Discrete Mathematics & Theoretical Computer Science |

- 1 Department of Mathematics

Motivated by the classical Frobenius problem, we introduce the Frobenius poset on the integers $\mathbb{Z}$, that is, for a sub-semigroup $\Lambda$ of the non-negative integers $(\mathbb{N},+)$, we define the order by $n \leq_{\Lambda} m$ if $m-n \in \Lambda$. When $\Lambda$ 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 $\Lambda$ is generated by the integers $\{a,a+d,a+2d,\ldots,a+(a-1)d\}$, the order complex is homotopy equivalent to a wedge of spheres.

Source: HAL:hal-01186244v1

Volume: DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)

Section: Proceedings

Published on: January 1, 2010

Imported on: January 31, 2017

Keywords: order complex,homotopy type,Morse matching,cylindrical posets,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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