Choices of power sum symmetric functions

Abstract

Let Sn be the symmetric group on [n] = {1,2, …n} and Dn be the set of derangements in Sn, i.e., Dn = {σ ∈ Sn:σ(i) ≠ i for i = 1,2, …, n}. A partition λ = (λ1, …, λr) of n, denoted by λ ⊢ n, is a weakly decreasing sequence, λ1 ≥ ⋯ ≥ λr ≥ 1 with ∑i=1rλi=n Let dλ=∑σ∈Dnχλ(σ), where χλ is the irreducible character of Sn corresponding to the partition λ. Let pk be the power sum symmetric polynomial of degree k in n variables. It is known by the Murnaghan-Nakayama Rule that dλ=n!sλ|p1=0,p2=⋯=pn=1, where Sλ is the Schur function. So, the set Dn arises from the choice of p1 = 0, p2 = ⋯ = pn = 1. In this paper, we consider sets arising from different choices of p1, …, pn. For example, if d1, …, dn are non-zero complex numbers, then Sn arises from the choice of pi = di for i = 1, …, n.