On linear intervals in the alt $ν$-Tamari lattices

TL;DR

This work extends the classical Tamari framework by introducing the alt $\nu$-Tamari posets $\operatorname{Tam}_{\nu}(\delta)$, a family of lattices parameterized by an increment vector $\delta$ relative to a fixed path $\nu$. It develops dual (path- and tree-based) representations, via $\nu$-paths, $\delta$-rotations, and $(\delta,\nu)$-trees, and shows each such poset is a lattice isomorphic to an interval in a larger $\check{\nu}$-Tamari lattice. Crucially, the authors prove that for fixed $\nu$, all alt $\nu$-Tamari lattices have the same number of linear intervals of any given length, attained through explicit bijections: horizontal flushings preserve left intervals, while reduced vertical flushings preserve right intervals. The results highlight a robust combinatorial structure across the entire family, with potential geometric realizations and connections to generalized diagonal harmonics and map enumerations. Overall, the paper provides a unified, lattice-theoretic and combinatorial framework for counting linear intervals in a broad class of $\nu$-Tamari-type posets.

Abstract

Given a lattice path $ν$, the $ν$-Tamari lattice and the $ν$-Dyck lattice are two natural examples of partial order structures on the set of lattice paths that lie weakly above $ν$. In this paper, we introduce a more general family of lattices, called alt $ν$-Tamari lattices, which contains these two examples as particular cases. Unexpectedly, we show that all these lattices have the same number of linear intervals.

Figures (23)

• Figure 1: The $\nu$-Tamari lattice and $\nu$-Dyck lattice for $\nu=ENEENN$. They are the alt $\nu$-Tamari lattices $\operatorname{Tam}_{\nu}(\delta)$ for $\delta=(2,0,0)$ and $\delta=(0,0,0)$, respectively.
• Figure 2: The alt $\nu$-Tamari lattice $\operatorname{Tam}_{\nu}(\delta)$ for $\nu=ENEENN$ and $\delta=(1,0,0)$.
• Figure 3: Examples of alt $\nu$-Tamari lattices $\operatorname{Tam}_{\nu}(\delta)$ for $\nu=ENEEN=(1,2,0)$. Left: the $\nu$-Dyck lattice, for $\delta=(0,0)$. Middle: the lattice for $\delta=(1,0)$. Right: the $\nu$-Tamari lattice, for $\delta=(2,0)$. In each case, the number of linear intervals of length $k$ is given by $\ell_k$ where $\ell=(\ell_0,\ell_1,\ell_2,\ell_3)=(7,8,4,1)$. For instance, 7 represents the trivial intervals of length 0, which are just the elements of each poset; there are 8 linear intervals of length 1, which correspond to the edges; 4 linear interval of length 2, and 1 linear interval of length 3.
• Figure 4: The rotation operation of a $\nu$-path. Each node is labelled with its $\nu$-altitude.
• Figure 5: The rotation operation of a $\nu$-tree.

Theorems & Definitions (74)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Proposition 2.6
• proof
• Corollary 2.7
• Definition 3.1