Signed Mahonian Polynomials on Colored Derangements
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917193776627712 |
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| author | Arslan, Hasan Ahmia, Moussa Alemdar, Nazmiye |
| author_facet | Arslan, Hasan Ahmia, Moussa Alemdar, Nazmiye |
| contents | The polynomial $\sum_{π\in W}q^{maj(π)}$ of major index over a classical Weyl group $W$ with a generating set $S$ is called the Mahonian polynomial over $W$, and also the polynomial $\sum_{π\in W}(-1)^{l(π)}q^{maj(π)}$ of major index together with sign over the group $W$ is called the signed Mahonian polynomial over the group $W$, where $l$ is the length function on $W$ defined in terms of the generating set $S$. We concern with the signed Mahonian polynomial $$\sum_{π\in D_{n}^{(c)}}(-1)^{L(π)}q^{fmaj(π)}$$ on the set $D_{n}^{(c)}$ of colored derangements in the group $G_{c,n}$ of colored permutations, where $L$ denotes the length function defined by means of a complex root system described by Bremke and Malle in $G_{c,n}$ and $fmaj$ defined by Adin and Roichman in $G_{c,n}$ represents the \textit{flag-major index}, which is a Mahonian statistic. As an application of the formula for signed Mahonian polynomials on the set of colored derangements, we will derive a formula to count colored derangements of even length in $G_{c,n}$ when $c$ is an even number. Finally, we conclude by providing a formula for the difference between the number of derangements of even and odd lengths in $G_{c,n}$ for every positive integer $c$, regardless of whether c is odd or even. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02404 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Signed Mahonian Polynomials on Colored Derangements Arslan, Hasan Ahmia, Moussa Alemdar, Nazmiye Combinatorics 05A05, 05A15, 05A19 The polynomial $\sum_{π\in W}q^{maj(π)}$ of major index over a classical Weyl group $W$ with a generating set $S$ is called the Mahonian polynomial over $W$, and also the polynomial $\sum_{π\in W}(-1)^{l(π)}q^{maj(π)}$ of major index together with sign over the group $W$ is called the signed Mahonian polynomial over the group $W$, where $l$ is the length function on $W$ defined in terms of the generating set $S$. We concern with the signed Mahonian polynomial $$\sum_{π\in D_{n}^{(c)}}(-1)^{L(π)}q^{fmaj(π)}$$ on the set $D_{n}^{(c)}$ of colored derangements in the group $G_{c,n}$ of colored permutations, where $L$ denotes the length function defined by means of a complex root system described by Bremke and Malle in $G_{c,n}$ and $fmaj$ defined by Adin and Roichman in $G_{c,n}$ represents the \textit{flag-major index}, which is a Mahonian statistic. As an application of the formula for signed Mahonian polynomials on the set of colored derangements, we will derive a formula to count colored derangements of even length in $G_{c,n}$ when $c$ is an even number. Finally, we conclude by providing a formula for the difference between the number of derangements of even and odd lengths in $G_{c,n}$ for every positive integer $c$, regardless of whether c is odd or even. |
| title | Signed Mahonian Polynomials on Colored Derangements |
| topic | Combinatorics 05A05, 05A15, 05A19 |
| url | https://arxiv.org/abs/2512.02404 |