Kronecker Coefficients for Some Near-rectangular Partitions

We give formulae for computing Kronecker coefficients occurring in the expansion of sμ ∗ sν , where both μ and ν are nearly rectangular, and have smallest parts equal to either 1 or 2. In particular, we study s(n,n−1,1) ∗ s(n,n), s(n−1,n−1,1) ∗ s(n,n−1), s(n−1,n−1,2) ∗ s(n,n), s(n−1,n−1,1,1) ∗ s(n,n) and s(n,n,1) ∗ s(n,n,1). Our approach relies on the interplay between manipulation of symmetric functions and the representation theory of the symmetric group, mainly employing the Pieri rule and a useful identity of Littlewood. As a consequence of these formulae, we also derive an expression enumerating certain standard Young tableaux of bounded height, in terms of the Motzkin and Catalan numbers. An outstanding open problem in algebraic combinatorics is to derive a combinatorial formula to compute the Kronecker product of two Schur functions. Given partitions λ, μ and ν, the Kronecker coefficients, g μν , occur in the decomposition of the Kronecker product sμ ∗ sν of Schur functions in the Schur basis. sμ ∗ sν = ∑