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arXiv · 2607.06351

Schur positivity of nabla on Petrie symmetric functions

Menghao Qu
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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's work.

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Menghao Qu. 2026-07-07. Schur positivity of nabla on Petrie symmetric functions. https://arxiv.org/abs/2607.06351

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