Combinatorics of q-Mahonian numbers of type B and log-concavity
Abstract
This paper is a continuation of earlier work of Arslan Ars, who introduced the Mahonian number of type B by using a new statistic on the hyperoctahedral group Bn, in response to questions he suggested in his paper entitled " A combinatorial interpretation of Mahonian numbers of type B" published in arXiv:2404.05099v1. We first give the Knuth-Netto formula and generating function for the subdiagonals on or below the main diagonal of the Mahonian numbers of type B, then its combinatorial interpretations by lattice path/partition and tiling. Next, we propose a q-analogue of Mahonian numbers of type B by using a new statistics on the permutations of the hyperoctahedral group Bn that we introduced, then we study their basic properties and their combinatorial interpretations by lattice path/partition and tiling. Finally, we prove combinatorially that the q-analogue of Mahonian numbers of type B form a strongly q-log-concave sequence of polynomials in k, which implies that the Mahonian numbers of type B form a log-concave sequence in k and therefore unimodal.