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.
Keywords
- Morse matching
- coin exchange
- cylindrical posets
- homotopy type
- order complex
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics