The Briggs inequality for partitions and overpartitions

TL;DR

This paper proves that various partition and overpartition functions, including regular and overpartition variants, satisfy the Briggs inequality, extending the inequality's applicability to these combinatorial sequences.

Contribution

It establishes the Briggs inequality for partition, overpartition, and k-regular partition functions, expanding its scope to new classes of combinatorial sequences.

Findings

Partition and overpartition functions satisfy Briggs inequality.

Regular k-partition functions satisfy Briggs inequality for 2 ≤ k ≤ 9.

Results are based on Chern's eta-quotient formula.

Abstract

A sequence of $\{a_n\}_{n\ge 0}$ satisfies the Briggs inequality if \begin{align*} a_n^2(a_n^2-a_{n-1}a_{n+1})>a_{n-1}^2(a_{n+1}^2-a_na_{n+2}) \end{align*} holds for any $n\ge 1$. In this paper we show that both the partition function $\{p(n+N_0)\}_{n\geq 0}$ and the overpartition function $\{\overline{p}(n+\overline{N}_0)\}_{n\ge 0}$ satisfy the Briggs inequality for some $N_0$ and $\overline{N}_{0}$. Based on Chern's formula for $\eta$-quotients, we further prove that the $k$-regular partition function $\{p_k(n+N_{k})\}_{n\geq 0}$ and the $k$-regular overpartition function $\{\overline{p}_k(n+\overline{N}_k)\}_{n\ge 0}$ also satisfy the Briggs inequality for $2\le k\le 9$ and some $N_k,\overline{N}_{k}$.