 
Summary: STAIRCASE SKEW SCHUR FUNCTIONS
ARE SCHUR PPOSITIVE
FEDERICO ARDILA AND LUIS G. SERRANO
Abstract. We prove Stanley's conjecture that, if n is the staircase shape,
then the skew Schur functions sn/µ are nonnegative sums of Schur Pfunctions.
We prove that the coefficients in this sum count certain fillings of shifted
shapes. In particular, for the skew Schur function sn/n2
, we discuss con
nections with Eulerian numbers and alternating permutations.
1. introduction
The Schur functions s, indexed by partitions , form a basis for the ring of
symmetric functions. These are very important objects in algebraic combinatorics.
They play a fundamental role in the study of the representations of the symmetric
group and the general linear group, and the cohomology ring of the Grassman
nian [4]. The Schur Pfunctions P, indexed by strict partitions, form a basis
for an important subring of . They are crucial in the study of the projective
representations of the symmetric group, and the cohomology ring of the isotropic
Grassmannian [10] [11].
The goal of this paper is to prove the following conjecture of Richard Stanley
[15]: If n is the staircase shape and µ n, then the staircase skew Schur function
