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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2401.08214 |
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| _version_ | 1866911798299459584 |
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| author | Shankar, Umesh Sivasubramanian, Sivaramakrishnan |
| author_facet | Shankar, Umesh Sivasubramanian, Sivaramakrishnan |
| contents | Reifegerste and independently, Petersen and Tenner studied a statistic $\mathrm{drops}()$ on permutations in $\mathfrak{S}_n$. Two other studied statistics on $\mathfrak{S}_n$ are $\mathrm{depth}$ and $\mathrm{exc}$. Using descents in ${\it canonical\ reduced\ words}$ of elements in $\mathfrak{S}_n$, we give an involution $f_A: \mathfrak{S}_n \mapsto \mathfrak{S}_n$ that leads to a neat formula for the signed trivariate enumerator of $\mathrm{drops},\mathrm{depth}, \mathrm{exc}$ in $\mathfrak{S}_n$. This gives a simple formula for the signed univariate drops enumerator in $\mathfrak{S}_n$. For the type-B Coxeter group $\mathfrak{B}_n$ as well, using similar techniques, we show analogous results. For the type D Coxeter group, we again get analogous results, but our proof is inductive.
Under the famous Foata-Zeilberger bijection $ϕ_{FZ}$ which takes permutations to restricted Laguerre histories, we show that permutations $π$ and $f_A(π)$ map to the same Motzkin path, but have different history components. Using the Foata-Zeilberger bijection, we also get a continued fraction for the generating function enumerating the pair of statistics $\mathrm{drops}$ and $\mathrm{MAD}$. Graham and Diaconis determined the mean and the variance of the Spearman metric of disarray $D(π)$ when one samples $π$ from $\mathfrak{S}_n$ at random. As an application of our results, we get the mean and variance of the statistic $\mathrm{drops}(π)$ when we sample $π$ from $\mathcal{A}_n$ at random. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_08214 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Canonical reduced words and signed descent length enumeration in Coxeter groups Shankar, Umesh Sivasubramanian, Sivaramakrishnan Combinatorics O5A05, 05A15 Reifegerste and independently, Petersen and Tenner studied a statistic $\mathrm{drops}()$ on permutations in $\mathfrak{S}_n$. Two other studied statistics on $\mathfrak{S}_n$ are $\mathrm{depth}$ and $\mathrm{exc}$. Using descents in ${\it canonical\ reduced\ words}$ of elements in $\mathfrak{S}_n$, we give an involution $f_A: \mathfrak{S}_n \mapsto \mathfrak{S}_n$ that leads to a neat formula for the signed trivariate enumerator of $\mathrm{drops},\mathrm{depth}, \mathrm{exc}$ in $\mathfrak{S}_n$. This gives a simple formula for the signed univariate drops enumerator in $\mathfrak{S}_n$. For the type-B Coxeter group $\mathfrak{B}_n$ as well, using similar techniques, we show analogous results. For the type D Coxeter group, we again get analogous results, but our proof is inductive. Under the famous Foata-Zeilberger bijection $ϕ_{FZ}$ which takes permutations to restricted Laguerre histories, we show that permutations $π$ and $f_A(π)$ map to the same Motzkin path, but have different history components. Using the Foata-Zeilberger bijection, we also get a continued fraction for the generating function enumerating the pair of statistics $\mathrm{drops}$ and $\mathrm{MAD}$. Graham and Diaconis determined the mean and the variance of the Spearman metric of disarray $D(π)$ when one samples $π$ from $\mathfrak{S}_n$ at random. As an application of our results, we get the mean and variance of the statistic $\mathrm{drops}(π)$ when we sample $π$ from $\mathcal{A}_n$ at random. |
| title | Canonical reduced words and signed descent length enumeration in Coxeter groups |
| topic | Combinatorics O5A05, 05A15 |
| url | https://arxiv.org/abs/2401.08214 |