Bottom Schur functions

Abstract

We give a basis for the space V spanned by the lowest degree part \hat{s}_λof the expansion of the Schur symmetric functions s_λin terms of power sums, where we define the degree of the power sum p_i to be 1. In particular, the dimension of the subspace V_n spanned by those \hat{s}_λfor which λis a partition of n is equal to the number of partitions of n whose parts differ by at least 2. We also show that a symmetric function closely related to \hat{s}_λhas the same coefficients when expanded in terms of power sums or augmented monomial symmetric functions. Proofs are based on the theory of minimal border strip decompositions of Young diagrams.