Some Observations on Dyson's New Symmetries of Partitions

Details Some Observations on Dyson's New Symmetries of Partitions Some Observations on Dyson's New Symmetries of Partitions

2002-03-12

arxiv.org/abs/math/0203111v4

Mathematics

Combinatorics

27 pages, 15 figures, appendix B added, additional references, some typos eliminated, to appear in Journal of Combinatorial Th

Scientific paper

We utilize Dyson's concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler's Pentagonal Number Theorem. We streamline Dyson's bijection relating partitions with crank <= k and those with k in the Rank-Set of partitions. Also, we extend Dyson's adjoint of a partition to MacMahon's ``modular'' partitions with modulus 2. This way we find a new combinatorial proof of Gauss'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.

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