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 sλ 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 sλ 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.
- Petrie symmetric functions
- modular complete symmetric functions
- signed multiplicity free
- truncated homogeneous symmetric functions
ASJC Scopus subject areas
- Applied Mathematics