Crossings and Nestings of Two Edges in Set Partitions
| dc.creator | Poznanovik, Svetlana | |
| dc.creator | Yan, Catherine | |
| dc.date | 2007-10-09 | |
| dc.date.accessioned | 2026-07-07T08:35:06Z | |
| dc.date.available | 2026-07-07T08:35:06Z | |
| dc.description | Let $π$ and $λ$ be two set partitions with the same number of blocks. Assume $π$ is a partition of $[n]$. For any integer $l, m \geq 0$, let $\mathcal{T}(π, l)$ be the set of partitions of $[n+l]$ whose restrictions to the last $n$ elements are isomorphic to $π$, and $\mathcal{T}(π, l, m)$ the subset of $\mathcal{T}(π,l)$ consisting of those partitions with exactly $m$ blocks. Similarly define $\mathcal{T}(λ, l)$ and $\mathcal{T}(λ, l,m)$. We prove that if the statistic $cr$ ($ne$), the number of crossings (nestings) of two edges, coincides on the sets $\mathcal{T}(π, l)$ and $\mathcal{T}(λ, l)$ for $l =0, 1$, then it coincides on $\mathcal{T}(π, l,m)$ and $\mathcal{T}(λ, l,m)$ for all $l, m \geq 0$. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings. | |
| dc.description | 19 pages, 2 figures | |
| dc.identifier | https://arxiv.org/abs/0710.1816 | |
| dc.identifier | http://arxiv.org/abs/0710.1816 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/139651 | |
| dc.subject | Combinatorics | |
| dc.subject | 05A18, 05A15 | |
| dc.title | Crossings and Nestings of Two Edges in Set Partitions | |
| dc.type | text |