Euler’s partition theorem and refinements without appeal to infinite products. (English) Zbl 07293156
Pillwein, Veronika (ed.) et al., Algorithmic combinatorics: enumerative combinatorics, special functions and computer algebra. Proceedings of the workshop on combinatorics, special functions and computer algebra (Paule60), Research Institute of Symbolic Computation (RISC), Hagenberg, Austria, May 17–18, 2018. In honour of Peter Paule on his 60th birthday. Cham: Springer. Texts Monogr. Symb. Comput., 9-23 (2020).
Summary: We present a new proof of Euler’s celebrated theorem that the number of partitions of an integer into odd parts equals the number of partitions of that integer into distinct parts without using infinite product representations but only certain “amalgamation” properties and dissections of their series generating functions. Our approach also yields a simpler and more direct proof of a well known refinement of Euler’s theorem due to Fine. We then use the same ideas along with conjugation of partitions into distinct parts to improve Fine’s theorem and also to obtain a dual of a refinement of Euler’s theorem due to Bessenrodt. Finally we discuss links between our approach and the Rogers-Fine identity.
For the entire collection see [Zbl 1455.68025].
For the entire collection see [Zbl 1455.68025].
MSC:
| 68W30 | Symbolic computation and algebraic computation |
Keywords:
partitions; odd parts; distinct parts; Euler’s theorem; series generating functions; amalgamation; even-odd split; 2-modular Ferrers graphs; refinementReferences:
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