Cambrian Acyclic Domains: Counting c-singletons

Abstract

We study the size of certain acyclic domains that arise from geometric and combinatorial constructions. These acyclic domains consist of all permutations visited by commuting equivalence classes of maximal reduced decompositions if we consider the symmetric group and, more generally, of all c-singletons of a Cambrian lattice associated to the weak order of a finite Coxeter group. For this reason, we call these sets Cambrian acyclic domains. Extending a closed formula of Galambos–Reiner for a particular acyclic domain called Fishburn’s alternating scheme, we provide explicit formulae for the size of any Cambrian acyclic domain and characterize the Cambrian acyclic domains of minimum or maximum size.

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Acknowledgements

The authors would like to thank Vic Reiner for pointing out his article with Galambos which initiated this work, and Cesar Ceballos and Vincent Pilaud for helpful discussions and their hospitality in Paris and Toronto. Finally, the authors thank the anonymous referees for their conscientious work and their numerous constructive suggestions that help to improve the article.

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Correspondence to Jean-Philippe Labbé.

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This work was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”. The first author was partially supported by a FQRNT Doctoral scholarship.

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Labbé, JP., Lange, C.E.M.C. Cambrian Acyclic Domains: Counting c-singletons. Order 37, 571–603 (2020). https://doi.org/10.1007/s11083-019-09520-4

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Keywords

  • Acyclic sets
  • Enumeration
  • Generalized permutahedra
  • Pseudoline arrangements
  • Sortable elements
  • Coxeter groups