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 prove that k-trace operators of SO(6)R irreps with Young tableau partitions {r1, r2, r3}, constructed out of k chiral primaries in the twenty dimensional SO(6)R irrep, lead to quasi primary fields with protected conformal dimensions under certain conditions. If r1, r2, r3 are all even, r1 + r2 + r3 = 2k and the field is a spacetime scalar, then the conformal dimension 2k is unrenormalized. I...

The notorious middle two levels problem is to nd a Hamilton cycle in the middle two levels, M 2k+1 , of the Hasse diagram of B 2k+1 (the partially ordered set of subsets of a 2k + 1-element set ordered by inclusion). Previously, the best known result, due to Moews and Reid 11] in 1990, was that M 2k+1 is Hamiltonian for all positive k through k = 11. We show that if a Hamilton path between two ...