zbMATH — the first resource for mathematics

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

11P81 Elementary theory of partitions
Full Text: DOI Numdam EuDML
[1] 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
[2] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.