Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Partitions, rooks, and symmetric functions in noncommuting variables   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.

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The categories of this paper are: comp-gas
1
The categories of this paper are: cs
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The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
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The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: stat
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The categories of this paper are: q-bio
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The categories of this paper are: q-fin