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We give combinatorial proofs of two identities from the representation theoryof the partition algebra $C A_k(n), n \ge 2k$. The first is $n^k = \sum_\lambdaf^\lambda m_k^\lambda$, where the sum is over partitions $\lambda$ of $n$,$f^\lambda$ is the number of standard tableaux of shape $\lambda$, and$m_k^\lambda$ is the number of "vacillating tableaux" of shape $\lambda$ andlength $2k$. Our proof uses a combination of Robinson-Schensted-Knuth insertionand jeu de taquin. The second identity is $B(2k) = \sum_\lambda(m_k^\lambda)^2$, where $B(2k)$ is the number of set partitions of $\{1, >...,2k\}$. We show that this insertion restricts to work for the diagram algebraswhich appear as subalgebras of the partition algebra: the Brauer,Temperley-Lieb, planar partition, rook monoid, planar rook monoid, andsymmetric group algebras.

RSK Insertion for Set Partitions and Diagram Algebras