Combinatorics of $q$-Mahonian numbers of type $B$ and log-concavity
Abstract
This paper is a continuation of earlier work of Arslan \cite{Ars}, who introduced the Mahonian number of type by using a new statistic on the hyperoctahedral group , in response to questions he suggested in his paper entitled "{\it A combinatorial interpretation of Mahonian numbers of type }" 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 , then its combinatorial interpretations by lattice path/partition and tiling. Next, we propose a -analogue of Mahonian numbers of type by using a new statistics on the permutations of the hyperoctahedral group 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 -analogue of Mahonian numbers of type form a strongly -log-concave sequence of polynomials in , which implies that the Mahonian numbers of type form a log-concave sequence in and therefore unimodal.
Keywords
Cite
@article{arxiv.2408.02424,
title = {Combinatorics of $q$-Mahonian numbers of type $B$ and log-concavity},
author = {Ali Kessouri and Moussa Ahmia and Hasan Arslan and Salim Mesbahi},
journal= {arXiv preprint arXiv:2408.02424},
year = {2024}
}