  We give combinatorial proofs of two identities from the representation theory
of the partition algebra $C A_k(n), n \ge 2k$. The first is $n^k = \sum_\lambda
f^\lambda m_k^\lambda$, 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 $\lambda$ and
length $2k$. Our proof uses a combination of Robinson-Schensted-Knuth insertion
and 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 algebras
which appear as subalgebras of the partition algebra: the Brauer,
Temperley-Lieb, planar partition, rook monoid, planar rook monoid, and
symmetric group algebras.
