2013/10/20 by William Y. C. Chen, Alan J. X. Guo, Chen, William Y. C. +7 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Combinatorial Mathematics #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems #math.CO
paper · pdf · doi:10.48550/arxiv.1310.5313
14 pages, 1 figure
arxiv created 2013/10/20 · openalex publication_date 2013/10/20 · arxiv updated 2013/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a sequence s=(s1,s2,…) 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 (e1,e2,…,en) of nonnegative integers is called an s-inversion sequence of length n if 0≤ ei < si for 1≤ i≤ n. Let I(n) be the set of s-inversion sequences of length n for s=(1,4,3,8,5,12,…), that is, s2i=4i and s2i-1=2i-1 for i≥1, and let Pn be the set of signed permutations on \12,22,…,n2\. Savage and Visontai conjectured that when n=2k, the ascent number over In is equidistributed with the descent number over Pk. 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 In. 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,…), that is, s2i=2i and s2i-1=4i-2 for i≥1. We find a set of signed permutations over which the descent number is equidistributed with the ascent number over I'n.