Some Observations on Dyson's New Symmetries of Partitions

Abstract

We utilize Dyson&#39;s concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler&#39;s Pentagonal Number Theorem. We streamline Dyson&#39;s bijection relating partitions with crank <= k and those with k in the Rank-Set of partitions. Also, we extend Dyson&#39;s adjoint of a partition to MacMahon&#39;s ``modular&#39;&#39; partitions with modulus 2. This way we find a new combinatorial proof of Gauss&#39;s famous identity. We give a direct combinatorial proof that for n>1 the partitions of n with crank k are equinumerous with partitions of n with crank -k.