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.
| Original language | English |
|---|---|
| Pages (from-to) | 215-232 |
| Number of pages | 18 |
| Journal | Annals of Combinatorics |
| Volume | 16 |
| Issue number | 2 |
| DOIs | |
| State | Published - Jun 2012 |
Bibliographical 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.
Funding
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.
| Funders | Funder number |
|---|---|
| National Science Foundation (NSF) | 0902063, DMS-0902063 |
Keywords
- Morse matching
- coin exchange
- cylindrical posets
- homotopy type
- order complex
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics