Some conjectures on the Schur expansion of Jack polynomials

TL;DR

This paper proposes positivity conjectures for the Schur expansion of Jack symmetric functions in binomial coefficient bases, revealing rich combinatorial structures and leading to new conjectures about quasisymmetric expansions.

Contribution

It introduces new positivity conjectures for Jack polynomials' Schur expansion and explores their combinatorial implications, including partial proofs for special cases.

Findings

Positivity conjectures for Schur expansion in binomial bases

Connections to Eulerian, Stirling numbers, and tableaux

Partial proofs for specific cases of quasisymmetric expansions

Abstract

We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian numbers, Stirling numbers, quasi-Yamanouchi tableaux, and rook boards. These results also lead to further conjectures about the fundamental quasisymmetric expansions of these bases, which we prove for special cases.