Abstract. 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.

We present a general construction of involutions on integer partitions which enable us to prove a number of modulo 2 partition congruences.

Abstract. Combinatorial proofs are given for certain entries in Ramanujan’s lost notebook. Bijections of Sylvester, Franklin, and Wright, and applications of Algorithm Z of Zeilberger are employed. A new bijection, involving the new concept of the parity sequence of a partition, is used to prove one of Ramanujan’s fascinating identities for a partial theta function. 1.

. We consider a new generalization of Euler's and Sylvester's identities for partitions. Our proof is based on an explicit bijection. 1. Main Results A partition of n is a sequence ( 1 ; 2 ; : : : ; l ) of positive integers such that 1 2 \Delta \Delta \Delta l ? 0 and P i = n. The numbers i are called parts of . Denote by l() the number l of parts in . One of the well-known facts in the theory of partitions is Euler's identity. Theorem (Euler, 1748). The number of partitions of n with odd parts is equal to the number of partitions of n with distinct parts. There exist several generalizations of Euler's identity (e.g. see [A2], [C]). One of them is Sylvester's identity. By A(n; k) denote the set of partitions of n into odd parts (repetitions allowed) with exactly k different parts. By B(n; k) denote the set of partitions = ( 1 ? 2 ? \Delta \Delta \Delta ? l ) of n such that the sequence ( 1 \Gammal; 2 \Gammal+1; : : : ; l \Gamma1) has exactly k different elements...

Abstract. In answer to a question of Andrews about finding combinatorial proofs of two identities in Ramanujan’s “Lost ” Notebook, we obtain weighted forms of Euler’s theorem on partitions with odd parts and distinct parts. This work is inspired by the insight of Andrews on the connection between Ramanujan’s identities and Euler’s theorem. Our combinatorial formulations of Ramanujan’s identities rely on the notion of rooted partitions. Iterated Dyson’s map and Sylvester’s bijection are the main ingredients in the weighted forms of Euler’s theorem. Keywords: partition, rooted partition, Euler’s theorem, Ramanujan’s identities, iterated Dyson’s map, Sylvester’s bijection