conclude, if α contains an element larger than 2 then AD(α) = ∅. Otherwise, the size of AD(α) depends only on the number of occurrences of 2 in α. Therefore, |AD(α)| = |AD(β)| for all α ∼ β N . By Lemma 2.7, we obtain that A is symmetric.
As the next step of the proof, we prove that A is Schur-positive and find its Schur coefficients. For this, we define a total order on partitions λ ⊢ N : i}| denote the length of the i-th column in the Young diagram of λ. We say that µ is larger than λ in the conjugate order, and denote µ > λ, if there exists i such that µ ′ j = λ ′ j for all j < i and µ ′ i > λ ′ i . Following Hamaker, Pawlowski and Sagan [14,Section 5], we define the column superstandard Young tableau of shape λ ⊢ N , obtained by filling the columns of the Young diagram of shape λ one by one. We denote it by T λ ∈ SYT(λ). Formally,  The power of these notions may be reflected by the following statement: Proof. The first assertion is obvious, so let us focus on the second assertion.
Let T ∈ SYT(µ), and assume that Des(T ) = Des(T λ ) and T = T λ . We aim to show that µ > λ. Let x ∈ [N ] be the minimal letter placed differently in T and T λ . We consider two cases based on the relative positions of x in T and T λ .
Case 1: x appears at position (i 1 , j 1 ) in T and (i 2 , j 2 ) in T λ , where i 1 < i 2 and j 1 > j 2 . Since (T λ ) i 2 ,j 2 = x and following the structure of T λ , it follows that all entries in the j 1 -th column of T λ are larger than x. Since any letter smaller than x is placed in the same position in both T and T λ , we obtain that all entries in the j 1 -th column of T are at least x, implying that i 1 = 1. Therefore, we obtain that x − 1 / ∈ Des(T ), since x − 1 does not appear in a higher position than x in T . However, since i 1 < i 2 , we obtain that i 2 > 1. Following the structure of T λ , for every letter y ∈ [N − 1] we have y ∈ Des(T λ ) if and only if y + 1 / ∈ row 1 (T λ ), implying that x − 1 ∈ Des(T λ ). Consequently, we obtain that Des(T ) = Des(T λ ), in contradiction to the assumptions.
Case 2: x appears at position (i 1 , j 1 ) in T and (i 2 , j 2 ) in T λ , where i 1 > i 2 and j 1 < j 2 . In this case, we observe that every letter y appearing in a column j < j 2 of T λ must satisfy y < x, as per the structure of T λ . Due to the minimality of x, we conclude that y appears in the j-th column of T as well. Since T ∈ SYT(µ) and T λ ∈ SYT(λ), we have λ ′ j ≤ µ ′ j for all j < j 1 and λ ′ j 1 + 1 ≤ µ ′ j 1 . By Definition 3.6, we then obtain λ < µ, as required.
Lemma 3.7 associates the Young diagram of shape λ with the set Des(T λ ). As we will see later, this association is powerful in analysing the Schur coefficients of symmetric sets.
Proof. The first assertion is obvious, so let us focus on the second assertion. Let T ∈ SYT(µ), and assume by contradiction that Des(T ) = Des(T λ ) and T = T λ . Denote by x ∈ [N ] the minimal letter placed differently in T and T λ , and assume that T i 1 ,j 1 = x and (T λ ) i 2 ,j 2 = x. Similarly to the proof of Lemma 3.7, the assumption that i 1 < i 2 and j 1 > j 2 leads to a contradiction. Thus, we may assume that i 1 > i 2 and j 1 < j 2 , implying that i 1 = 2 and i 2 = 1.
Notably, x − 1 appears in the first row of both T and T λ . Thus, we obtain x − 1 ∈ Des(T ) and x − 1 / ∈ Des(T λ ), contradicting the assumption Des(T ) = Des(T λ ). Therefore, we may conclude that if Des(T ) = Des(T λ ) then T = T λ . Now we are ready to prove that if a set is symmetric with respect to a sparse statistic then it is Schur-positive: Lemma 3.9. Let A be a symmetric set with respect to a sparse statistic D : Then A is Schur-positive, and its Schur expansion is Proof. The set A is assumed to be symmetric, so by Theorem 2.5, we have where c λ are the Schur coefficients of A. It suffices to show that if there exists k such that λ = (N − k, k) then c λ = |A(D = odds(k))|, and otherwise c λ = 0. If A = ∅, then Q N,D (A) = 0, and the statement holds. Therefore, we may assume that A = ∅, and consequently, there exists c λ = 0 for some partition λ ⊢ N . Let µ ⊢ N be a partition with c µ = 0, maximal in the conjugate order (i.e., such that c λ = 0 for every λ ⊢ N with λ > µ).
