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On Schur's second partition theorem

Published online by Cambridge University Press:  18 May 2009

George E. Andrews
George E. Andrews
Affiliation:
The Pennsylvania State UniversityUniversity Park, Pennsylvania
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In 1926, I. J. Schur proved the following theorem on partitions [3].

The number of partitions of n into parts congruent to ±1 (mod 6) is equal to the number of partitions of n of the form 1 + …+bs = n, where bi–bi+1 ≧ 3 and, if 3 ∣ bi, then bi–bi+1 > 3.

Schur's proof was based on a lemma concerning recurrence relations for certain polynomials. In 1928, W. Gleissberg gave an arithmetic proof of a strengthened form of Schur's theorem [2]; however, the combinatorial reasoning in Gleissberg's paper becomes very intricate.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 8 , Issue 2 , July 1967 , pp. 127 - 132
Copyright
Copyright © Glasgow Mathematical Journal Trust 1967

References

1.Dienes, P., The Taylor series, Dover (New York, 1957).Google Scholar
2.Gleissberg, W., Über einen Satz von Herrn I. Schur, Math. Z. 28 (1928), 372–382.Google Scholar
3.Schur, I. J., Zur additiven Zahlentheorie, S.-B. Akad. Wiss. Berlin (1926), 488–495.Google Scholar
4.Slater, L. J., Generalized hypergeometric functions, Cambridge University Press (Cambridge, 1966).Google Scholar