A combinatorial formula for certain binomial coefficients for Jack polynomials

TL;DR

This paper introduces a new combinatorial decomposition of binomial coefficients related to Jack polynomials, providing explicit formulas and revealing symmetries in special cases.

Contribution

It offers a novel decomposition into a stem and leaf, with an explicit formula for the leaf in specific cases, advancing understanding of Jack polynomial binomial coefficients.

Findings

Decomposition into stem and leaf factors

Explicit combinatorial formula for the leaf when partitions differ by at most two rows

Discovery of symmetry related to row lengths in special cases

Abstract

We present a decomposition of the generalized binomial coefficients associated with Jack polynomials into two factors: a stem, which is described explicitly in terms of hooks of the indexing partitions, and a leaf, which inherits various recurrence properties from the binomial coefficients and depends exclusively on the skew diagram. We then derive a direct combinatorial formula for the leaf in the special case where the two indexing partitions differ by at most two rows. This formula also exhibits an unexpected symmetry with respect to the lengths of the two rows.