A Unifying Framework for the ν -Tamari Lattice and Principal Order Ideals in Young’s Lattice

Matias von Bell; Rafael S. González D’León; Francisco A. Mayorga Cetina; Martha Yip

doi:10.1007/s00493-023-00022-x

A Unifying Framework for the ν -Tamari Lattice and Principal Order Ideals in Young’s Lattice

Matias von Bell
, Rafael S. González D’León
, Francisco A. Mayorga Cetina
, Martha Yip

Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

We present a unifying framework in which both the ν -Tamari lattice, introduced by Préville-Ratelle and Viennot, and principal order ideals in Young’s lattice indexed by lattice paths ν , are realized as the dual graphs of two combinatorially striking triangulations of a family of flow polytopes which we call the ν -caracol flow polytopes. The first triangulation gives a new geometric realization of the ν -Tamari complex introduced by Ceballos et al. We use the second triangulation to show that the h^∗ -vector of the ν -caracol flow polytope is given by the ν -Narayana numbers, extending a result of Mészáros when ν 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.

Original language	English
Pages (from-to)	479-504
Number of pages	26
Journal	Combinatorica
Volume	43
Issue number	3
DOIs	https://doi.org/10.1007/s00493-023-00022-x
State	Published - Jun 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature.

Funding

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.

Funders	Funder number
Simons Collaboration	429920

Keywords

Flow polytope
Triangulation
Young’s lattice
ν-Dyck path
ν-Tamari lattice

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Computational Mathematics

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10.1007/s00493-023-00022-x

Cite this

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title = "A Unifying Framework for the ν -Tamari Lattice and Principal Order Ideals in Young{\textquoteright}s Lattice",

abstract = "We present a unifying framework in which both the ν -Tamari lattice, introduced by Pr{\'e}ville-Ratelle and Viennot, and principal order ideals in Young{\textquoteright}s lattice indexed by lattice paths ν , are realized as the dual graphs of two combinatorially striking triangulations of a family of flow polytopes which we call the ν -caracol flow polytopes. The first triangulation gives a new geometric realization of the ν -Tamari complex introduced by Ceballos et al. We use the second triangulation to show that the h∗ -vector of the ν -caracol flow polytope is given by the ν -Narayana numbers, extending a result of M{\'e}sz{\'a}ros when ν 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.",

keywords = "Flow polytope, Triangulation, Young{\textquoteright}s lattice, ν-Dyck path, ν-Tamari lattice",

author = "\{von Bell\}, Matias and \{Gonz{\'a}lez D{\textquoteright}Le{\'o}n\}, \{Rafael S.\} and \{Mayorga Cetina\}, \{Francisco A.\} and Martha Yip",

note = "Publisher Copyright: {\textcopyright} 2023, The Author(s), under exclusive licence to J{\'a}nos Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature.",

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T1 - A Unifying Framework for the ν -Tamari Lattice and Principal Order Ideals in Young’s Lattice

AU - von Bell, Matias

AU - González D’León, Rafael S.

AU - Mayorga Cetina, Francisco A.

AU - Yip, Martha

PY - 2023/6

Y1 - 2023/6

N2 - We present a unifying framework in which both the ν -Tamari lattice, introduced by Préville-Ratelle and Viennot, and principal order ideals in Young’s lattice indexed by lattice paths ν , are realized as the dual graphs of two combinatorially striking triangulations of a family of flow polytopes which we call the ν -caracol flow polytopes. The first triangulation gives a new geometric realization of the ν -Tamari complex introduced by Ceballos et al. We use the second triangulation to show that the h∗ -vector of the ν -caracol flow polytope is given by the ν -Narayana numbers, extending a result of Mészáros when ν 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.

AB - We present a unifying framework in which both the ν -Tamari lattice, introduced by Préville-Ratelle and Viennot, and principal order ideals in Young’s lattice indexed by lattice paths ν , are realized as the dual graphs of two combinatorially striking triangulations of a family of flow polytopes which we call the ν -caracol flow polytopes. The first triangulation gives a new geometric realization of the ν -Tamari complex introduced by Ceballos et al. We use the second triangulation to show that the h∗ -vector of the ν -caracol flow polytope is given by the ν -Narayana numbers, extending a result of Mészáros when ν 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.

KW - Flow polytope

KW - Triangulation

KW - Young’s lattice

KW - ν-Dyck path

KW - ν-Tamari lattice

UR - https://www.scopus.com/pages/publications/85163111544

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JO - Combinatorica

JF - Combinatorica

IS - 3

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A Unifying Framework for the ν -Tamari Lattice and Principal Order Ideals in Young’s Lattice

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

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