Sur les sous-sommes d’une partition. (On subsums of a partition).

*(French)*Zbl 0684.10047Let \(\Pi =\{a_ 1+a_ 2+...+a_ s=n\); \(a_ 1\geq a_ 2\geq...\geq a_ s>0\}\) 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.

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 |

##### Keywords:

subsums of partition; nonpractical partitions; asymptotic expansion; gaps; asymptotic results##### References:

[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 |

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