Combinatorial interpretations of Somos's Dedekind $\eta$-function identities

Abstract

Michael Somos used computer experimentation via the PARI/GP system to discover a large number of conjectural identities of the $\eta$-function type. He identified around 6200 such identities of varying levels. He did not provide rigorous proofs for them and they remained conjectural from the standpoint of the publication of his list. Among these he discovered nearly 15 Dedekind $\eta$-function identities. In the present work, we interpret them combinatorially by showing that they arise as generating functions for suitable colored partitions with suitable examples.