Staircase skew Schur functions are Schur P-positive

摘要

We prove Stanley’s conjecture that, if δ n is the staircase shape, then the skew Schur functions $s_{delta_{n} / mu}$ are non-negative sums of Schur P-functions. We prove that the coefficients in this sum count certain fillings of shifted shapes. In particular, for the skew Schur function $s_{delta_{n} / delta _{n-2}}$ , we discuss connections with Eulerian numbers and alternating permutations.