ON SIGNED MULTIPLICITIES OF SCHUR EXPANSIONS SURROUNDING PETRIE SYMMETRIC FUNCTIONS

For k ≥ 1, the homogeneous symmetric functions G(k, m) of degree m defined by Formula Presented are called Petrie symmetric functions. As derived by Grinberg and Fu–Mei independently, the expansion of G(k, m) in the basis of Schur functions s_λ turns out to be signed multiplicity free, i.e., the coefficients are −1, 0 and 1. In this paper we give a combinatorial interpretation of the coefficient of s_λ in terms of the k-core of λ and a sequence of rim hooks of size k removed from λ. We further study the product of G(k, m) with a power sum symmetric function pn. For all n ≥ 1, we give necessary and sufficient conditions on the parameters k and m in order for the expansion of G(k, m) · pn in the basis of Schur functions to be signed multiplicity free. This settles affirmatively a conjecture of Alexandersson as the special case n = 2.