---
abstract: 'We study integral ratios of hook products of quotient partitions. This question is motivated by an analogous question in number theory concerning integral factorial ratios. We prove an analogue of a theorem of Landau that already applied in the factorial case. Under the additional condition that the ratio has one more factor on the denominator than the numerator, we provide a complete classification. Ultimately this relies on Kneser’s theorem in additive combinatorics.'
address: 'D-MATH, ETH Zürich, 101 Rämistrasse, 8092 Zürich, Switzerland'
author:
- 'Paul-Olivier Dehaye'
bibliography:
- 'references.bib'
title: Integrality of hook ratios
---

Introduction
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Questions
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This paper starts with the following (unsolved) question, a classic in number theory:

For which vectors of positive integers $(\gamma_k:1 \le k \le K)$ and $(\delta_l:1 \le l \le L)$ is $$\label{eqn.alt}
v(n;\vec{\gamma},\vec{\delta}) := \frac{\prod_{k=0}^K {{\left\lfloor {\displaystyle\frac{n}{\gamma_k}} \right\rfloor}}!}{\prod_{l=0}^L{{\left\lfloor {\displaystyle\frac{ n}{\delta_l}} \right\rfloor}}!}$$ an integer for all integers $n$? \[q.alt\]

From now on, we assume $k$ runs between 1 and $K$ and $l$ between 1 and $L$. We call $L-K$ the *height*. Some easy reductions are immediately in order. We will require $\gamma_k \ne \delta_l$ for all $k$ and $l$. We also assume a *balancing* condition: $\sum_k \frac{1}{\gamma_k} = \sum_l \frac{1}{\delta_l}$. This 