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....

We generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota’s theory of Möbius inversion. The technique works for a large class of semigroups including: inverse semigroups, semigroups with commuting idempotents, idempotent semigroups and semigroups with basic algebras. Using these tools we are able to give a complete descr...

A new family of Dirichlet series having interesting combinatorial properties is introduced. Although they have no functional equation or Euler product, under the Riemann Hypothesis it is shown that these functions have no zeros in Re(s) > 1/2. Some identities in the ring of formal power series involving rook theory and continued fractions are developed.