Partitions, rooks, and symmetric functions in noncommuting variables

Mahir Bilen Can ; Bruce E. Sagan

The electronic journal of combinatorics, The Zeilberger Festschrift volume, Tome 18 (2011) no. 2

Résumé

Let $\Pi_n$ denote the set of all set partitions of $\{1,2,\ldots,n\}$. We consider two subsets of $\Pi_n$, one connected to rook theory and one associated with symmetric functions in noncommuting variables. Let ${\cal E}_n\subseteq\Pi_n$ be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, ${\cal T}_{n-1}$. Given $\pi\in\Pi_m$ and $\sigma\in\Pi_n$, define their slash product to be $\pi|\sigma=\pi\cup(\sigma+m)\in\Pi_{m+n}$ where $\sigma+m$ is the partition obtained by adding $m$ to every element of every block of $\sigma$. Call $\tau$ atomic if it can not be written as a nontrivial slash product and let ${\cal A}_n\subseteq\Pi_n$ denote the subset of atomic partitions. Atomic partitions were first defined by Bergeron, Hohlweg, Rosas, and Zabrocki during their study of $NCSym$, the symmetric functions in noncommuting variables. We show that, despite their very different definitions, ${\cal E}_n={\cal A}_n$ for all $n\ge0$. Furthermore, we put an algebra structure on the formal vector space generated by all rook placements on upper triangular boards which makes it isomorphic to $NCSym$. We end with some remarks.

Détail

DOI : 10.37236/1999

Classification : 05A18, 05E05
Mots-clés : noncommuting variables, rook, set partition, symmetric function