Some observations on Dyson's new symmetries of partitions

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:100

Article

Berkovich, A ; Garvan, FG

关键词: partitions;    congruences;    Dyson's rank;    cranks;    Euler's pentagonal number theorem;    modular partitions;    q-series;    polynomial analogs;   

DOI  :  10.1006/jcta.2002.3281

来源: Elsevier

【 摘 要 】

We utilize Dyson's concept of the adjoint of a partition to derive an infinite family of new polynomial analogs of Euler's Pentagonal Number Theorem. We streamline Dyson's bijection relating partitions with crank less than or equal tok 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 colubinatorial proof that for n > 1 the partitions of n with crank k are equinumerous with partitions of n with crank -k. (C) 2002 Elsevier Science (USA).

【 授权许可】

Free