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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.

In this paper, we propose the number of permutations of length $n$ having $k$ overlined inversions, which we call over-Mahonian number. We study useful properties and some combinatorial interpretations by lattice paths/overpartitions and tilings. Furthermore, we prove combinatorially that these numbers form a log-concave sequence and therefore unimodal.

In this paper, we consider a generalization of the Stirling number sequence of both kinds by using a specialization of a new family of symmetric functions. We give combinatorial interpretations for this symmetric functions by means of weighted lattice path and tilings. We also present some new convolutions involving the complete and elementary symmetric functions. Additionally, we introduce different families of set partitions to give combinatorial interpretations for the modular $s$-Stirling numbers., Comment: 2 figures

In this paper, we consider the generating functions of the complete and elementary symmetric functions and provide a new generalization of these classical symmetric functions. Some classical relationships involving the complete and elementary symmetric functions are reformulated in a more general context. Combinatorial interpretations of these generalized symmetric functions are also introduced.