Partitions, rooks, and symmetric functions in noncommuting variables

TL;DR

This paper establishes a surprising equivalence between two combinatorial subsets related to rook placements and atomic partitions, and constructs an algebraic structure linking rook theory with symmetric functions in noncommuting variables.

Contribution

It proves that the rook placement subset equals the atomic partitions set and develops an algebra isomorphism with noncommuting symmetric functions.

Findings

$ ext{E}_n = ext{A}_n$ for all } n \

An algebra structure on rook placements isomorphic to $NCSym$ is constructed.

The paper connects rook theory with symmetric functions in noncommuting variables.

Abstract

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 $\cE_n\sbe\Pi_n$ be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, $\cT_{n-1}$. Given $\pi\in\Pi_m$ and $\si\in\Pi_n$, define their {\it slash product\/} to be $\pi|\si=\pi\cup(\si+m)\in\Pi_{m+n}$ where $\si+m$ is the partition obtained by adding $m$ to every element of every block of $\si$. Call $\tau$ {\it atomic\/} if it can not be written as a nontrivial slash product and let $\cA_n\sbe\Pi_n$ 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, $\cE_n=\cA_n$ 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 and an open problem.