On signed multiplicities of Schur expansions surrounding Petrie symmetric functions

Abstract

For k ≥ 1 k\ge 1 , the homogeneous symmetric functions G ( k , m ) G(k,m) of degree m m defined by ∑ m ≥ 0 G ( k , m ) z m = ∏ i ≥ 1 ( 1 + x i z + x i 2 z 2 + ⋯ + x i k − 1 z k − 1 ) \sum _{m\ge 0} G(k,m) z^m=\prod _{i\ge 1} \big (1+x_iz+x^2_iz^2+\cdots +x^{k-1}_iz^{k-1}\big ) are called Petrie symmetric functions. As derived by Grinberg and Fu–Mei independently, the expansion of G ( k , m ) G(k,m) in the basis of Schur functions s λ s_{\lambda } turns out to be signed multiplicity free, i.e., the coefficients are − 1 -1 , 0 0 and 1 1 . In this paper we give a combinatorial interpretation of the coefficient of s λ s_{\lambda } in terms of the k k -core of λ \lambda and a sequence of rim hooks of size k k removed from λ \lambda . We further study the product of G ( k , m ) G(k,m) with a power sum symmetric function p n p_n . For all n ≥ 1 n\ge 1 , we give necessary and sufficient conditions on the parameters k k and m m in order for the expansion of G ( k , m ) ⋅ p n G(k,m)\cdot p_n in the basis of Schur functions to be signed multiplicity free. This settles affirmatively a conjecture of Alexandersson as the special case n = 2 n=2 .