The Stable Tamari Lattice
Abstract
We introduce the stable Tamari lattice 𝒮𝒯n, an infinite lattice on ℕn. The r-Tamari lattices on r-Dyck paths of height n+1 occur as a nested family of principal lower order ideals whose union is 𝒮𝒯n. Partition-indexed principal lower order ideals recover the ν-Tamari lattices associated with partition boundaries. We give an explicit coordinate description of the stable Tamari order in terms of interval vectors determined by the lower endpoint. These coordinates characterize comparability and covering relations, and their sum gives the maximum length of a chain between two comparable elements. Analogous coordinates determined by the upper endpoint give an explicit formula for the Möbius function. We also construct an EL-labeling of every interval and show that every interval in 𝒮𝒯n is isomorphic to an interval in a classical Tamari lattice. We classify the possible h-vectors by Dyck paths and show that each h-vector fiber is parametrized by a unimodular basis of interval vectors. Combining this parametrization with the lower-endpoint coordinates, we obtain a finite rational formula for the multivariate generating function of the height generating functions of all principal lower order ideals. Under the r-Tamari realization, these height generating functions recover the interval generating functions of Bergeron and Préville-Ratelle.
Related
BibTeX
@misc{chau2026stable,
title = {{The Stable Tamari Lattice}},
author = {Herman Chau and Spencer Daugherty and Sean Grate and Mark D. Haiman and Juan Carlos Martínez Mori and Anna Pun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39257},
eprint = {2610.39257},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39257},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Stable Tamari Lattice
AU - Herman Chau
AU - Spencer Daugherty
AU - Sean Grate
AU - Mark D. Haiman
AU - Juan Carlos Martínez Mori
AU - Anna Pun
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39257
UR - https://arxiv.org/abs/2610.39257
AB - We introduce the stable Tamari lattice 𝒮𝒯_n, an infinite lattice on ℕ^n. The r-Tamari lattices on r-Dyck paths of height n+1 occur as a nested family of principal lower order ideals whose union is 𝒮𝒯_n. Partition-indexed principal lower order ideals recover the ν-Tamari lattices associated with partition boundaries. We give an explicit coordinate description of the stable Tamari order in terms of interval vectors determined by the lower endpoint. These coordinates characterize comparability and covering relations, and their sum gives the maximum length of a chain between two comparable elements. Analogous coordinates determined by the upper endpoint give an explicit formula for the Möbius function. We also construct an EL-labeling of every interval and show that every interval in 𝒮𝒯_n is isomorphic to an interval in a classical Tamari lattice. We classify the possible h-vectors by Dyck paths and show that each h-vector fiber is parametrized by a unimodular basis of interval vectors. Combining this parametrization with the lower-endpoint coordinates, we obtain a finite rational formula for the multivariate generating function of the height generating functions of all principal lower order ideals. Under the r-Tamari realization, these height generating functions recover the interval generating functions of Bergeron and Préville-Ratelle.
ER -