Partitions, rooks, and symmetric functions in noncommuting variables

Let II(n) denote the set of all set partitions of {1,2, ... ,n}. We consider two subsets of II(n), one connected to rook theory and one associated with symmetric functions in noncommuting variables. Let E(n) subset of II(n) be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, T(n-1). Given pi is an element of II(m) and sigma is an element of II(n), define their slash product to be pi/sigma = pi boolean OR(sigma+m) is an element of II(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 cannot be written as a nontrivial slash product and let An subset of II(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, E(n) = A(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.