Some Observations on Dysonʹs New Symmetries of Partitions

We utilize Dysonʹ concept of the adjoint of a partition to derive an infinite family of new polynomial analogs of Eulerʹs Pentagonal Number Theorem. We streamline Dysonʹs bijection relating partitions with crank ⩽k and those with k in the Rank-Set of partitions. Also, we extend Dysonʹs adjoint of a partition to MacMahonʹs “modular” partitions with modulus 2. This way we find a new combinatorial proof of Gaussʹs 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.