On Signed Multiplicities of Schur Expansions Surrounding Petrie Symmetric Functions

For $k\ge 1$, the homogeneous symmetric functions $G(k,m)$ of degree $m$ defined by ${\sum }_{m\ge 0}G(k,m)z^m={\prod }_{i\ge 1}(1+x_{i}z+x_i^2{z}^2+\cdots +x_i^{k-1}{z}^{k-1})$ are called \emph{Petrie symmetric functions}. As derived by Grinberg and Fu–Mei independently, the expansion of $G(k,m)$ in the basis of Schur functions $s_{\lambda }$ 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 $s_{\lambda }$ in terms of the $k$-core of $\lambda$ and a sequence of rim hooks of size $k$ removed from $\lambda$. We further study the product of $G(k,m)$ with a power sum symmetric function $p_n$. For all $n\ge 1$, we give necessary and sufficient conditions on the parameters $k$ and $m$ in order for the expansion of $G(k,m)\cdot p_n$ in the basis of Schur functions to be signed multiplicity free. This settles affirmatively a conjecture of Alexandersson as the special case $n=2$.