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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

11P82 Analytic theory of partitions
Number Theory
Full Text: DOI
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