Second proof of Lemma 3.2. The lemma follows directly from Lemma 3.5 together with Lemma 3.9.

Short chords of matchings
In this section, we analyze the set of matchings M N,f with respect to short chords (Recall Definition 1.3 and Definition 1.5). First, we apply Theorem 1.8 to establish Theorem 1.6, which asserts that M N,f is Schur-positive. Next, we provide a bijective proof of Theorem 1.6, which will be utilized in Section 5 to refine the Schur-positivity result. Finally, we demonstrate that the Schur expansion of Q Short (M N,f ) may be explicitly interpreted in terms of the Bessel polynomials. Clearly, the function Short : be a sparse set, and let us enumerate the elements of M N,f (Short ⊇ J). In every matching in M N,f (Short ⊇ J), the vertices j i and j i + 1 are matched for all 1 ≤ i ≤ k, and the remaining N − 2k vertices can be matched in any way, subject to the condition that exactly f vertices remain unmatched. Therefore, we have and thus it depends only on the size of J.
By applying Theorem 1.8, we conclude that M N,f is Schur-positive with respect to Short, with the following Schur expansion: However, the extra summands equal 0 and do not affect the expression.) Clearly, and we obtain the required Schur expansion.

Bijective proof of Theorem 1.6
Before presenting the bijective proof of Theorem 1.6, we give some definitions and notations for matchings:

Now let us present a bijection
that sends matchings m ∈ M N,f to pairs (m 0 , T ), where m 0 is a short-chord-free matching on [N − 2k] with f unmatched vertices and T ∈ SYT(N − k, k) for some 0 ≤ k ≤ n, such that Des T = Short m for all m. By Theorem 2.5, the existence of such a bijection implies Theorem 1.6.

Constructing the bijection
First, we define the core of a matching: We define the bijection by F (m) = (core(m), T (m)). Next, we turn to provide examples of the bijection. Then, in the remaining of the subsection, we will prove that the bijection is well-defined and explore some of its properties. Section 4.2.2 will be devoted to proving that F is bijective by constructing its inverse function.
Moving on to proving that the bijection F is well-defined, first let us prove that the core of a matching is well-defined: It suffices to prove that e ℓ ∈ {e ′ 1 , . . . , e ′ k ′ } for all 1 ≤ ℓ ≤ k (i.e., every chord that is removed during the first reduction process is removed during the second process as well). Assume by contradiction that e ℓ / ∈ {e ′ 1 , . . . , e ′ k ′ } for some ℓ, and denote by ℓ 0 the minimal such ℓ. That is, e 1 , . . . , e ℓ 0 −1 ∈ {e ′ 1 , . . . , e ′ k ′ }. Removing the chords e 1 , . . . , e k from m constitutes a valid reduction process, so the chord e ℓ 0 becomes short before it is removed. That is, i ∈ {i 1 , j 1 , . . . , i ℓ 0 −1 , j ℓ 0 −1 } for all i ℓ 0 < i < j ℓ 0 . Therefore, the chord e ℓ 0 is a short chord of the core obtained by removing e ′ 1 , . . . , e ′ k ′ of m, contradicting the requirement that the reduction process continues until there are no short chords remaining. Therefore, we may conclude that if a chord is removed during a reduction process then it is removed during any reduction process, and core(m) and Stable(m) are well-defined. is also well-defined. In order to prove that F is well-defined, it remains to show that T (m) ∈ SYT(N − k, k), where k is the number of unstable chords of m.
