Bottom Schur Functions

Abstract

We give a basis for the space spanned by the sum $\hat{s}_\lambda$ of the lowest degree terms in the expansion of the Schur symmetric functions $s_\lambda$ in terms of the power sum symmetric functions $p_\mu$, where deg$(p_i)=1$. These lowest degree terms correspond to minimal border strip tableaux of $\lambda$. The dimension of the space spanned by $\hat{s}_\lambda$, where $\lambda$ is a partition of $n$, is equal to the number of partitions of $n$ into parts differing by at least 2. Applying the Rogers-Ramanujan identity, the generating function also counts the number of partitions of $n$ into parts $5k+1$ and $5k-1$.

We also show that a symmetric function closely related to $\hat{s}_\lambda$ has the same coefficients when expanded in terms of power sums or augmented monomial symmetric functions.