Some inequalities for $k$-colored partition functions

TL;DR

This paper establishes new inequalities for $k$-colored partition functions, extending their multiplicative properties and identifying conditions for maximum values, with a conjecture proposing further strengthening of these inequalities.

Contribution

It introduces inequalities for $k$-colored partition functions, extending their multiplicative extension and characterizing maximum points, building on prior work by Bessenrodt and Ono.

Findings

Derived inequalities for $p_{-k}(n)$ for all $k\u2265 2

Extended the $k$-colored partition function multiplicatively

Identified conditions for the function's unique maximum

Abstract

Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$-colored partition functions $p_{-k}(n)$ for all $k\geq2$. This enables us to extend the $k$-colored partition function multiplicatively to a function on $k$-colored partitions, and characterize when it has a unique maximum. We conclude with one conjectural inequality that strengthens our results.