The Frobenius Complex

Eric Clark; Richard Ehrenborg

doi:10.1007/s00026-012-0127-8

The Frobenius Complex

Eric Clark
, Richard Ehrenborg

Mathematics

Producción científica: Article › revisión exhaustiva

6 Citas (SciVal)

Resumen

Motivated by the classical Frobenius problem, we introduce the Frobenius poset on the integers ℤ, that is, for a sub-semigroup Λ of the non-negative integers (ℕ, +), we define the order by n ≤ _Λm if m-n ∈ Λ. When Λ is generated by two relatively prime integers a and b, we show that the order complex of an interval in the Frobenius poset is either contractible or homotopy equivalent to a sphere. We also show that when Λ is generated by the integers {a, a + d, a + 2d, . . ., a + (a-1)d}, the order complex is homotopy equivalent to a wedge of spheres.

Idioma original	English
Páginas (desde-hasta)	215-232
Número de páginas	18
Publicación	Annals of Combinatorics
Volumen	16
N.º	2
DOI	https://doi.org/10.1007/s00026-012-0127-8
Estado	Published - jun 2012

Nota bibliográfica

Funding Information:
Acknowledgments. The second author was partially funded by National Science Foundation grant DMS-0902063. The authors thank Richard Stanley for pointing out the references [5,10, 14,16], Vic Reiner for pointing out [18], and Volkmar Welker for suggesting Corollary 6.2. We also thank Benjamin Braun, Margaret Readdy, and the referee who read earlier versions of this paper.

Financiación

Acknowledgments. The second author was partially funded by National Science Foundation grant DMS-0902063. The authors thank Richard Stanley for pointing out the references [5,10, 14,16], Vic Reiner for pointing out [18], and Volkmar Welker for suggesting Corollary 6.2. We also thank Benjamin Braun, Margaret Readdy, and the referee who read earlier versions of this paper.

Financiadores	Número del financiador
National Science Foundation (NSF)	0902063, DMS-0902063

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

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title = "The Frobenius Complex",

abstract = "Motivated by the classical Frobenius problem, we introduce the Frobenius poset on the integers ℤ, that is, for a sub-semigroup Λ of the non-negative integers (ℕ, +), we define the order by n ≤ Λm if m-n ∈ Λ. When Λ is generated by two relatively prime integers a and b, we show that the order complex of an interval in the Frobenius poset is either contractible or homotopy equivalent to a sphere. We also show that when Λ is generated by the integers \{a, a + d, a + 2d, . . ., a + (a-1)d\}, the order complex is homotopy equivalent to a wedge of spheres.",

keywords = "Morse matching, coin exchange, cylindrical posets, homotopy type, order complex",

author = "Eric Clark and Richard Ehrenborg",

note = "Funding Information: Acknowledgments. The second author was partially funded by National Science Foundation grant DMS-0902063. The authors thank Richard Stanley for pointing out the references [5,10, 14,16], Vic Reiner for pointing out [18], and Volkmar Welker for suggesting Corollary 6.2. We also thank Benjamin Braun, Margaret Readdy, and the referee who read earlier versions of this paper.",

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AU - Clark, Eric

AU - Ehrenborg, Richard

N1 - Funding Information: Acknowledgments. The second author was partially funded by National Science Foundation grant DMS-0902063. The authors thank Richard Stanley for pointing out the references [5,10, 14,16], Vic Reiner for pointing out [18], and Volkmar Welker for suggesting Corollary 6.2. We also thank Benjamin Braun, Margaret Readdy, and the referee who read earlier versions of this paper.

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N2 - Motivated by the classical Frobenius problem, we introduce the Frobenius poset on the integers ℤ, that is, for a sub-semigroup Λ of the non-negative integers (ℕ, +), we define the order by n ≤ Λm if m-n ∈ Λ. When Λ is generated by two relatively prime integers a and b, we show that the order complex of an interval in the Frobenius poset is either contractible or homotopy equivalent to a sphere. We also show that when Λ is generated by the integers {a, a + d, a + 2d, . . ., a + (a-1)d}, the order complex is homotopy equivalent to a wedge of spheres.

AB - Motivated by the classical Frobenius problem, we introduce the Frobenius poset on the integers ℤ, that is, for a sub-semigroup Λ of the non-negative integers (ℕ, +), we define the order by n ≤ Λm if m-n ∈ Λ. When Λ is generated by two relatively prime integers a and b, we show that the order complex of an interval in the Frobenius poset is either contractible or homotopy equivalent to a sphere. We also show that when Λ is generated by the integers {a, a + d, a + 2d, . . ., a + (a-1)d}, the order complex is homotopy equivalent to a wedge of spheres.

KW - Morse matching

KW - coin exchange

KW - cylindrical posets

KW - homotopy type

KW - order complex

UR - https://www.scopus.com/pages/publications/84862149677

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U2 - 10.1007/s00026-012-0127-8

DO - 10.1007/s00026-012-0127-8

M3 - Article

AN - SCOPUS:84862149677

SN - 0218-0006

VL - 16

SP - 215

EP - 232

JO - Annals of Combinatorics

JF - Annals of Combinatorics

IS - 2

ER -

The Frobenius Complex

Resumen

Nota bibliográfica

Financiación

ASJC Scopus subject areas

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