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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

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