Minimal Bar Tableaux

Abstract

Motivated by Stanley's results in \cite{St02}, we generalize the rank of a partition $\lambda$ to the rank of a shifted partition $S(\lambda)$. We show that the number of bars required in a minimal bar tableau of $S(\lambda)$ is max$(o, e + (\ell(\lambda) \mathrm{mod} 2))$, where $o$ and $e$ are the number of odd and even rows of $\lambda$. As a consequence we show that the irreducible projective characters of $S_n$ vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur's $Q_{\lambda}$ symmetric functions in terms of the power sum symmetric functions.