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A Unifying Framework for the \(\nu \)-Tamari Lattice and Principal Order Ideals in Young’s Lattice

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Abstract

We present a unifying framework in which both the \(\nu \)-Tamari lattice, introduced by Préville-Ratelle and Viennot, and principal order ideals in Young’s lattice indexed by lattice paths \(\nu \), are realized as the dual graphs of two combinatorially striking triangulations of a family of flow polytopes which we call the \(\nu \)-caracol flow polytopes. The first triangulation gives a new geometric realization of the \(\nu \)-Tamari complex introduced by Ceballos et al. We use the second triangulation to show that the \(h^*\)-vector of the \(\nu \)-caracol flow polytope is given by the \(\nu \)-Narayana numbers, extending a result of Mészáros when \(\nu \) is a staircase lattice path. Our work generalizes and unifies results on the dual structure of two subdivisions of a polytope studied by Pitman and Stanley.

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Acknowledgements

The second and fourth authors are extremely grateful to AIM and the SQuaRE group “Computing volumes and lattice points of flow polytopes” as some of the ideas of this work came from discussions within the group. In particular, we want to thank Alejandro Morales for the many enlightening discussions and explanations on triangulations of flow polytopes. Martha Yip is partially supported by Simons Collaboration Grant 429920. Rafael S. González D’León is very grateful with Universidad Sergio Arboleda and Pontificia Universidad Javeriana since part of this work happened under their support.

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Authors and Affiliations

  1. Institute for Geometry, Graz University of Technology, Graz, Austria

    Matias von Bell

  2. Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL, USA

    Rafael S. González D’León

  3. Escuela de Ciencias Exactas e Ingeniería, Universidad Sergio Arboleda, Bogotá, Colombia

    Francisco A. Mayorga Cetina

  4. Department of Mathematics, University of Kentucky, Lexington, KY, USA

    Martha Yip

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  1. Matias von Bell
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  2. Rafael S. González D’León
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  3. Francisco A. Mayorga Cetina
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  4. Martha Yip
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Correspondence to Rafael S. González D’León.

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von Bell, M., González D’León, R.S., Mayorga Cetina, F.A. et al. A Unifying Framework for the \(\nu \)-Tamari Lattice and Principal Order Ideals in Young’s Lattice. Combinatorica 43, 479–504 (2023). https://doi.org/10.1007/s00493-023-00022-x

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