Partitions with fixed largest hook length

Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub’s analogue of Euler’s Odd-Distinct partition theorem, derive a generalization in the spirit of Alder’s conjecture, as well as a curious analogue of the first Rogers–Ramanujan identity. Moreover, we obtain a partition theorem that is the counterpart of Euler’s pentagonal number theorem in this setting, and connect it with the Rogers–Fine identity. We conclude with some congruence properties.