---
title: "Proof of a conjecture of Stanley-Zanello"
authors:
  - "Levent Alpoge"
year: 2014
venue: "Journal of Combinatorial Theory, Series A"
arxiv: "1308.2358"
doi: "10.1016/j.jcta.2014.03.004"
source: "https://arxiv.org/abs/1308.2358"
topics:
  - "number-theory"
seed_rank: 1213
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 8
lineage: arithmetic-geometry
cites:
  []
see:
  []
---

# Proof of a conjecture of Stanley-Zanello

## One-sentence takeaway

The number of partitions of an integer into at most b distinct parts of size at most n is unimodal for n much larger than b — Stanley-Zanello, proved as an undergraduate.

## Why it's lovely

Quanta later wrote about "new proofs of ultra old theorems, one of which i came up w in undergrad."

## Problem

Unimodality of a truncated distinct-parts partition function in n, for each fixed b.

## Design

Generating-function / asymptotic unimodality argument.

## Evidence

J. Combin. Theory Ser. A 125 (2014), 166-176. DOI 10.1016/j.jcta.2014.03.004. arXiv:1308.2358.

## Limitations

n large vs b; not a q-series identity paper.

## Implications for Broadside

Early published Alpoge.

## Bottom line

Mint the JCTA version.

## Links

- arXiv: [1308.2358](https://arxiv.org/abs/1308.2358)
- DOI: [10.1016/j.jcta.2014.03.004](https://doi.org/10.1016/j.jcta.2014.03.004)