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 = (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.

  AMS Classification:  05A05, 05A15.


  Keywords:
  inversion sequence, signed permutation, type B P-partition, equidistribution.

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