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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
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 $S(n) = {S_1}(n)$, 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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Article copyright: © Copyright 1995 American Mathematical Society