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Monotonicity of partition functions. (English) Zbl 0074.03502
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 $f_k(X) = \sum_{n=0}^\infty p^{(k)} (n) X^n=(1-X)^k \sum_{n=0}^\infty p(n) X^n =$ $= (1-X)^k \prod_{a \in A} (1-X^a)^{-1}.$ 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 $$n \to \infty$$. Then $$p^{(k)} (n) n^{-c} \to +\infty$$ for any fixed $$c$$. Also, if $\varrho^{(k)}(n) = p^{(k+1)}(n)/p^{(k)}(n) = 1-p^{(k)}(n-1)/p^{(k)} \mid (n),$ then $$\varrho^{(k)}(n) \to 0$$, $$n\varrho^{(k)}(n)$$ is unbounded above and $$n\varrho^{(k-1)}(n) \to +\infty$$.
Reviewer: E.M.Wright

##### MSC:
 11P82 Analytic theory of partitions
Number Theory
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##### References:
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