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Monotonicity of partition functions

Published online by Cambridge University Press:  26 February 2010

P. T. Bateman  and
P. Erdös
P. T. Bateman
Affiliation:
University of Illinois, Urbana, Illinois
P. Erdös
Affiliation:
University of Notre Dame, Notre Dame, Indiana.
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Let A be an arbitrary set of positive integers (finite or infinite) other than the empty set or the set consisting of the single element unity. Let p(n) = pA(n) denote the number of partitions of the integer n into parts taken from the set A, repetitions being allowed. In other words, p(n) is the number of ways n can be expressed in the form n1a1 + n2a2 + …, where a1, a2, … are the distinct elements of A and n1, n2, … are arbitrary non-negative integers. In this paper we shall prove that p(n) is a strictly increasing function of n for sufficiently large n if and only if A has the following property (which we shall subsequently call property P1): A contains more than one element, and if we remove any single element from A, the remaining elements have greatest common divisor unity.

Type
Research Article
Information
Mathematika , Volume 3 , Issue 1 , June 1956 , pp. 1 - 14
Copyright
Copyright © University College London 1956

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References

1.Auluck, F. C. and Haselgrove, C. B., “On Ingham's Tauberian theorem for partitions”, Proc. Cambridge Phil. Soc., 48 (1952), 566–570.CrossRefGoogle Scholar
2.Bachmann, P., Niedere Zahlentheorie: Zweiter Teil, Additive Zahlentheorie, Ch. 3 (Leipzig, Teubner, 1910).Google Scholar
3.Brauer, A., “On a problem of partitions”, American J. of Math., 64 (1942), 299–312.CrossRefGoogle Scholar
4.Brigham, N. A., “A general asymptotic formula for partition functions”, Proc. American Math. Soc., 1 (1950), 182–191, especially p. 183.CrossRefGoogle Scholar
5.Ingham, A. E., “A Tauberian theorem for partitions”, Ann. of Math. (2), 42 (1941), 1075–1090, especially pp. 1084–1086.CrossRefGoogle Scholar
6.Knopp, K., “Asymptotische Formeln der additiven Zahlentheorie”, Schriften der Konigsherger gelehrten Gesellschaft (Naturwissenschaftliche Klasse), 2 (1925), 45–74, especially pp. 60–63.Google Scholar
7.Netto, E., Lehrbuch der Combinatorik, Ch. 6 (Leipzig, Teubner, 1901 and 1927).Google Scholar
8.Pólya, G. and Szegö, G., Aufgaben und Lehrsäze aus der Analysis, Vol. 1, Part I, Problem 27 (Berlin, Springer, 1925).Google Scholar
9.Rademacher, H., “On the expansion of the partition function in a series”, Ann. of Math. (2), 44 (1943), 416–422.Google Scholar
10.Roth, K. F. and Szekeres, G., “Some asymptotic formulae in the theory of partitions”, Quart. J. of Math. (2), 5 (1954), 241–259.CrossRefGoogle Scholar