A q -analogue of a result of Carlitz, Scoville and Vaughan via the homology of posets

Let f ( z ) = P ∞ n =0 ( − 1) n z n /n ! n !. In their 1975 paper, Carlitz, Scoville and Vaughan provided a combinatorial interpretation of the coefficients in the power series 1 /f ( z ) = P ∞ n =0 ω n z n /n ! n !. They proved that ω n counts the number of pairs of permutations of the n th symmetric group S n with no common ascent. This paper gives a combinatorial interpretation of a natural q -analogue of ω n by studying the top homology of the Segre product of the subspace lattice B n ( q ) with itself. We also derive an equation that is analogous to a well-known symmetric function identity: P n i =0 ( − 1) i e i h n − i = 0, which then generalizes our q -analogue to a symmetric group representation result.


Introduction
Consider the power series f (z) = Two permutations have no common ascent if they do not rise at the same position when written in one-line notation. For example, in one-line notation (12,21), (21, 12), (21,21) are all the pairs of permutations of {1, 2} with no common ascent, so we have ω 2 = 3. Since the Bessel function J 0 (z) is essentially f (z 2 ), Carlitz, Scoville and Vaughan's result provided a combinatorial interpretation of the coefficient ω k in the reciprocal Bessel function.

Y. Li
and that n k q := [n]q!
In Section 2, we will prove Theorem 1.2 by studying the top homology of the Segre product of the subspace lattice B n (q) with itself. From a poset homology perspective, the coefficient W n (q) is a signless Euler characteristic and counts the number of decreasing maximal chains of this Segre product poset. All definitions will be reviewed in this section.
In Section 3, we define the product Frobenius characteristic map to serve as a useful tool in studying representations of the product group S n × S n . We then further generalize our q-analogue to a symmetric group representation result in Section 4 (see Theorem 4.1) using the Whitney homology technique. This generalization is an analogue of the well-known symmetric function identity: Finally, in Section 5 we point out that an alternative proof of Theorem 1.1 can be obtained by specializing our proof of Theorem 1.2 at q = 1.

The q-analogue of a result of Carlitz, Scoville, and Vaughan
We recall the definition of B n (q), which is a q-analogue of the subset lattice B n . Let q be a prime power and F q the finite field of q elements. Consider the n-dimensional linear vector space F n q and its subspaces. Then B n (q) is the lattice of those subspaces ordered by inclusion. The poset B n (q) is a geometric lattice, so every element is a join of atoms ([10, Example 3.10.2]). The poset B n (q) is graded with a rank function ρ(W ) := the dimension of the subspace W , where a poset is said to be graded if it is pure and bounded.
An edge labeling of a bounded poset P is a map λ : E(P ) → Λ, where E(P ) is the set of covering relations x < · y of P and Λ is some poset. If P is a poset with an edge labeling λ, then a maximal chain c If λ(c 1 ) lexicographically precedes λ(c 2 ), we say that c 1 lexicographically precedes c 2 and we denote this by c 1 < L c 2 .

