Record statistics in integer compositions

Arnold Knopfmacher; Toufik Mansour

Record statistics in integer compositions

Department of Mathematics

Research output: Contribution to conference › Paper › peer-review

Abstract

A composition σ = a ₁a ₂. .. a _m of n is an ordered collection of positive integers whose sum is n. An element ai in σ is a strong (weak) record if a _i > a _j (a _i ≥ a _j ) for all j = 1; 2;. . ., i - 1. Furthermore, the position of this record is i. We derive generating functions for the total number of strong (weak) records in all compositions of n, as well as for the sum of the positions of the records in all compositions of n, where the parts a _i belong to a fixed subset A of the natural numbers. In particular when A = ℕ, we find the asymptotic mean values for the number, and for the sum of positions, of records in compositions of n.

Original language	English
Pages	527-536
Number of pages	10
State	Published - 2009
Event	21st International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'09 - Linz, Austria Duration: 20 Jul 2009 → 24 Jul 2009

Conference

Conference	21st International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'09
Country/Territory	Austria
City	Linz
Period	20/07/09 → 24/07/09

Keywords

Asymptotic estimates
Composition
Generating function
Left-to-right maxima
Mellin transforms
Record

ASJC Scopus subject areas

Algebra and Number Theory

Cite this

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title = "Record statistics in integer compositions",

abstract = "A composition σ = a 1a 2. .. a m of n is an ordered collection of positive integers whose sum is n. An element ai in σ is a strong (weak) record if a i > a j (a i ≥ a j ) for all j = 1; 2;. . ., i - 1. Furthermore, the position of this record is i. We derive generating functions for the total number of strong (weak) records in all compositions of n, as well as for the sum of the positions of the records in all compositions of n, where the parts a i belong to a fixed subset A of the natural numbers. In particular when A = ℕ, we find the asymptotic mean values for the number, and for the sum of positions, of records in compositions of n.",

keywords = "Asymptotic estimates, Composition, Generating function, Left-to-right maxima, Mellin transforms, Record",

author = "Arnold Knopfmacher and Toufik Mansour",

year = "2009",

language = "English",

pages = "527--536",

note = "21st International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'09 ; Conference date: 20-07-2009 Through 24-07-2009",

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T1 - Record statistics in integer compositions

AU - Knopfmacher, Arnold

AU - Mansour, Toufik

PY - 2009

Y1 - 2009

N2 - A composition σ = a 1a 2. .. a m of n is an ordered collection of positive integers whose sum is n. An element ai in σ is a strong (weak) record if a i > a j (a i ≥ a j ) for all j = 1; 2;. . ., i - 1. Furthermore, the position of this record is i. We derive generating functions for the total number of strong (weak) records in all compositions of n, as well as for the sum of the positions of the records in all compositions of n, where the parts a i belong to a fixed subset A of the natural numbers. In particular when A = ℕ, we find the asymptotic mean values for the number, and for the sum of positions, of records in compositions of n.

AB - A composition σ = a 1a 2. .. a m of n is an ordered collection of positive integers whose sum is n. An element ai in σ is a strong (weak) record if a i > a j (a i ≥ a j ) for all j = 1; 2;. . ., i - 1. Furthermore, the position of this record is i. We derive generating functions for the total number of strong (weak) records in all compositions of n, as well as for the sum of the positions of the records in all compositions of n, where the parts a i belong to a fixed subset A of the natural numbers. In particular when A = ℕ, we find the asymptotic mean values for the number, and for the sum of positions, of records in compositions of n.

KW - Asymptotic estimates

KW - Composition

KW - Generating function

KW - Left-to-right maxima

KW - Mellin transforms

KW - Record

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M3 - Paper

AN - SCOPUS:84860486488

SP - 527

EP - 536

T2 - 21st International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'09

Y2 - 20 July 2009 through 24 July 2009

ER -

Record statistics in integer compositions

Abstract

Conference

Keywords

ASJC Scopus subject areas

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