$s$-Inversion Sequences and $P$-Partitions of Type $B$

Abstract

Given a sequence $s=(s_1,s_2,\ldots )$ of positive integers, the notion of inversion sequences with respect to $s$, or $s$-inversion sequences, was introduced by Savage and Schuster in their study of lecture hall polytopes. A sequence $(e_1,e_2,\ldots,e_n)$ of nonnegative integers is called an $s$-inversion sequence of length $n$ if $0\leq e_i < s_i$ for $1\leq i\leq n$. Let $I_n$ be the set of $s$-inversion sequences of length $n$ for $s=(1,4,3,8,5,12,\ldots)$---that is, $s_{2i-1}=2i-1$ and $s_{2i}=4i$ for $i\geq1$---and let $P_n$ be the set of signed permutations on the multiset $\{1^2,2^2,\ldots,n^2\}$. Savage and Visontai conjectured that the descent number over $P_n$ is equidistributed with the ascent number over $I_{2n}$. In this paper, we give a proof of this conjecture by using $P$-partitions of type $B$. (Lin independently obtained a proof based on recurrence relations.) Moreover, we find a set of signed permutations over which the descent number is equidistributed with the ascent number over $I_{2n-1}$. Let $I'_n$ be the set of $s$-inversion sequences of length $n$ for $s=(2,2,6,4,10,6,\ldots)$; that is, $s_{2i-1}=4i-2$ and $s_{2i}=2i$ for $i\geq1$. We also find two sets of signed permutations over which the descent number is equidistributed with the ascent number over $I'_n$, depending on the parity of $n$.