On Signed Multiplicities of Schur Expansions Surrounding Petrie Symmetric Functions

TL;DR

This paper explores the signed multiplicity free expansion of Petrie symmetric functions in Schur basis, providing combinatorial interpretations and conditions for product expansions with power sum functions, confirming a conjecture.

Contribution

It offers a combinatorial interpretation of Schur coefficients for Petrie symmetric functions and characterizes when their products with power sums are signed multiplicity free.

Findings

Coefficients relate to k-core and rim hooks of partitions.

Conditions for signed multiplicity free expansion of G(k,m)·p_n.

Confirmed a conjecture for n=2 case.

Abstract

For $k\ge 1$, the homogeneous symmetric functions $G(k,m)$ of degree $m$ defined by $\sum_{m\ge 0} G(k,m) z^m=\prod_{i\ge 1} \big(1+x_iz+x^2_iz^2+\cdots+x^{k-1}_iz^{k-1}\big)$ 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 $n=2$.