2002/03/12 by Alexander Bérkovich, Alexander Berkovich, Berkovich, Alexander +3 · 2 citations
Mathematics · #05A17 #11P81 #11P83 #33D15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Quantum Algebra (math.QA) #math.CO #math.NT #math.QA #msc:05A17 #msc:11P81 #msc:11P83 #msc:33D15
paper · pdf · doi:10.48550/arxiv.math/0203111
27 pages, 15 figures, appendix B added, additional references, some typos eliminated, to appear in Journal of Combinatorial Theory, Series A
openalex publication_date 2002/03/12 · arxiv created 2002/04/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.