Integrality of hook ratios

TL;DR

This paper investigates when ratios of hook products of quotient partitions are integers, drawing parallels to factorial ratios in number theory, and classifies such ratios using additive combinatorics.

Contribution

It extends Landau's theorem to hook ratios and provides a complete classification under specific conditions, utilizing Kneser's theorem.

Findings

Established an analogue of Landau's theorem for hook ratios

Classified integral hook ratios with an extra denominator factor

Applied Kneser's theorem to solve the classification problem

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.