s-Inversion Sequences and P-Partitions of Type B

Given a sequence s = (s1, s2, . . .) 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 (e1, e2, . . . , en) of nonnegative integers is called an s-inversion sequence of length n if 0 ≤ ei < si for 1 ≤ i ≤ n. Let In be the set of s-inversion sequences of length n for s = (1, 4, 3, 8, 5, 12, . . .), that is, s2i−1 = 2i − 1 and s2i = 4i for i ≥ 1, and let Pn be the set of signed permutations on the multiset {12, 22, . . . , n2}. Savage and Visontai conjectured that the descent number over Pn is equidistributed with the ascent number over I2n. 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 I2n−1. Let I ′ n be the set of s-inversion sequences of length n for s = (2, 2, 6, 4, 10, 6, . . .), that is, s2i−1 = 4i−2 and s2i = 2i for i ≥ 1. We also find two sets of signed permutations over which the descent number is equidistributed with the ascent number over I ′ n, depending on whether n is even or odd.