Combinatorics of $q$-Mahonian numbers of type $B$ and log-concavity

TL;DR

This paper extends the combinatorial understanding of Mahonian numbers of type B, introduces a new q-analogue, and proves their log-concavity and unimodality, enriching the algebraic and combinatorial theory of these numbers.

Contribution

It introduces a new q-analogue of Mahonian numbers of type B and proves their strong q-log-concavity, establishing log-concavity and unimodality of the original numbers.

Findings

Derived the Knuth-Netto formula and generating function for subdiagonals of type B Mahonian numbers.

Proposed a new q-analogue of Mahonian numbers of type B using a novel statistic.

Proved that the q-analogue sequence is strongly q-log-concave, implying log-concavity and unimodality of the original numbers.

Abstract

This paper is a continuation of earlier work of Arslan \cite{Ars}, who introduced the Mahonian number of type $B$ by using a new statistic on the hyperoctahedral group $B_{n}$, in response to questions he suggested in his paper entitled "{\it 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 $B_n$ 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.