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) = pA(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) = pA(k)(n) by the formal power-series relation

fk(X) = sumn = 0oo p(k) (n) Xn = (1-X)k sumn = 0oo p(n) Xn =

= (1-X)k proda in A (1-Xa)-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 Pk, 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 Pk 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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