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Sur les sous-sommes d’une partition. (On subsums of a partition). (French) Zbl 0684.10047
Let $$\Pi =\{a_ 1+a_ 2+...+a_ s=n$$; $$a_ 1\geq a_ 2\geq...\geq a_ s>0\}$$ be a generic partition of n where $$s=s(\Pi)$$ and $$a_ j$$ are integers. Each sum $$a_{i_ 1}+...+a_{i_ t}$$ $$(i_ 1<...<i_ t$$; $$t\geq 0)$$ is said to be a subsum of $$\Pi$$. Let $$\Sigma(\Pi)\subseteq [0,n]$$ be the set of integers representable by subsums of $$\Pi$$. The maximal sets of consecutive integers of $$\Sigma(\Pi)$$ (resp. of [0,n]$$\setminus \Sigma (\Pi))$$ are called the components (resp. the gaps) of $$\Pi$$. A partition $$\Pi$$ of n is said to be practical if $$\Sigma (\Pi)=[0,n]$$. Let p(n) denote the number of partitions of n. P. Erdős and the reviewer [Studies in pure mathematics, Mem. P. Turan, 187-212 (1983; Zbl 0523.10029)] proved that the number M(n) of nonpractical partitions of n is $$(1+O(n^{-1/2} \log^{30} n))\pi (6n)^{-1/2} p(n)$$. The author and J.-L. Nicolas [Coll. Math. Soc. J. Bolyai 51, 9-33 (1990)] obtained an asymptotic expansion for M(n)/p(n) in terms of powers of $$n^{-1/2}.$$
The paper under review is the first part of a series dealing with the structure of $$\Sigma(\Pi)$$; for parts II, III see the following reviews. In Chapter I of the present paper the author investigates the following generalization of the practical partitions. For a fixed positive integer a, $$\Pi$$ is said to be a-practical if the cardinalities of the gaps of $$\Pi$$ are less than a. Some characterizations are new for $$a=1$$, too. The main result refers to the number M(n,a) of partitions of n which are not a -practical. The author obtains an asymptotic expansion for M(n,a)/p(n) in terms of powers of $$n^{-1/2}$$, as $$n\to \infty$$. The first term is $$a!(\pi (6n)^{-1/2})^ a.$$
In chapters II and III the author studies the first two components of $$\Sigma(\Pi)$$ and the partitions having one or two gaps. Chapters IV and V contain asymptotic results for the number of partitions (of n) not representing certain given integers by subsums. In Chapter VI the above- mentioned estimations of $$M(n)/p(n)$$ are generalized for partitions without small parts. The following corollary refers to $$n\to \infty$$ and fixed positive a. Almost all partitions $$\Pi$$ of n such that $$\Sigma (\Pi)\cap \{1,2,...,a\}=\emptyset$$ verify $$\Sigma (\Pi)=\{a+1,a+2,a+3,...,n-a-1\}$$. The paper is closed with 7 problems.
Reviewer: M.Szalay

##### MSC:
 11P81 Elementary theory of partitions
Full Text:
##### References:
  J. DIXMIER et J.L. NICOLAS , Partitions without small parts , to be published in the proceedings of Colloquium in number theory, Budapest, July 1987 . Zbl 0707.11072 · Zbl 0707.11072  P. ERDÖS and M. SZALAY , On some problems of J. Dénes and P. Turán , Studies in pure Mathematics to the memory of P. Turán, Editor P. Erdös, Budapest 1983 , p. 187-212. MR 87g:11131 | Zbl 0523.10029 · Zbl 0523.10029
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