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The Frobenius Complex

Abstract : 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.
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Eric Clark, Richard Ehrenborg. The Frobenius Complex. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.649-660, ⟨10.46298/dmtcs.2816⟩. ⟨hal-01186244⟩

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