  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 $n=2$.
