Proof of a conjecture of Stanley-Zanello

TL;DR

This paper proves that the sequence counting partitions of an integer into at most b distinct parts of size at most n becomes unimodal for large enough n, confirming a recent conjecture by Stanley and Zanello.

Contribution

It establishes the unimodality of a specific partition sequence, resolving a conjecture posed by Stanley and Zanello.

Findings

Sequence is unimodal for sufficiently large n

Confirms Stanley-Zanello conjecture

Provides combinatorial proof of unimodality

Abstract

We prove that the number of partitions of an integer into at most b distinct parts of size at most n forms a unimodal sequence for n sufficiently large with respect to b. This resolves a recent conjecture of Stanley and Zanello.