Tur\'an inequalities for the plane partition function

TL;DR

This paper proves the conjecture that the plane partition function is log-concave for all n ≥ 12 and establishes Turán inequalities for it, extending known inequalities from the classical partition function.

Contribution

It confirms the log-concavity conjecture for the plane partition function and proves Turán inequalities of all degrees for large n, advancing understanding of its combinatorial properties.

Findings

Proved the log-concavity of the plane partition function for all n ≥ 12.

Established Turán inequalities of arbitrary degree for sufficiently large n.

Extended known inequalities from the classical partition function to plane partitions.

Abstract

Heim, Neuhauser, and Tr\"oger recently established some inequalities for MacMahon's plane partition function $\mathrm{PL}(n)$ that generalize known results for Euler's partition function $p(n)$. They also conjectured that $\mathrm{PL}(n)$ is log-concave for all $n\geq 12.$ We prove this conjecture. Moreover, for every $d\geq 1$, we prove their speculation that $\mathrm{PL}(n)$ satisfies the degree $d$ Tur\'an inequality for sufficiently large $n$. The case where $d=2$ is the case of log-concavity.