  We prove Stanley's conjecture that, if delta_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.
