Log-concavity and q-Log-convexity Conjectures on the Longest Increasing Subsequences of Permutations
| dc.creator | Chen, William Y. C. | |
| dc.date | 2008-06-20 | |
| dc.date | 2008-06-22 | |
| dc.date.accessioned | 2026-07-07T09:45:53Z | |
| dc.date.available | 2026-07-07T09:45:53Z | |
| dc.description | Let $P_{n,k}$ be the number of permutations $π$ on [n]={1, 2,..., n} such that the length of the longest increasing subsequences of $π$ equals k, and let $M_{2n, k}$ be the number of matchings on [2n] with crossing number k. Define $P_n(x)= \sum_k P_{n,k}x^k$ and $M_{2n}(x)=\sum_{k} M_{2n,k}x^k$. We propose some conjectures on the log-concavity and q-log-convexity of the polynomials $P_n(x)$ and $M_{2n}(x)$. We also introduce the notions of $\infty$-q-log-convexity and $\infty$-q-log-concavity, and the notion of higher order log-concavity with respect to $\infty$-q-log-convex or $\infty$-q-log-concavity. A conjecture on the $\infty$-q-log-convexity of the Boros-Moll polynomials is presented. It seems that $M_{2n}(x)$ are log-concave of any order with respect to $\infty$-q-log-convexity. | |
| dc.description | 7 pages | |
| dc.identifier | https://arxiv.org/abs/0806.3392 | |
| dc.identifier | http://arxiv.org/abs/0806.3392 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/163345 | |
| dc.subject | Combinatorics | |
| dc.title | Log-concavity and q-Log-convexity Conjectures on the Longest Increasing Subsequences of Permutations | |
| dc.type | text |