Obviously, every letter i ∈ [N ] appears exactly once in T (m), and the rows are increasing. Notice that for every entry j ∈ row 2 (T (m)) there exists i < j such that (i, j) ∈ m is an unstable chord. Therefore, every entry j ∈ row 2 (T (m)) is associated to an entry i ∈ row 1 (T (m)) such that i < j, so the columns of T (m) are increasing and T (m) is a standard Young tableau. Finally, | row 2 (T (m))| = k, implying that T (m) ∈ SYT(N − k, k).
After establishing that F is a valid function, we turn our attention to exploring some of its properties.
Lemma 4.6. The reduction process has the following properties: (1) If a chord intersects another chord then it is stable.

Proof.
(1) Assume that (i 1 , i 3 ), (i 2 , i 4 ) ∈ m for i 1 < i 2 < i 3 < i 4 . As long as the chord (i 2 , i 4 ) is not removed, the chord (i 1 , i 3 ) does not become short and cannot be removed. On the other hand, as long as the chord (i 1 , i 3 ) is not removed, the chord (i 2 , i 4 ) cannot be removed. Therefore, both chords cannot be removed during the reduction process. (2) If the chord (i, j) is unstable, then after removing some unstable vertices it becomes short, implying that every vertex between i and j is unstable. On the other hand, if all the vertices between i and j are unstable, then they will eventually be removed, making the chord (i, j) short, so the chord (i, j) is unstable too. with i ≤ i 1 , i 2 ≤ j) is short-chord-free, non-crossing and perfect. The only such a matching is the empty matching ∅ ∈ M 0 , so every i ≤ i 1 ≤ j is unstable.
As we proceed to apply Theorem 2.5 and establish the Schur-positivity of M N,f with respect to short chords, let us prove that F sends short chords of matchings to descents of SYTs, in the following sense: Proof. Let i ∈ Short m. That is, (i, i + 1) ∈ m. The chord (i, i + 1) is unstable, so we may deduce from the definition of T (m) that i ∈ row 1 (T ) and i + 1 ∈ row 2 (T ). Therefore, i ∈ Des T .
On the other hand, let i ∈ Des T . It implies that i ∈ row 1 (T ) and i + 1 ∈ row 2 (T ). Since i + 1 ∈ row 2 (T ), we obtain that there exists j < i + 1 such that (j, i + 1) ∈ m is an unstable chord. Since the chord (j, i + 1) is unstable and j ≤ i ≤ i + 1, we obtain by Lemma 4.6 (part 2) that i is an endpoint of an unstable chord of m. Since i ∈ row 1 (T ), we obtain that i opens an unstable chord (i, j ′ ) of m for some j ′ > i. If (i, i + 1) / ∈ m then j < i and j ′ > i + 1, and we obtain that the unstable chord (j, i + 1) intersects the chord (i, j ′ ), contradicting Lemma 4.6 (part 1). Therefore, we may conclude that (i, i + 1) ∈ m and i ∈ Short m.

Proof of bijection
In this section we will prove that the transformation F defined in Section 4.2.1 is indeed a bijection, by constructing its inverse function.
In order to construct the inverse function, we will establish a correspondence between standard Young tableaux of two rows and ballot paths. We define ballot paths as follows: Definition 4.8. Let N ≥ 0. A ballot path of length N is a sequence of N steps, where each step is either (1, 1) or (1, −1). The first step is (1, 1), and the path starts at the origin (0, 0). Each subsequent step either moves one unit up and one unit right, or one unit down and one unit right. The path is said to be valid if it never crosses below the x-axis, i.e., the y-coordinate of the path is always non-negative.
The set of ballot paths from (0, 0) to (N, t) is denoted P N,t . Given a ballot path p ∈ P N,t , denote by p i the y-coordinate of p after i steps. The set UP(p) ⊆ [N ] (DOWN(p) ⊆ [N ]) consists of the indices i such that p i > p i−1 (respectively, p i < p i−1 ). Finally, define the height of the i-th step of a path p to be the maximum height of its two endpoints, and denote it by height p (i) := max(p i−1 , p i ).  Figure 6. A ballot path p ∈ P 10,4 The bijection between SYTs of two rows and ballot paths is direct: Associate T ∈ SYT(N − k, k) with the path p(T ) ∈ P N,N −2k such that UP(p(T )) = row 1 (T ) and DOWN(p(T )) = row 2 (T ).
