Some Observations on Dyson's New Symmetries of Partitions
Combinatorics
2007-05-23 v4 Number Theory
Quantum Algebra
Abstract
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.
Keywords
Cite
@article{arxiv.math/0203111,
title = {Some Observations on Dyson's New Symmetries of Partitions},
author = {Alexander Berkovich and Frank G. Garvan},
journal= {arXiv preprint arXiv:math/0203111},
year = {2007}
}
Comments
27 pages, 15 figures, appendix B added, additional references, some typos eliminated, to appear in Journal of Combinatorial Theory, Series A