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- W2950871341 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 $cE_nsbePi_n$ be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, $cT_{n-1}$. Given $piinPi_m$ and $siinPi_n$, define their {it slash product/} to be $pi|si=picup(si+m)inPi_{m+n}$ 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 $cA_nsbePi_n$ 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 $nge0$. 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." @default.
- W2950871341 created "2019-06-27" @default.
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- W2950871341 date "2010-08-17" @default.
- W2950871341 modified "2023-09-27" @default.
- W2950871341 title "Partitions, rooks, and symmetric functions in noncommuting variables" @default.
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