Next, for a given ballot path p ∈ P N,t , we construct a set Stable(p) and a perfect matching m unstable (p) on [N ]\Stable(p) as follows: For every vertex j ∈ DOWN(p), we match it in m unstable (p) to the maximal i < j such that height p (i) = height p (j). Since p is a valid ballot path, it can be deduced that for every j ∈ DOWN(p), there exists a unique i < j satisfying height p (i) = height p (j) such that i is maximal among all such elements in [N ]. Additionally, it can be inferred from the discrete continuity of the path that this i is necessarily a part of UP(p). The set Stable(p) consists of all i ∈ UP(p) that are not involved in any chord in m unstable (p). Notice that i ∈ Stable(p) if and only if i ∈ UP(p) and height p (j) > height p (i) for all j > i, so | Stable(p)| = t.
We are now ready to describe the inverse bijection of F , denoted F : Given a short-chord-free matching m 0 ∈ M N −2k,f (Short = ∅) and a tableau T ∈ SYT(N − k, k) for some k, denote p = p(T ) ∈ P N,N −2k . We will construct a matching m stable on Stable(p) and a matching m unstable on [N ] \ Stable(p), and then apply Observation 4.2 to obtainF (m 0 , T ) ∈ M N,f . We construct these sub-matchings as follows: • m stable : Since p ∈ P N,N −2k , we infer that | Stable(p)| = N − 2k. Therefore, we may rename the vertices of m 0 ∈ M N −2k,f to Stable(p) as follows: There exists a unique bijection ϕ : [N − 2k] → Stable(p) such that i < j if and only if ϕ(i) < ϕ(j) for all i, j. The matching m stable on Stable(p) consists of the chords (ϕ(i), ϕ(j)) for all (i, j) ∈ m 0 . • m unstable = m unstable (p) is the matching described earlier.
It remains to show thatF is indeed the inverse function of F . We will do so in two steps.
Step 1:F • F = Id.  m) is a perfect matching. Therefore, in order to prove that Equation (9) holds, it suffices to prove that any unstable chord of m belongs to m unstable (p) too. Let (i, j) ∈ m be an unstable chord with i < j. Thus, we may deduce that i ∈ UP(p) and j ∈ DOWN(p). Let i ′ ∈ UP(p) such that i < i ′ < j. By Lemma 4.6 (part 2), i ′ is an endpoint of an unstable chord (i ′ , j ′ ) ∈ m with i ′ < j ′ . By Lemma 4.6 (part 1), we obtain that the chord (i, j) does not intersect (i ′ , j ′ ), implying that i < i ′ < j ′ < j. On the other hand, every j ′ ∈ DOWN(p) is an endpoint of an unstable chord (i ′ , j ′ ) ∈ m with i < i ′ < j ′ < j. Therefore, we obtain a bijection from , the ascending steps of the path appear before the associated descending steps), so height p (j) < height p (i ′ ) for all i ′ ∈ UP(p) ∩ [i + 1, j − 1]. We may conclude that (i, j) ∈ m unstable (p) for every unstable chord (i, j) of m and prove that Equation (9) holds. Next, we denote m ′ =F (m 0 , T ) and prove that m = m ′ . From Equation (9) we deduce that the supports of these matchings are identical, and therefore Stable(m) = Stable(p). Thus, the set [N ] \ Stable(m) is both m-invariant and m ′ -invariant, and restricting each of these matchings to [N ] \ Stable(m) results in m unstable (p). In addition, it can be easily verified from the descriptions of F andF that res Stable(m) (m) = res Stable(m) (m ′ ) is the matching obtained by relabeling the vertices of core(m) with the elements of Stable(m) in increasing order. By Observation 4.2, we may deduce that m = m ′ .
