Refined product formulas for Tamari intervals

TL;DR

The paper proves two explicit product formulas for the f-vectors of two Tamari- and associahedron-derived complexes: $a_{n,k}$ for the canonical complex of the Tamari lattice and $b_{n,k}$ for the cellular diagonal of the associahedron, with $a_{n,k} = \frac{2}{n(n+1)} \binom{n+1}{k+2} \binom{3n}{k}$ and $b_{n,k} = \frac{2}{(3n+1)(3n+2)} \binom{n-1}{k} \binom{4n+1-k}{n+1}$. It develops both analytic and bijective viewpoints: an analytic proof using grafting decompositions, a Lagrange-inversion extraction, and a binomial-identity derivation, plus multiple bijective interpretations via canopy agreements, Dyck paths, and triangulations with Schnyder woods. The paper also links these counts to well-studied combinatorial families (Tamari intervals, rooted triangulations, and cellular diagonals) and discusses q-analogues, offering conjectured $q$-identities and highlighting potential proofs via computer-aided techniques. Overall, the work provides compact, explicit descriptions of two fundamental f-vectors tied to Tamari and associahedron structures, enriching the combinatorial understanding of these polytopal and lattice-theoretic objects.

Abstract

We provide short product formulas for the $f$-vectors of the canonical complexes of the Tamari lattices and of the cellular diagonals of the associahedra.

Figures (7)

• Figure 1: The canonical complex of the Tamari lattice. Left: The Tamari lattice $\mathrm{Tam}(2)$ seen on binary trees (top) and on semi-crossing arc bidiagrams (middle), and the canonical complex of $\mathrm{Tam}(2)$ (bottom), with $f$-vector~$(1, 6, 6)$. Right: The canonical complex of $\mathrm{Tam}(3)$, with $f$-vector~$(1,12, 33, 22)$.
• Figure 2: Left: The $2$-dimensional associahedron with its faces labeled by Schröder trees with $4$ leaves (in particular, its vertices correspond to binary trees). Middle: The cellular diagonal $\Delta_2$ with its faces labeled by pairs of Schröder trees given by the magical formula (in particular, its vertices correspond to Tamari intervals). Right: The decomposition of the cellular diagonal $\Delta_2$ obtained by associating each face $(F,G)$ to the Tamari interval $\max(F) \le \min(G)$. The $f$-vector is $(13,18,6)$.
• Figure 3: All grafting decompositions of a binary tree.
• Figure 4: A grafting decomposition of a Tamari interval.
• Figure 5: The binary trees $S_k'$ for $0 \le k \le 3$ obtained from the binary tree $S$ of \ref{['fig:graftingDecompositionTree']} in the proof of \ref{['prop:quadraticEquationA']}.

Theorems & Definitions (40)

• Theorem 1
• Theorem 2
• Example 3
• Proposition 4
• proof
• Proposition 5: MasudaThomasTonksVallette
• Example 6
• Proposition 7
• proof
• Remark 8