RSK insertion for set partitions and diagram algebras

Abstract

We give combinatorial proofs of two identities from the representation theory of the partition algebra ℂAk(n),n ≥ 2k. The first is nk = ∑λfλmkλ where the sum is over partitions λ of n, f λ is the number of standard tableaux of shape λ, and mkλ is the number of "vacillating tableaux" of shape λ and length 2k. Our proof uses a combination of Robinson-Schensted-Knuth insertion and jeu de taquin. The second identity is B(2k) = ∑;λ(mkλ)2, where B(2k) is the number of set partitions of {1,..., 2k}. We show that this insertion restricts to work for the diagram algebras which appear as subalgebras of the partition algebra: the Brauer, Temperley-Lieb, planar partition, rook monoid, planar rook monoid, and symmetric group algebras.