Step 2: F •F = Id. Proof. Denote p = p(T ) ∈ P N,N −2k and Stable(p) = {i 1 , . . . , i N −2k } where i 1 < · · · i N −2k . We first prove that Stable(m) = Stable(p). Notice that the matching m unstable (p) is non-crossing. This can be viewed visually from Figure 6, where m unstable (p) is denoted by horizontal dotted lines that cross the path only in their endpoints. Indeed, let j 1 < j 2 < j 3 < j 4 , and assume, by contradiction, that both (j 1 , j 3 ) and (j 2 , j 4 ) belong to m unstable (p). By the definition of m unstable (p), the assumption (j 1 , j 3 ) ∈ m unstable (p) implies that height p (j 1 ) = height p (j 3 ), and height p (j) = height p (j 1 ) for all j 1 < j < j 3 . Since height p (j 1 + 1) ≥ height p (j 1 ) and due to the discrete continuity of the path, we obtain that height p (j) > height p (j 1 ) for all j 1 < j < j 3 . Consequently, we obtain height p (j 2 ) > height p (j 1 ) = height p (j 3 ). Similarly, the assumption (j 2 , j 4 ) ∈ m unstable (p) implies that height p (j 3 ) > height p (j 2 ), in contradiction. Therefore, we may conclude that the matching m unstable (p) is non-crossing.
Next, we may infer that for every 1 ≤ ℓ < N − 2k, the segment [i ℓ + 1, i ℓ+1 − 1] is m-invariant and the restricted matching res [i ℓ +1,i ℓ+1 −1] (m) is perfect and non-crossing. By Lemma 4.6 (part 3), we obtain that j / ∈ Stable(m) for all i ℓ < j < i ℓ+1 , and therefore Stable(m) ⊆ Stable(p). Thus, a valid reduction process of m may begin with removing every vertex not in Stable(p). We may deduce from the description ofF that res Stable(p) (m) is the matching obtained by relabeling the vertices of m 0 with the elements of Stable(p) in increasing order. This matching is short-chord-free, so Stable(m) = Stable(p) and core(m) = m 0 .
It remains to prove that p ′ = p, where p ′ := p(T (m)). Since Stable(m) = Stable(p), we may infer from the description ofF that DOWN(p) = {j | the chord (i, j) ∈ m is unstable and i < j}.
Finally, we conclude the bijective proof: Bijective proof of Theorem 1.6. By Theorem 2.5, it suffices to prove that

Analysis of the coefficients and relations with Bessel polynomials
The Bessel polynomials θ n (x), sometimes called the reverse Bessel polynomials, are given by the generating function They also have the explicit formula For more information about the Bessel polynomials, the reader is referred to [13].
Therefore, we can reformulate Theorem 1.6 and obtain: Corollary 4.13. Let n ∈ N and f ∈ N 0 , and denote N = 2n + f . Then the Schur expansion of the set M N,f with respect to Short is given by the formula where h(P N +f −2k , f ) is the coefficient of (x + 1) f in the Taylor expansion of the Bessel polynomial θ n+f −k (x) around x = −1

Refinements of the bijection
In this section, we will utilize the bijection F discussed in Section 4.2 to refine Theorem 1.6 and find many Schur-positive sets of matchings with respect to the set of short chords. Indeed, given a non-negative integer N , for every set of short-chord-free matchings M 0 ⊆ n k=0 M N −2k (Short = ∅), the set {m ∈ M N | core(m) ∈ M 0 } is Schur-positive (as we will see later in Corollary 5.10).

Sets closed under the Knuth-like equivalence
As a first refinement of the Schur-positivity of M N,f , we present an equivalence relation on matchings, motivated by the Knuth equivalence on permutations [11]: Definition 5.1. We say that two matchings m 1 , m 2 ∈ M N are Knuth-like equivalent if one can be obtained from the other by a sequence of elementary Knuth-like transformations: • Replace the chords (i, i + 1), (i + 2) with the chords (i), (i + 1, i + 2) or vice versa (i.e., interchange a short chord with an adjacent unmatched vertex). • Replace the chords (i, i + 1), (i + 2, j) (regardless of whether i + 2 < j or i + 2 > j) with the chords (i, j), (i + 1, i + 2) or vice versa (i.e., interchange a short chord with an adjacent endpoint of another chord).
