Combinatorics of $q$-Mahonian numbers of type $B$ and log-concavity
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929449617850368 |
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| author | Kessouri, Ali Ahmia, Moussa Arslan, Hasan Mesbahi, Salim |
| author_facet | Kessouri, Ali Ahmia, Moussa Arslan, Hasan Mesbahi, Salim |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_02424 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Combinatorics of $q$-Mahonian numbers of type $B$ and log-concavity Kessouri, Ali Ahmia, Moussa Arslan, Hasan Mesbahi, Salim Combinatorics 05A05, 05A15, 05A19, 05A30 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. |
| title | Combinatorics of $q$-Mahonian numbers of type $B$ and log-concavity |
| topic | Combinatorics 05A05, 05A15, 05A19, 05A30 |
| url | https://arxiv.org/abs/2408.02424 |