A Refinement of the Eulerian Numbers, and the Joint Distribution of $π(1)$ and Des($π$) in $S_n$
| dc.creator | Conger, Mark | |
| dc.date | 2005-08-05 | |
| dc.date.accessioned | 2026-07-07T05:22:14Z | |
| dc.date.available | 2026-07-07T05:22:14Z | |
| dc.description | Given a permutation $π$ chosen uniformly from $S_n$, we explore the joint distribution of $π(1)$ and the number of descents in $π$. We obtain a formula for the number of permutations with $\des(π)=d$ and $π(1)=k$, and use it to show that if $\des(π)$ is fixed at $d$, then the expected value of $π(1)$ is $d+1$. We go on to derive generating functions for the joint distribution, show that it is unimodal if viewed correctly, and show that when $d$ is small the distribution of $π(1)$ among the permutations with $d$ descents is approximately geometric. Applications to Stein's method and the Neggers-Stanley problem are presented. | |
| dc.description | 21 pages, 4 figures | |
| dc.identifier | https://arxiv.org/abs/math/0508112 | |
| dc.identifier | http://arxiv.org/abs/math/0508112 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/75984 | |
| dc.subject | Combinatorics | |
| dc.subject | Probability | |
| dc.subject | 05A15 (Primary) 20B30, 60C05 (Secondary) | |
| dc.title | A Refinement of the Eulerian Numbers, and the Joint Distribution of $π(1)$ and Des($π$) in $S_n$ | |
| dc.type | text |