The power of this notion is reflected by the following theorem: We will prove Theorem 5.2 in two steps: Step 1: If two matchings are equivalent then they have the same core. Proof. Since m 1 and m 2 are equivalent, we deduce that m 2 can be obtained from m 1 by a sequence of elementary Knuth-like transformations. Notably, given a matching m ∈ M N and a matching ϕ(m) obtained from m by applying an elementary Knuth-like transformation, the matchings m and ϕ(m) differ only in the relative position of a certain short chord, and therefore core(ϕ(m)) = core(m). A direct induction shows that core(m 1 ) = core(m 2 ).
Step 2: If two matchings have the same core then they are equivalent.
Lemma 5.4. Let m 1 , m 2 ∈ M N and assume that core(m 1 ) = core(m 2 ). Then m 1 and m 2 are Knuth-like equivalent.
Before proving Lemma 5.4, we introduce the notion of inserting a short chord into a matching: Definition 5.5. Let m ∈ M N and let 1 ≤ i ≤ N + 1. Denote by insert i (m) ∈ M N +2 the matching obtained by inserting a short chord that matches the vertices i and i + 1, while pushing every vertex j ≥ i to position j + 2. Formally, denote by f i : [N ] → [N + 2] \ {i, i + 1} the function that is described as follows: Then the matching insert i (m) consists of the chords (f i (j 1 ), f i (j 2 )) for all (j 1 , j 2 ) ∈ m together with (i, i + 1), and consists of the unmatched vertices (f i (j)) for all (j) ∈ m. Proof. We may assume without loss of generality that i ≤ j, and prove the statement by induction on j − i. The statement is obvious for i = j.
Assume that i < j. By Definition 5.5, the matching insert i+1 (m) is Knuth-like equivalent to insert i (m). By the induction hypothesis, insert i+1 (m) is Knuth-like equivalent to insert j (m) as well. Therefore, insert i (m) is Knuth-like equivalent to insert j (m). Proof. By Lemma 5.6, we may assume that i = N + 1. Assume that m 2 = ϕ 1 · · · ϕ ℓ (m 1 ) for some elementary Knuth-like transformations ϕ 1 , . . . , ϕ ℓ . We prove the statement by induction on ℓ. The statement is obvious for ℓ = 0.
We can combine Lemma 5.6 with Lemma 5.7 to obtain the following: Lemma 5.8. Let m ∈ M n and let i 1 , . . . , i k and j 1 , . . . , j k be two sequences. Then the matchings insert i 1 · · · insert i k (m) and insert j 1 · · · insert j k (m) are Knuth-like equivalent.
Proof. We prove the statement by induction on k. If k = 0 then the statement is obvious.
Assume that k > 0, and denote m i = insert i 2 · · · insert i k (m) and m j = insert j 2 · · · insert j k (m). We aim to prove that the matchings insert i 1 (m i ) and insert j 1 (m j ) are Knuth-like equivalent. By the induction hypothesis, m i and m j are equivalent. Therefore, by Lemma 5.7, the matchings insert i 1 (m i ) and insert i 1 (m j ) are equivalent too. In addition, by Lemma 5.6, the matchings insert i 1 (m j ) and insert j 1 (m j ) are equivalent. Therefore, the matchings insert i 1 (m i ) and insert j 1 (m j ) are equivalent. Lemma 5.9. Let m ∈ M N , and assume that m has k unstable chords. Then there exist indices i 1 , . . . , i k such that m = insert i 1 · · · insert i k (core(m)).
Proof. We prove the statement by induction on k. If k = 0 then core(m) = m and the statement is obvious.
Assume that a matching m ∈ M N has k > 0 unstable chords. Therefore, m has at least one short chord, denoted (i 1 , i 1 + 1) ∈ m. Thus, there exists a matching m 1 ∈ M N −2 such that m = insert i 1 (m 1 ). Notice that core(m) = core(m 1 ), so m 1 has k − 1 unstable chords. Thus, by the induction hypothesis, there exist indices i 2 , . . . , i k such that m 1 = insert i 2 · · · insert i k (core(m)). Therefore, m = insert i 1 · · · insert i k (core(m)). Now we are ready to prove Lemma 5.4.
