Staircase skew Schur functions are Schur P-positive

TL;DR

This paper proves Stanley's conjecture that certain skew Schur functions decompose positively into Schur P-functions, with coefficients linked to combinatorial fillings, and explores connections to Eulerian numbers and permutations.

Contribution

It establishes the Schur P-positivity of staircase skew Schur functions and relates coefficients to combinatorial fillings, advancing understanding of symmetric functions.

Findings

Proves non-negativity of skew Schur functions in Schur P-basis.

Connects coefficients to combinatorial fillings of shifted shapes.

Links specific skew Schur functions to Eulerian numbers and permutations.

Abstract

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.