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The Enumeration of the Partitions of Multipartite Numbers

Published online by Cambridge University Press:  24 October 2008

P. A. Macmahon
P. A. Macmahon
Affiliation:
St John's College
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Extract

This paper is a study of a new method of enumeration of the partitions of multipartite numbers.

Incidentally an algebraic function, which is derived from the repetitional exponents of partitions of unipartite numbers, presents itself. The generating function which enumerates the partitions of unipartite numbers is expressible in terms of these functions and finds in such expression its fullest connection with the divisors of numbers. There are also similarly derived functions connected directly with bipartite, tripartite, etc. numbers. It has not been necessary to study these for the purposes of this paper.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 6 , November 1925 , pp. 951 - 963
Copyright
Copyright © Cambridge Philosophical Society 1925

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References

* Combinatory Analysis, vol. I, pp. 264el seq.Google Scholar

* Mém. Ac. Sc. St Pétersbourg (7), 4, 1862, No. 2, 35 pp.Google Scholar

† Proc. Lond. Math. Soc. 34, 1901, pp. 3–15Google Scholar

* This result is not difficult to arrive at, but we may if we please derive it from

by referring to Laguerre, Bull. Soc. Math. France, 1, 1872–3, pp. 77–81Google Scholar, who showed that if

then F (n) = Σf (d), d a divisor of n.

* Journ. für Math. 54, 1857, pp. 21, 25.Google Scholar

* This number is given, erroneously, as 336 in Combinatory Analysis, vol. I, p. 269. Bead ‘The multipartite 55 has 339 partitions.’Google Scholar