Integral Factorial Ratios

Abstract

This paper describes a new approach to classifying integral factorial ratio, obtaining in particular a direct proof of a result of Bober. These results generalize an observation going back to Chebyshev that $(30n)!n!/((15n)!(10n)!(6n)!)$ is an integer for all $n$. Due to the work of Rodriguez-Villegas and Beukers and Heckman, this problem is closely related to classifying hypergeometric functions with finite monodromy groups, and the result of Bober was originally derived as a consequence of the work of Beukers--Heckman. The new proof is elementary and makes partial progress on other related questions.