Partitions, rooks, and symmetric functions in noncommuting variables

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Let Πₙ denote the set of all set partitions of {1,2,,n}. We consider two subsets of Πₙ, one connected to rook theory and one associated with symmetric functions in noncommuting variables. Let ₙΠₙ be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, ₙ₋₁. Given π∈Πₘ and ∈Πₙ, define their slash product to be π|=π∪(+m)∈Πₘ₊ₙ where +m is the partition obtained by adding m to every element of every block of. Call τ atomic if it can not be written as a nontrivial slash product and let ₙΠₙ 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, ₙ=ₙ for all n≥0. 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.