s-Inversion Sequences and P-Partitions of Type B

Given a sequence $s=(s_1,s_2,\ldots)$ of positive integers, the inversion sequences with respect to $s$, or $s$-inversion sequences, were 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, $s_{2i}=4i$ and $s_{2i-1}=2i-1$ for $i\geq1$, and let $P_n$ be the set of signed permutations on $\{1^2,2^2,\ldots,n^2\}$. Savage and Visontai conjectured that when $n=2k$, the ascent number over $I_n$ is equidistributed with the descent number over $P_k$. For a positive integer $n$, we use type $B$ $P$-partitions to give a characterization of signed permutations over which the descent number is equidistributed with the ascent number over $I_n$. When $n$ is even, this confirms the conjecture of Savage and Visontai. Moreover, let $I'_n$ be the set of $s$-inversion sequences of length $n$ for $s=(2,2,6,4,10,6,\ldots)$, that is, $s_{2i}=2i$ and $s_{2i-1}=4i-2$ for $i\geq1$. We find a set of signed permutations over which the descent number is equidistributed with the ascent number over $I'_n$.

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