Bijections for the Rogers-Ramanujan Reciprocal

Abstract

The Rogers-Ramanujan identities
The famous Rogers-Ramanujan identities are the following
1+∑qⁿ²/(1-q)(1-q²)...(1-qⁿ)=Π 1/(1-q⁵ⁿ⁺¹)(1-q⁵ⁿ⁺⁴)                   (1)
1+∑qⁿ²⁺ⁿ/(1-q)(1-q²)...(1-qⁿ)=Π1/(1-q⁵ⁿ⁺¹)(1-q⁵ⁿ⁺⁴)                 (2)
The history of the discovery of these marvelous identities by Rogers [14], Ramanujan [12] and Schur [15] is well known and has been related many times (see for example Andrews [1; Ch. 7] [2; Ch. 3], Hardy [12; Ch. 6]). Although this remarkable history starts in 1894, there have been recently a lot of activity among combinatorists around these identities. Adding to the fascination of this history, these same identities have recently appeared in statistical Physics in the beautiful exact solution by Baxter [5], [6] of the hard hexagon model.