Schur positivity of nabla on Petrie symmetric functions

The Petrie symmetric function $G(k,n)$, introduced by Grinberg, is defined as the sum of monomial symmetric functions $m_{\lambda }$ indexed by partitions $\lambda ⊢n$ satisfying ${\lambda }_1<k$. This article demonstrates that the Schur positivity pattern of ${\mathrm{\nabla }}^{r}G(k,n)$ for all $r\ge 1$ depends exclusively on whether $k$ divides $n$, thus answering an open problem of Bergeron noted in Grinberg’s work.