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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 074.03502

**Autor: ** Bateman, Paul T.; Erdös, Pál

**Title: ** Monotonicity of partition functions. (In English)

**Source: ** Mathematika, London 3, 1-14 (1956).

**Review: ** Let A be an arbitrary set of different positive integers (finite or infinite) other than the empty set or the set consisting of the single element unity. Let p(n) = p_{A}(n) denote the number of partitions of the integer n into parts taken from the set A, repetitions being allowed. Let k be any integer and suppose we define p^{(k)}(n) = p_{A}^{(k)}(n) by the formal power-series relation f_{k}(X) = **sum**_{n = 0}^{oo} p^{(k)} (n) X^{n} = (1-X)^{k} **sum**_{n = 0}^{oo} p(n) X^{n} =

= (1-X)^{k} **prod**_{a in A} (1-X^{a})^{-1}. For k \geq 0, the authors prove that p^{(k)}(n) is positive for all sufficiently large positive integers n if and only if A has the property P_{k}, viz. there are more than k elements in A and, if we remove an arbitrary subset of k elements from A, the remaining elements have greatest common divisor unity. Among a number of other results we may select as typical: For any k, let A infinite and have property P_{k} and let n ––> oo. Then p^{(k)} (n) n^{-c} ––> +oo for any fixed c. Also, if

\rho^{(k)}(n) = p^{(k+1)}(n)/p^{(k)}(n) = 1-p^{(k)}(n-1)/p^{(k)} | (n), then \rho^{(k)}(n) ––> 0, n\rho^{(k)}(n) is unbounded above and n\rho^{(k-1)}(n) ––> +oo.

**Reviewer: ** E.M.Wright

**Classif.: ** * 11P82 Analytic theory of partitions

**Index Words: ** Number Theory

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