An Injection for the Ehrenpreis Rogers–Ramanujan Problem

Garsia and Milne used their elegant involution principle to give a bijective proof of the first Rogers–Ramanujan identity. We give an injection as called for by Ehrenpreis of the partitions ofninto parts of the forms 5m+2 and 5m+3 into the partitions ofninto parts of the forms 5m+1 and 5m+4. As observed by Ehrenpreis, Andrews and Baxter, this gives a potential start to a Rogers–Ramanujan bijection and a new partition identity involving partitions into parts ⩾3 with difference between parts at least 2. Our potential bijection does not agree with the Garsia–Milne bijection.