Proof of Lemma 5.4. Let m 1 , m 2 ∈ M N and assume that m 0 := core(m 1 ) = core(m 2 ). Thus, m 1 and m 2 have the same number of unstable chords, denoted k. By Lemma 5.9, we can write m 1 = insert i 1 · · · insert i k (m 0 ) and m 2 = insert j 1 · · · insert j k (m 0 ) for some i 1 , . . . , i k and j 1 , . . . , j k . The statement now follows directly from Lemma 5.8. is a statistic-preserving bijection. Thus, applying Theorem 2.5 completes the proof.

Other constructions of Schur-positive sets
We may apply Corollary 5.10 to obtain other Schur-positive sets of matchings. For example, we can filter M N by the isomorphism class of the intersection graph: Definition 5.11. Let m be a matching. Its intersection graph, denoted G(m), is defined to be the undirected simple graph with the chords of m as its vertices, and with an edge between two vertices if the associated chords of m intersect.
It can be easily seen that applying an elementary Knuth-like transformation on a matching preserves its intersection graph up to graph-isomorphism. Therefore, by Corollary 5.10:

Pattern avoidance in matchings
Sagan and Woo [24], motivated by Elizalde and Roichman [9], posed the problem of determining which sets Π of permutations satisfy the property that for all n, the set of permutations in S n that avoid every pattern in Π is Schur-positive. This problem has been extensively studied since then [14,6,19].
An analogous question may be asked about Schur-positivity of pattern-avoiding matchings as well. Extensive research has been conducted on pattern avoidance in perfect matchings by Simion and Schmidt [25], Jelínek and Mansour [16], Bloom and Elizalde [5], and others, leading to multiple definitions in the literature. Fang, Hamaker, and Troyka [10] explore some of these definitions and provide a comparison. Additionally, various conventions exist for generalizing pattern avoidance to non-perfect matchings [18,10,20]. We adopt the definition from McGovern [20], which directly generalizes the definition for perfect matchings from [16] and [7].
Definition 5.13. Let m 1 ∈ M N 1 and m 2 ∈ M N 2 for some positive integers N 1 ≤ N 2 . We say that m 2 contains the pattern m 1 , if there exist indices 1 ≤ i 1 < · · · < i N 1 ≤ N 2 such that the following holds: • For all 1 ≤ j < j ′ ≤ N 1 , • For all 1 ≤ j ≤ N 1 , (j) ∈ m 1 ⇐⇒ (i j ) ∈ m 2 . Otherwise, we say that m 2 avoids m 1 . For two sets of matchings M 1 ⊆ M N 1 and M 2 ⊆ M N 2 , denote by M 2 (M 1 ) the set of matchings in M 2 that avoid every matching in M 1 . In addition, denote M 2 (m) := M 2 ({m}) for a matching m.
The following problem is analogous to the problem of Sagan and Woo [24] regarding patternavoiding permutations: Problem 5.14. Determine which sets M ⊆ M N of matchings satisfy the property that for all N ′ , f ′ , the pattern avoiding set M N ′ ,f ′ (M) is Schur-positive with respect to Short.
While a complete solution of Problem 5.14 seems challenging, we are able to solve the problem in the case where |M| = 1. In the proof of Proposition 5.15, we will apply the following result: Proof. Let m ′ 1 ∈ M N ′ ,f ′ , and let m ′ 2 ∈ M N ′ ,f ′ be the result of applying an elementary Knuthlike transformation on m ′ 1 . Assume that m ′ 1 contains the pattern m. It suffices to prove that m 2 contains m too. By Definition 5.13, there exist indices 1 ≤ i 1 < · · · < i N ≤ N ′ such that (j, j ′ ) ∈ m ⇐⇒ (i j , i j ′ ) ∈ m ′ 1 for all 1 ≤ j < j ′ ≤ N and (j) ∈ m ⇐⇒ (i j ) ∈ m ′ 1 for all 1 ≤ j ≤ N . Without loss of generality, we may assume that (i, i + 1) ∈ m ′ 1 and m ′ 2 is obtained from m ′ 1 by interchanging the chord (i, i + 1) with the vertex i + 2. The pattern m is short-chord-free, so Similarly to Proposition 6.1, Theorem 1.8 provides a simple criterion under which Q N,D (A) is non-negatively spanned by the Schur functions of two-row shape {s N −k,k | 0 ≤ k ≤ n}. In the spirit of discovering new methods and expanding our understanding, we pose the problem of finding other such criteria, hopeful that they will contribute to further advancements in the study of Schur-positivity and Schur coefficients. Schur-positive sets of particular interest are pattern avoiding sets. As discussed earlier in Section 5.3, Schur-positivity of pattern avoiding sets was extensively studied in the context of permutations. Recall the problem stated in Section 5.3: Problem (Problem 5.14 above). Determine which sets M ⊆ M N of matchings satisfy the property that for all N ′ , f ′ , the pattern avoiding set M N ′ ,f ′ (M) is Schur-positive with respect to Short. Proposition 5.15 solves this problem for pattern sets of size 1. We do not know a general answer for larger pattern sets.
