Kronecker coefficients for some near-rectangular partitions

We give formulae for computing Kronecker coefficients occurring in the expansion of sμ⁎sνsμ⁎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,n−1,1)⁎s(n,n), s(n−1,n−1,1)⁎s(n,n−1)s(n−1,n−1,1)⁎s(n,n−1), s(n−1,n−1,2)⁎s(n,n)s(n−1,n−1,2)⁎s(n,n), s(n−1,n−1,1,1)⁎s(n,n)s(n−1,n−1,1,1)⁎s(n,n) and s(n,n,1)⁎s(n,n,1)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.