Fractional partitions and conjectures of Chern-Fu-Tang and Heim-Neuhauser

TL;DR

This paper proves the Chern-Fu-Tang conjecture and partially confirms the Heim-Neuhauser conjecture regarding inequalities of partition function coefficients, providing explicit error bounds and advancing understanding of additive-multiplicative partition inequalities.

Contribution

It establishes the Chern-Fu-Tang conjecture and verifies the Heim-Neuhauser conjecture within a certain range, offering explicit error terms for future research.

Findings

Proved the Chern-Fu-Tang conjecture.

Confirmed the Heim-Neuhauser conjecture in a specific range.

Provided explicit error bounds for partition coefficient inequalities.

Abstract

Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern-Fu-Tang and Heim-Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov-Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern-Fu-Tang conjecture and to show the Heim-Neuhauser conjecture in a certain range. The explicit error terms provided will also be useful in the future study of partition inequalities. These are laid out in a user-friendly way for the researcher in combinatorics interested in such analytic questions.