We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises.

Partition bijections, a survey

The quintuple product identity was first discovered about 90 years ago. It has been published in many different forms, and at least 29 proofs have been given. We shall give a comprehensive survey of the work on the quintuple product identity, and a detailed analysis of the many proofs.

The quintuple product identity

In this paper we give a simple proof of the Jacobi triple product identity by using basic properties of cube roots of unity. Then we give a new proof of the quintuple product identity, the septuple product identity and Winquist’s identity by using the Jacobi triple product identity and basic properties of cube and fifth roots of unity. Furthermore, we derive some new product identities by this uniform method. Later, we give some generalizations of those identities. Lastly, we derive some modular equations.

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Uniform proofs of q-Series-product identities

This paper takes up again the study of the Jacobi triple and Watson quintuple identities that have been derived combinatorially in several manners in the classical literature. It also contains a proof of the recent Farkas-Kra septuple product identity that makes use only of “smanipulatorics” methods.

The Triple, Quintuple and Septuple Product Identities Revisited

Let N(i,m;n) be the number of partitions of n with rank (Dyson) congruent to i (mod m) and let M(j,m;n) be the number of partitions of n with crank (Andrews, Garvan) congruent to j (mod m). I give here the generating functions for the numbers N(i,8;n) and M(j,8;n). I suggest forms for the one hundred power series 

$$R(i,j,k)=\sum_{n\ge0}(N(i,8;4n+k)-M(j,8;4n+k))q^{n},$$

from which, if true, a number of inequalities follow.

R. P. Lewis

Papers

A Combinatorial Proof of the Triple Product Identity