s-Inversion Sequences and P-Partitions of Type B
William Y.C. Chen, Alan J.X. Guo, Peter L. Guo, Harry H.Y. Huang, Thomas Y.H. Liu
Abstract: Given a sequence s = (s_{1}, s_{2},...) 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},..., e_{n}) of nonnegative integers is called an s-inversion sequence of length n if 0 ≤ e_{i} < s_{i} 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, s_{2i-1} = 2i - 1 and s_{2i} = 4i for i ≥ 1, and let P_{n} be the set of signed permutations on the multiset {1^{2}, 2^{2},..., 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,...), that is, s_{2i-1 }= 4i-2 and s_{2i} = 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. AMS Classification: 05A05, 05A15. Keywords: inversion sequence, signed permutation, type B P-partition, equidistribution. Download: pdf |