Abstract
In this paper we define and study for a finite partially ordered set P a class of simplicial complexes on the set \(P_r\) of r-element multichains of P. The simplicial complexes depend on a strictly monotone function from [r] to [2r]. We show that there are exactly \(2^r\) such functions which yield subdivisions of the order complex of P, of which \(2^{r-1}\) are pairwise different. Within this class are, for example, the order complexes of the intervals in P, the zig-zag poset of P, and the \(r{\hbox {th}}\) edgewise subdivision of the order complex of P. We also exhibit a large subclass for which our simplicial complexes are order complexes and homotopy equivalent to the order complex of P.
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Acknowledgements
We are grateful to the referees for their careful reading of the manuscript and their helpful suggestions. In particular, the open questions on connections to commutative algebra are inspired by their comments.
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Communicated by Kolja Knauer.
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Nazir, S., Welker, V. On the Homeomorphism and Homotopy Type of Complexes of Multichains. Ann. Comb. 27, 229–247 (2023). https://doi.org/10.1007/s00026-022-00626-y
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DOI: https://doi.org/10.1007/s00026-022-00626-y