q-analogue of C-S-V result
A bounded poset that admits an EL-labeling is said to be EL-shellable. We only need to consider pure shellability in this paper since both B n (q) and the Segre product of B n (q) with itself (see Definition 2.4) are pure and bounded. It is well known that B n (q) is EL-shellable (see [14,Exercise 3.4.7]) and a general edge-labeling for semimodular lattices is given in [10]. Here we define a specific EL-labeling of B n (q), which will be used to prove our results. Let A be the set of all atoms of B n (q). For a subspace of F n q , X ∈ B n (q), we define A(X) := {V ∈ A | V ⩽ X}. The following two steps define an edge-labeling on the graded poset B n (q).
2. In the case of B 3 (3), if X = span{⟨1, 0, 1⟩, ⟨2, 1, 0⟩}, then A(X) also contains span{⟨0, 1, 1⟩} and span{⟨2, 2, 1⟩}. But any vector whose right-most non-zero coordinate is the first coordinate will not be in X. So f (A(X)) = {2, 3} and | f (A(X)) |= 2. For a k-dimensional subspace X of F n q , Gaussian elimination implies the existence of a basis of X whose elements have distinct right-most non-zero coordinates and this in turn implies that f (A(X)) has dim(X) elements. Let Y be an element of B n (q) that covers X, then dim(Y ) = dim(X) + 1. The set f (A(Y ))\f (A(X)) is a subset of [n] and has exactly one element. This element will be the label of the edge (X, Y ).
Proposition 2.2. The edge labeling described above is an EL-labeling on the subspace lattice B n (q).
Proof. Edges in the same chain cannot take duplicate labels since F n q is ndimensional and any maximal chain must take all labels in {1, 2, . . . , n}. Let [X, Y ] be a closed interval in B n (q). All maximal chains of [X, Y ] will take labels from the set f (A(Y ))\f (A(X)). Let a 1 < a 2 < · · · < a l be all the elements of f (A(Y ))\f (A(X)) arranged in increasing order. Given a i ∈ f (A(Y ))\f (A(X)), there exists an atom V i ∈ A(Y ) with f (V i ) = a i . Note that V i is a 1-dimensional subspace of F n q and the join of V i and X is in [X, Y ]. We build a chain according to the increasing order of a i 's, each time adjoining one 1-dimensional subspace. Then the chain c = ( For the uniqueness of the increasing maximal chain, it suffices to show the uniqueness of the selection of X ∨ V 1 since X and Y are arbitrary.
We can find a basis vector v 1 of V 1 and a basis vector v ′ 1 of V ′ 1 so that the a 1 th coordinate of both vectors are 1.
Under this EL-labeling, to each maximal chain of the subspace lattice B n (q), one can assign a permutation σ of S n . See Section 1 for the definition of the inversion statistic inv(σ). Proof. For each 1-dimensional subspace of F n q , we can pick a basis vector that has 1 on its right-most non-zero coordinate. Given σ ∈ S n , for each i ∈ [n − 1], let inv σ(i) denote the number of j such that 1 ⩽ i < j ⩽ n and σ(j) < σ(i). The number of ways to choose an atom W 1 such that the edge (0, and V k is an atom. Pick a basis vector for V k , call it v k , that has 1 on the σ(k)th coordinate and all 0's after the σ(k)th coordinate. For all j such that 1 ⩽ k < j ⩽ n and σ(j) < σ(k), W k−1 contains no vector whose right-most non-zero coordinate is the σ(j)th. Thus, any variation of the values on those σ(j)th coordinates of v k will result in a different W k . Then there are q inv σ(k) ways to choose a W k . Therefore, the number Let us review a simplified definition of the Segre product poset. A general definition can be found in [4].
Definition 2.4. Let P be a graded poset with a rank function ρ. Then the Segre product poset of P with itself, denoted by P • P , is defined to be the induced subposet of the product poset P ×P consisting of the pairs (x, y) ∈ P ×P such that ρ(x) = ρ(y). Now consider the Segre product of B n (q) with itself. Using the EL-labeling of B n (q) described right after Definition 2.1, the Segre product poset B n (q) • B n (q) admits the following edge labeling. Given two elements X = (X 1 , In the EL-labeling of B n (q), suppose the edge connecting X 1 and Y 1 admits a label i and the edge connecting X 2 and Y 2 admits a label j, then the edge connecting X and Y in B n (q) • B n (q) is labeled by (i, j).
are closed intervals in B n (q). Since the labeling for B n (q) is an EL-labeling, there is a unique increasing maximal chain c 1 in [X 1 , Y 1 ] that lexicographically precedes all other chains in the same interval. There is also a unique increasing maximal chain c 2 in [X 2 , Y 2 ]. Then the chain in [X, Y ] formed by pairing elements of c 1 and c 2 of the same rank must be the unique increasing maximal chain in [X, Y ]. Any other chain would have non-increasing labels in [X 1 , . This unique increasing maximal chain of [X, Y ] must also satisfy part (2) of Definition 2.1 because c 1 and c 2 both satisfy this condition. □ The following theorem of Björner and Wachs connects the permutations in S n with the maximal chains of the Segre product poset B n (q) • B n (q). LetP be the bounded extension of P . That is,P = P ∪ {0,1} and0 and1 are attached even if P already has a bottom or a top element.
Since P has the homotopy type of a wedge of spheres, H i (P ; Z) ∼ = H i (P ; Z) (Wachs [14, Theorem 1.5.1]). We will use H i (P ; Z), the reduced homology of the order complex ∆(P ), instead of the cohomology group in this paper.
Proposition 2.8. Let W n (q) = (σ,τ )∈Dn q (inv(σ)+inv(τ )) , where D n denotes the set of pairs of permutations (σ, τ ) ∈ S n × S n with no common ascent. Then W n (q) equals the total number of decreasing maximal chains of P n (q) := B n (q) • B n (q) ∖ {0,1} with respect to the labeling described above.
Proof. An edge label (i, j) ∈ [n] × [n] ⩽ (k, l) if and only if i ⩽ k and j ⩽ l. By the definition of our labeling for B n (q) • B n (q), there cannot be repeat edge labels along any one chain. So a chain label is decreasing as long as the two components of any two consecutive edge labels do not increase at the same time. Each maximal chain labeling of B n (q) • B n (q) corresponds to a pair of permutations of S n . Then labels of decreasing maximal chains are all pairs of permutations with no common ascent. Given a pair of permutations (σ, τ ), the number of maximal chains assigned label (σ, τ ) is q inv(σ) · q inv(τ ) = q (inv(σ)+inv(τ )) by Lemma 2.3. Then the total number of decreasing maximal chains of P n (q) is □ Remark 2.9. The Segre product poset B n (q) • B n (q) is the q-analogue of the Segre product poset B n • B n , agreeing with the formal definition of a q-analogue in R. Simion's paper [8]. She showed that the q-analogue of an EL-shellable poset is also EL-shellable. The EL-labeling of B n (q) • B n (q) we use in this paper provides intuition and a combinatorial interpretation for W n (q).
Björner and Welker proved that if two pure posets are Cohen-Macaulay, then their Segre product is also Cohen-Macaulay (see [4,Theorem 1]). This result in particular proves that the poset B n (q)•B n (q) is Cohen-Macaulay because B n (q) is. Corollary 2.5 Algebraic Combinatorics, Vol. 6 #2 (2023) Y. Li says that B n (q) • B n (q) is shellable, which is a stronger property than the Cohen-Macaulayness. Later in Section 4 we will use the fact that the Segre product poset B n • B n is Cohen-Macaulay.
On the other hand, by the definition of the Möbius function, Each x in P n (q) is the product of two k-dimensional subspaces X 1 , X 2 of F n q , for some k with 0 ⩽ k < n. The intervals [0, X 1 ] and [0,

