The Frobenius Complex

Eric Clark, Richard Ehrenborg

Research output: Contribution to journalArticlepeer-review

6 Citations (SciVal)


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 languageEnglish
Pages (from-to)215-232
Number of pages18
JournalAnnals of Combinatorics
Issue number2
StatePublished - 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.


  • Morse matching
  • coin exchange
  • cylindrical posets
  • homotopy type
  • order complex

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


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