A central limit theorem for integer partitions into small powers

Gabriel F. Lipnik; Manfred G. Madritsch; Robert F. Tichy

doi:10.1007/s00605-023-01926-y

A central limit theorem for integer partitions into small powers

Gabriel F. Lipnik, Manfred G. Madritsch^*, Robert F. Tichy

^*Korrespondierende/r Autor/-in für diese Arbeit

Institut für Analysis und Zahlentheorie (5010)

Publikation: Beitrag in einer Fachzeitschrift › Artikel › Begutachtung

Abstract

The study of the well-known partition function p(n) counting the number of solutions to n= a₁+ ⋯ + a_ℓ with integers 1 ≤ a₁≤ ⋯ ≤ a_ℓ has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into n=⌊a1α⌋+⋯+⌊aℓα⌋ with 1 ≤ a₁< ⋯ < a_ℓ and some fixed 0 < α< 1 . In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle-point method.

Originalsprache	englisch
Seiten (von - bis)	149-173
Seitenumfang	25
Fachzeitschrift	Monatshefte fur Mathematik
Jahrgang	203
Ausgabenummer	1
DOIs	https://doi.org/10.1007/s00605-023-01926-y
Publikationsstatus	Veröffentlicht - Jan. 2024

ASJC Scopus subject areas

Allgemeine Mathematik

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10.1007/s00605-023-01926-yLizenz: CC BY 4.0

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AU - Tichy, Robert F.

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N2 - The study of the well-known partition function p(n) counting the number of solutions to n= a1+ ⋯ + aℓ with integers 1 ≤ a1≤ ⋯ ≤ aℓ has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into n=⌊a1α⌋+⋯+⌊aℓα⌋ with 1 ≤ a1< ⋯ < aℓ and some fixed 0 < α< 1 . In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle-point method.

AB - The study of the well-known partition function p(n) counting the number of solutions to n= a1+ ⋯ + aℓ with integers 1 ≤ a1≤ ⋯ ≤ aℓ has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into n=⌊a1α⌋+⋯+⌊aℓα⌋ with 1 ≤ a1< ⋯ < aℓ and some fixed 0 < α< 1 . In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle-point method.

KW - Central limit theorem

KW - Integer partitions

KW - Mellin transform

KW - Partition function

KW - Saddle-point method

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A central limit theorem for integer partitions into small powers

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