Partition congruences by involutions

We present a general construction of involutions on integer partitions which enable us to prove a number of modulo 2 partition congruences. Introduction The theory of partitions is a beautiful subject introduced by Euler over 250 years ago and is still under intense development [2]. Arguably, a turning point in its history was the invention of the “constructive partition theory” symbolized by Franklin’s involution [10] and commemorated in Sylvester’s magnum opus [17]. Based on explicit constructions of bijections and involutions, this approach was taken to a new high by Schur’s proof of RogersRamanujan’s identities and led to numerous new proofs and identities. We refer to [14] for an extensive survey of history and recent developments of the subject. By themselves, partition congruences became a subject of intense interest ever since Ramanujan’s celebrated discovery of the congruence p(5n−1) ≡ 0 mod 5. Despite various proofs, extensions and even Dyson’s ‘rank’ combinatorial interpretation [7], there is still no bijective proof of Ramanujan’s congruences. In fact, the few partition congruences which are known to have