## 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
Full Text:

### References:

 [1] DOI: 10.2307/2371684 · Zbl 0061.06801 · doi:10.2307/2371684 [2] Bachmann, Niedere Zahlentheorie: Zweiter Teil, Additive Zahlentheorie (1910) [3] DOI: 10.1017/S0305004100076325 · doi:10.1017/S0305004100076325 [4] DOI: 10.2307/1968973 · Zbl 0060.10005 · doi:10.2307/1968973 [5] DOI: 10.2307/1970462 · Zbl 0063.02973 · doi:10.2307/1970462 [6] Pólya, Aufgaben und Lehrsäze aus der Analysis (1925) [7] DOI: 10.2307/2031921 · Zbl 0037.16903 · doi:10.2307/2031921 [8] Netto, Lehrbuch der Combinatorik (1901) [9] Knopp, Schriften der Konigsherger gelehrten Gesellschaft (Naturwissenschaftliche Klasse) 2 pp 45– (1925) [10] DOI: 10.1093/qmath/5.1.241 · Zbl 0057.03902 · doi:10.1093/qmath/5.1.241
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