  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.
