\(1/k\)-Eulerian polynomials and \(k\)-inversion sequences

Ting-Wei Chao¹ ; Jun Ma ; Shi-Mei Ma² ; Yeong-Nan Yeh³

¹Institute of Mathematics, Academia Sinica
²School of Mathematics and Statistics, Northeastern University at Qinhuangdao
³Institute of Mathematics, Academia Sinica, Taipei

The electronic journal of combinatorics, Tome 26 (2019) no. 3

Résumé

Let ${\bf s} = (s_1, s_2, \ldots, s_n,\ldots)$ be a sequence of positive integers. An ${\bf s}$-inversion sequence of length $n$ is a sequence ${\bf e} = (e_1, e_2, \ldots, e_n)$ of nonnegative integers such that $0 \leq e_i < s_i$ for $1\leq i\leq n$. When $s_i=(i-1)k+1$ for any $i\geq 1$, we call the ${\bf s}$-inversion sequences the $k$-inversion sequences. In this paper, we provide a bijective proof that the ascent number over $k$-inversion sequences of length $n$ is equidistributed with a weighted variant of the ascent number of permutations of order $n$, which leads to an affirmative answer of a question of Savage (2016). A key ingredient of the proof is a bijection between $k$-inversion sequences of length $n$ and $2\times n$ arrays with particular restrictions. Moreover, we present a bijective proof of the fact that the ascent plateau number over $k$-Stirling permutations of order $n$ is equidistributed with the ascent number over $k$-inversion sequences of length $n$.

Détail

DOI : 10.37236/8466

Classification : 05A05, 05A15
Mots-clés : Eulerian polynomials, \(s\)-inversion sequences, ascents, permutations