## Abstract

For k ≥ 1, the homogeneous symmetric functions G(k, m) of degree m defined by Formula Presented are called Petrie symmetric functions. As derived by Grinberg and Fu–Mei independently, the expansion of G(k, m) in the basis of Schur functions s_{λ} 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_{λ} in terms of the k-core of λ and a sequence of rim hooks of size k removed from λ. We further study the product of G(k, m) with a power sum symmetric function pn. For all n ≥ 1, we give necessary and sufficient conditions on the parameters k and m in order for the expansion of G(k, m) · pn in the basis of Schur functions to be signed multiplicity free. This settles affirmatively a conjecture of Alexandersson as the special case n = 2.

Original language | English |
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Pages (from-to) | 1839-1854 |

Number of pages | 16 |

Journal | Proceedings of the American Mathematical Society |

Volume | 151 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2023 May 1 |

## Keywords

- Petrie symmetric functions
- modular complete symmetric functions
- signed multiplicity free
- truncated homogeneous symmetric functions

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics