In honor of Richard Stanley on his 60th birthday.
AbstractWe give combinatorial proofs of two identities from the representation theory of the partition algebra CA k (n), n ≥ 2k. The first is n k = λ f λ m λ k , where the sum is over partitions λ of n, f λ is the number of standard tableaux of shape λ, and m λ k 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.
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