d-Separated permutations and q-Stirling numbers of the first kind

Abstract

Let d be a nonnegative integer, a d-separated permutation is a permutation in which every two left-to-right minima are at distance greater than d. More precisely, for $\pi ={\pi }_1{\pi }_2\cdots {\pi }_n\in S_n$, suppose that ${\pi }_{i_1},{\pi }_{i_2},\dots ,{\pi }_{i_k}$ are the left-to-right minima of π with $k\ge 1$ and $1=i_1<i_2<\cdots <i_k\le n$, then π is d-separated if $i_j-i_{j-1}>d$ for each j, $1<j\le k$. In this paper we study different enumerative properties on d-separated permutations. We first give a recurrence formula of the numbers $c^d(n,k)$ of d-separated permutations in $S_n$ with exactly k left-to-right minima. Then we study the inversion and co-inversion polynomials of d-separated permutations, and give q-analogue $c_q^d(n,k)$ and $(p,q)$-analogue $c_{p,q}^d(n,k)$ of $c^d(n,k)$ for any d. Note that when $d=0$, 0-separated permutations are just all permutations in $S_n$, $c^0(n,k)$ is Stirling number of the first kind, and the two polynomials $c_q^d(n,k)$ and $c_{p,1}^d(n,k)$ are both generalizations of q-Stirling numbers of the first kind. The q-Stirling numbers of the first kind when $d=0$ have been well studied by Médicis-Leroux and Cai-Readdy, who gave nice combinatorial interpretations via rook placements on a staircase chessboard. We give a bijection between permutations and rook placements in staircase chessboards.