Abstract
We study three dimensional array of numbers $B(n,k,j)$, $0\le j,k\le n$, where $B(n,k,j)$ is the number of type $B$ permutations of order $n$ with $k$ descents and $j$ minus signs. We prove in particular, that $b(n,k,j):=B(n,k,j)/\binom{n}{j}$ is an integer and provide two combinatorial interpretations for these numbers.
Highlights
Let B(n, k, j) denote the number of type B permutations (0, σ1, . . . , σn) which have k descents and j minus signs
That b(n, k, j) := B(n, k, j)/
We study properties of the three-dimensional array B(n, k, j), 0 j, k n
Summary
Let B(n, k, j) denote the number of type B permutations (0, σ1, . . . , σn) which have k descents and j minus signs. Let B(n, k, j) denote the number of type B permutations Some of these properties appear in the work of Brenti [4]. In particular he computed the three-variable generating function and proved real rootedness of some linear combinations of the polynomials Pn,j(x) :=. Τn) such that τ1 = j and τ has k descents He proved many interesting properties of these numbers, like direct formula, asymptotic. The array b(n, k, 1), 1 k n, appears in OEIS [8] as A120434 It counts permutations σ ∈ An which have k − 1 big descents, i.e. such descents σi > σi+1 that σi − σi+1 2. Pn,j (x) admit the same property, which is a generalization of Corollary 3.7 in [4] and of Corollary 6.9 in [2]
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