2022/06/28 by Yen-Jen Cheng, Cheng, Yen-Jen, Meng-Chien Chou +7 · 1 citation
Mathematics · #05A17 #05E05 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2206.14023
openalex publication_date 2022/06/28 · openalex created_date 2022/07/01 · openalex updated_date 2026/07/28
For k≥ 1, the homogeneous symmetric functions G(k,m) of degree m defined by ∑m≥ 0 G(k,m) zm=∏i≥ 1 (1+xiz+x2iz2+⋯+xk-1izk-1) 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.