Compositions with distinct parts
type: Item
instance of: scholarly article
title: Compositions with distinct parts
zbMATH Open document ID: 0824.11065
DOI: 10.1007/BF01827930
author: V. Pereyra
published in: Aequationes Mathematicae
publication date: 1995-11-15T00:00:00Z
full work available at URL: https://eudml.org/doc/182473
review text: Let \(q(n,k)\) denote the number of partitions of \(n\) into \(k\) distinct parts. The function \(q(n,k)\) has been studied in detail by \textit{G. Szekeres} [Q. J. Math., Oxf. II. Ser. 2, 85-108 (1951; Zbl 0042.041); ibid. 4, 96-111 (1953; Zbl 0050.041)]. In the paper under review the authors prove some analogous results for \(C(n,k)= k! q(n,k)\), the number of compositions of \(n\) with \(k\) distinct parts. Let \(n= m(m+1)/2+ j\), \(0\leq j\leq m\), \(n\geq n_ 0\). Theorem 1 asserts the unimodality of the sequence \(\{C (n, k)\}\): there is an integer \(k_ 1\) such that \[ C(n,1)< C(n,2)< \dots< C(n, k_ 1)\geq C(n, k_ 1+ 1)> \dots > C(n,m). \] Theorem 2 yields an asymptotic estimate for \(C(n)= \sum_{k=1}^ m C(n,k)\). The proof of Theorem 2 shows that the \(C(n,k)/ C(n)\) tend to a normal distribution with mean \(k_ 1\) and standard deviation \(\sim (2m/ \log m)^{1/2}\).
Mathematics Subject Classification ID: 11P82
Mathematics Subject Classification ID: 05A16
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 729147
zbMATH Keywords: unequal partitions
zbMATH Keywords: unequal compositions
zbMATH Keywords: unimodality
reviewed by: Mihály Szalay
MaRDI profile type: Publication
cites work: The asymptotic number of set partitions with unequal block sizes
cites work: SOME ASYMPTOTIC FORMULAE IN THE THEORY OF PARTITIONS
cites work: Q4294628
cites work: AN ASYMPTOTIC FORMULA IN THE THEORY OF PARTITIONS
cites work: SOME ASYMPTOTIC FORMULAE IN THE THEORY OF PARTITIONS (II)
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description: scientific article; zbMATH DE number 729147