Schur positivity of nabla on Petrie symmetric functions

Abstract

The Petrie symmetric function $G(k,n)$, introduced by Grinberg, is defined as the sum of monomial symmetric functions $m_λ$ indexed by partitions $λ\vdash n$ satisfying $λ_1<k$. This article demonstrates that the Schur positivity pattern of $\nabla^r G(k,n)$ for all $r\geq 1$ depends exclusively on whether $k$ divides $n$, thus answering an open problem of Bergeron noted in Grinberg&#39;s work.