The product Frobenius characteristic map
The Frobenius characteristic map is often used to study representations of the symmetric group. Here we will define a product Frobenius characteristic map to help understand representations of S n × S n . Therefore, let us consider two sets of variables x = (x 1 , x 2 , ...) and y = (y 1 , y 2 , ...). Following Sagan's notations [7], R n denotes the space of class functions on S n and R = ⊕ n R n . We will use R m,n to denote the space of class functions on S m × S n and let R 2d = ⊕ m,n R m,n . Let Λ n be the space of homogeneous degree n symmetric functions. Then Λ(x) = ⊕ n Λ n (x) and Λ(y) = ⊕ n Λ n (y) denote the rings of symmetric functions in variables ( where χ µ is the value of χ on the class µ and p µ is the power sum symmetric function. Define ch := ⊕ n ch n . Now we define a product characteristic map. Definition 3.2. Let χ be a class function on S m × S n . The product Frobenius characteristic map ch : R 2d → Λ(x) ⊗ Λ(y) is defined as: where χ (µ,λ) is the value of χ on the class (µ, λ) and p µ , p λ are power sum symmetric functions. The class (µ, λ) is indexed by a partition µ ⊢ m and a partition λ ⊢ n that tell us the cycle types of elements of S m and S n respectively. Proof. Equation (4) gives us For a conjugacy class (µ, λ) ⊢ (m, n), let σ ∈ S m have cycle type µ and τ ∈ S n have cycle type λ. The character value where the second equality is by [7,Theorem 1.11.2]. Then = ch(f )(x)ch(g)(y).

□
Because the product Frobenius characteristic map is an extension of the usual (Frobenius) characteristic map, we keep the notation ch for product Frobenius characteristic map even though ch was previously defined to be ⊕ n ch n in various literature (Sagan [7], Stanley [9]). The meaning of ch will be clear in the given context.
Recall that the induction product f • g is the induction of f ⊗ g from S m × S n to S m+n . A fundamental property of the usual characteristic map is the following:

Y. Li
We would like the product Frobenius characteristic map to be a homomorphism as well. Given an S k × S l -module V with character ψ and an S m × S n -module W with character ϕ, ψ ⊗ ϕ is the character of V ⊗ W , which is a representation of (S k × S l ) × (S m × S n ). We want to produce a character of S k+m × S l+n .
Definition 3.6. For ψ and ϕ as given above, we define the induction product ψ • ϕ to be ψ ⊗ ϕ ↑ S k+m ×S l+n (S k ×S l )×(Sm×Sn) . The induction product on characters extends to all class functions on R 2d by (bi)linearity.
Proposition 3.7. Let ψ be a class function on S k × S l , and ϕ a class function on S m × S n . The product Frobenius characteristic map ch : R 2d → Λ(x) ⊗ Λ(y) is a bijective ring homomorphism, i.e., ch is one-to-one and onto, and satisfies ch(ψ • ϕ) = ch(ψ)ch(ϕ).
For the second and fourth equalities, see [7,Theorem 1.11.2]. □ Proof of Proposition 3.7. The bijectiveness of the product Frobenius characteristic map follows from the definition of ch and the fact that the power sums p µ (x)p λ (y) form a Q-basis for Λ(x) ⊗ Λ(y). Next we will show that the map is a homomorphism.
l 's are irreducible characters of representations of S k and S l respectively. Similarly, n . For any σ k ∈ S k , σ l ∈ S l , τ m ∈ S m , and τ n ∈ S n , we have Algebraic Combinatorics, Vol. 6 #2 (2023) 464 q-analogue of C-S-V result by Lemma 3.8. Now take the product Frobenius characteristic of both sides of the above equation. For clarity, we keep track of variables x and y. By Proposition 3.3 and then Proposition 3.4 we get

