$1/k$-Eulerian Polynomials and $k$-Inversion Sequences

Yeong-Nan Yeh,Jun Ma,Ting-Wei Chao,Shi-Mei Ma

doi:10.37236/8466

Abstract

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 $n$.

Highlights

Let Sn be the symmetric group on the set [n] = {1, 2, . . . , n}
Let Asc (π) and Lrm (π) denote the set of ascents and left-to-right minima of π, respectively
In [11], Savage and Schuster introduced the concept of s-inversion sequences in study of lecture hall polytopes

Summary

Introduction

Let Asc (π) and Lrm (π) denote the set of ascents and left-to-right minima of π, respectively. Exc (π), asc (π), lrmin(π), cyc (π)) denote the number of descents In [11], Savage and Schuster introduced the concept of s-inversion sequences in study of lecture hall polytopes. Let I(ns) denote the set of s-inversion sequences of length n. Savage [10] gave a survey for the study of lecture hall partitions A bijective proof of (1) may arouse interests in the study of the connections between s-lecture hall polytope and other structures. The method is to present a series of three bijections: the first bijection maps k-inversion sequences to 2 × n arrays with particular restrictions.

If ei

We have

For any e

Given an array