They also defined the involutive weak order ≤ I on I 2n , as the reflexive and transitive closure of the relation w ≺ I s i ws i ifl(s i ws i ) =l(w) + 1, where s i = (i, i + 1) is a simple reflection. The involutive order is motivated by the Bruhat order defined for Coxeter groups [4,Chapter 2].
Since fixed-point-free involutions in S 2n naturally correspond to perfect matchings on 2n vertices, we will adopt the involutive length and the involutive weak order for perfect matchings in PM 2n := M 2n,0 and use the same notation for them.
Consider the natural action of S 2n on the set PM 2n , and denote m 0 = {(1, 2), (3,4), . . . , (2n − 1, 2n)}. Notably, the stabilizer of m 0 is isomorphic to the hyperoctahedral group H n = S 2 ≀ S n . Furthermoer, Consider the Schreier graph associated with this action with respect to the simple reflections s i = (i, i + 1) ∈ S 2n . For an illustration of this Schreier graph, see Figure 7 which depicts it when 2n = 4. Avni [3] proved that this graph is bipartite when ignoring loops. Therefore, we can create a layered graph by partitioning the vertices (matchings) into layers based on their distance from m 0 . Specifically, a matching in the ℓ-th layer is at a distance of ℓ from m 0 . Furthermore, a matching in one layer is connected to matchings in the adjacent layers, but not within its own layer or to matchings in non-adjacent layers.
It turns out that this graph corresponds to the invoultive length and the involutive weak order. First, the layer of a matching m is equal to its involutive lengthl(m). Moreover, for every m 1 , m 2 ∈ PM 2n we have m 1 ≤ I m 2 if and only if there exists a geodesic path from m 0 to m 2 passing through m 1 .
Based on the structure of the Schreier graph, we define three natural set-valued functions on perfect matchings, denoted as Asc(m), Loop(m), and Des(m), where m is a matching in PM 2n . Given a matching m ∈ PM 2n and an index 1 ≤ i ≤ 2n − 1, we define: • i ∈ Asc(m) if s i · m > I m, where the dot denotes the group action, The algebraic interest of the graph motivates the analysis of these three statistics, specifically focusing on the question of the Schur-positivity of PM 2n with respect to each of them. As a first step in this direction, it is noteworthy that the definition of Asc coincides with the standard ascents of permutations when restricted to I 2n . Consequently, the Schur-positivity of PM 2n with respect to Asc can be derived directly from classical properties of the Robinson-Schensted correspondence, as noted for example in Gessel-Reutenauer [12, end of Section 7]. Moreover, we have the observation Loop(m) = Short(m) for a perfect matching m. This observation, when combined with Theorem 1.6, implies that PM 2n is Schur-positive with respect to Loop.
In contrast, the Schur-positivity of PM 2n with respect to Des remains a mystery.
Question 6.4. Is PM 2n Schur-positive with respect to Des?
Despite extensive analysis of the Des statistic, we are currently unable to provide an answer. However, based on experimental results from simulations, we have discovered that PM 2n is Schurpositive with respect to Des when 2n ≤ 14. Moreover, intriguingly, the statistics Asc and Des are found to be equidistributed for all 2n ≤ 14. These findings lead us to propose the following conjecture: If this conjecture can be proven, it would establish the Schur-positivity of PM 2n with respect to Des. Furthermore, a bijection on PM 2n that maps ascents to descents may unveil hidden symmetries within the Schreier graph.