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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Schur’s partition theorem, companions, refinements and generalizations
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by Krishnaswami Alladi and Basil Gordon PDF
Trans. Amer. Math. Soc. 347 (1995), 1591-1608 Request permission

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
  • © Copyright 1995 American Mathematical Society
  • 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