Schur's partition theorem, companions, refinements and generalizations

Authors:
Krishnaswami Alladi and Basil Gordon

Journal:
Trans. Amer. Math. Soc. **347** (1995), 1591-1608

MSC:
Primary 11P83; Secondary 05A17, 05A19, 11P81

DOI:
https://doi.org/10.1090/S0002-9947-1995-1297520-X

MathSciNet review:
1297520

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Abstract: Schur's partition theorem asserts the equality , where is the number of partitions of into distinct parts and is the number of partitions of into parts with minimal difference and no consecutive multiples of . Using a computer search Andrews found a companion result , where is the number of partitions of whose parts satisfy according as or . By means of a new technique called the method of weighted words, a combinatorial as well as a generating function proof of both these theorems are given simultaneously. It is shown that and are only two of six companion partition functions , all equal to . A three parameter refinement and generalization of these results is obtained.

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DOI:
https://doi.org/10.1090/S0002-9947-1995-1297520-X

Article copyright:
© Copyright 1995
American Mathematical Society