Inequalities for the Broken $k$-Diamond Partition Function

TL;DR

This paper investigates the inequalities and positivity properties of the broken k-diamond partition function, proving specific cases and conjecturing broader positivity results for large n, with implications for partition function inequalities.

Contribution

The paper proves new inequalities for the broken k-diamond partition function and conjectures general positivity properties for all k and r, extending known results in partition theory.

Findings

Proved $D^3\,\log\,\Delta_1(n-1)>0$ for $n\geq 5$.

Established $D^3\,\log\,\Delta_2(n-1)>0$ for $n\geq 7$.

Both sequences satisfy higher order Turán inequalities for $n\geq 6$.

Abstract

In 2007, Andrews and Paule introduced the broken $k$-diamond partition function $\Delta_k(n)$, which has received a lot of researches on the arithmetic propertises. In this paper, we prove that $D^3\log \Delta_1(n-1)>0$ for $n\geq 5$ and $D^3 \log \Delta_2(n-1)>0$ for $n\geq 7$, where $D$ is the difference operator with respect to $n$. We also conjecture that for any $k\geq 1$ and $r\geq 1$, there exists a positive integer $n_k(r)$ such that for $n\geq n_{k}(r)$, $(-1)^r D^r \log \Delta_k(n)>0$. This is analogous to the positivity of finite differences of the logarithm of the partition function, which has been proved by Chen, Wang and Xie. Furthermore, we obtain that both $\{\Delta_1(n)\}_{n\geq 0}$ and $\{\Delta_2(n)\}_{n\geq 0}$ satisfy the higher order Tur\'an inequalities for $n \geq 6$.