RSK Insertion for Set Partitions and Diagram Algebras

We give combinatorial proofs of two identities from the representation theory of the partition algebra CAk(n), n ≥ 2k. The first is nk = ∑ λ f mk , where the sum is over partitions λ of n, fλ is the number of standard tableaux of shape λ, and mk 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(m λ k) 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.