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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 1297520
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Abstract: Schur's partition theorem asserts the equality $ S(n) = {S_1}(n)$, where $ S(n)$ is the number of partitions of $ n$ into distinct parts $ \equiv 1,2(\mod 3)$ and $ {S_1}(n)$ is the number of partitions of $ n$ into parts with minimal difference $ 3$ and no consecutive multiples of $ 3$. Using a computer search Andrews found a companion result $ S(n) = {S_2}(n)$, where $ {S_2}(n)$ is the number of partitions of $ n$ whose parts $ {e_i}$ satisfy $ {e_i} - {e_{i + 1}} \geqslant 3,2or5$ according as $ {e_i} \equiv 1,2$ or $ (\bmod 3)$. 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 $ {S_1}(n)$ and $ {S_2}(n)$ are only two of six companion partition functions $ {S_j}(n),j = 1,2, \ldots 6$, all equal to $ S(n)$. A three parameter refinement and generalization of these results is obtained.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1297520-X
PII: S 0002-9947(1995)1297520-X
Article copyright: © Copyright 1995 American Mathematical Society