A symmetric function analogue
Using the product Frobenius characteristic map, we derive an equation that is analogous to a well-known symmetric function identity (see Stanley In the proof of our analogue, we use the Whitney homology technique, which was introduced by Sundaram [12] for pure posets and then generalized by Wachs [13] for semipure posets.
Let Q be a poset with a bottom element0 and G an automorphism group of Q. Suppose Q is a Cohen-Macaulay G-poset, for each integer r, the r-th Whitney homology of Q is defined as where Q r := {x ∈ Q | rank(x) = r}. Algebraic Combinatorics, Vol. 6 #2 (2023)

Y. Li
For the subset lattice B n , let P n be the proper part of the Segre product poset B n • B n . The action of S n × S n on P n induces a representation on the reduced top homology of P n .

q-analogue of C-S-V result
Thus, equation (6) becomes It is known that the Frobenius characteristic of the trivial representation of S n is h n (Stanley [9]). Multiplying both sides of equation (7) by (−1) n−1 , we get Finally, we conclude that  (y 1 , y 2 , ...). The product Frobenius characteristic of the S n × S n -modules H n−2 (P n ) is a symmetric function in two sets of variables. Then it is natural to ask what we can say about its specialization.
Recall that P n is the proper part of the Segre product of the subset lattice B n with itself. The product Frobenius characteristic of the S n × S n -module H n−2 (P n ) has an innate connection with the Euler characteristic of B n (q) • B n (q). From Corollary 2.10, W n (q) is the signless Euler characteristic of B n (q)•B n (q). The following theorem gives us a connection between the stable principal specialization of ch( H n−2 (P n )) and the Euler characteristic W n (q).
Theorem 4.2. Let W n (q) be the signless Euler characteristic of B n (q) • B n (q). For a symmetric function f in two sets of variables x = (x 1 , x 2 , . . . ) and y = (y 1 , y 2 , . . . ), the stable principal specialization ps(f ) specializes both x i and y i to q i−1 . Then where ch(V ) is the product Frobenius characteristic of V .

Y. Li
Assume that the statement is true for P i , i = 1, ..., n − 1. Now let us consider the reduced top homology of P n . Equation (5) gives us a way to express ch( H n−2 (P n )) in terms of the product Frobenius characteristic of smaller posets. We get (8) ch( H n−2 (P n )) = n−1 i=0 (−1) n−1+i h n−i (x)h n−i (y)ch( H i−2 (P i )).
Then we take the stable principal specialization of both sides of equation (8). We know from Stanley [9] that ps(h n ) = n i=1 1 1−q i . It follows from our induction hypothesis that ps(ch( H n−2 (P n ))) = n−1 i=0 (−1) n−1+i ps(ch( H i−2 (P i ))) Finally, using the identity involving the signless Euler characteristic W n (q) given in Theorem 1.2, we obtain ps(ch( H n−2 (P n ))) = W n (q) n j=1 (1 − q j ) 2 . □ Theorem 4.1 was motivated by our initial findings regarding the q-analogue of equation (1). Once we formulated the specialization of ch( H i−2 (P i )), the q-analogue can be retrieved by taking the stable principal specialization of equation (5).

Alternative proof of the result of Carlitz-Scoville-Vaughan
Carlitz, Scoville and Vaughan's result, Theorem 1.1, provides a combinatorial explanation for the coefficients ω k in the reciprocal Bessel function. They showed that ω k is the number of pairs of k-permutations with no common ascent. When letting q = 1 in our q-analogue (2), the subspaces of F n q become subsets of {1, 2, ..., n}. The value W n (1) = (σ,τ )∈Dn 1 inv(σ)+inv(τ ) simply counts the number of pairs of permutations of [n] with no common ascent, i.e. ω n . The proof of Theorem 1.2 is then easily adapted into an alternative proof of Carlitz, Scoville and Vaughan's result (1). Carlitz, Scoville and Vaughan's proof in [5] includes general cases where occurrences of common ascent are allowed. Our proof does not account for those general cases, but it gives a less technical approach by utilizing Björner and Wachs' work on shellability and poset homology [3].