T-POP in Tamari Lattices

• T-POP is the specialization of Defant’s pop-stack-sorting operator applied to Tamari lattices, characterized by Dyck paths and 312-avoiding permutations.
• It yields a rational generating function for t-Pop sortable elements and identifies the image with a Motzkin-number family.
• The analysis leverages combinatorial models, lattice congruence techniques, and pattern avoidance to extend classical pop-stack sorting to structured lattices.

“T-POP” (Editor’s term) denotes the specialization of Defant’s pop-stack-sorting operator to Tamari lattices, that is, the dynamics of $\mathsf{Pop}_{\mathrm{Tam}_n}$ on the $n$-th Tamari lattice. Defant defined, for each complete meet-semilattice $M$, the operator

$$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}),$$

where $\lessdot$ is the covering relation. In the Tamari setting, $\mathrm{Tam}_n$ is the set of Dyck paths of semilength $n$, or equivalently, 312-avoiding permutations under weak order, and the resulting dynamics connect Catalan structures, lattice-theoretic covers, pattern avoidance, and orbit enumeration. The central results are an explicit rational generating function for $t$-$\mathsf{Pop}$-sortable elements and the identification of the image of $\mathsf{Pop}_{\mathrm{Tam}_n}$ with a Motzkin-number family (Hong, 2022).

1. Definition in the Tamari setting

For $n$0, the operator $n$1 is the meet of $n$2 together with all elements it covers. The minimal element of $n$3 is denoted $n$4. An element $n$5 is called $n$6-$n$7-sortable if

$n$8

and $n$9 denotes the number of $M$0-$M$1-sortable elements in $M$2 (Hong, 2022).

This formulation places T-POP in the broader program of extending classical pop-stack sorting from the symmetric group to lattice-theoretic and Coxeter-theoretic settings. In the Tamari case, the operator is especially tractable because the lattice admits several interchangeable models—Dyck paths, 312-avoiding permutations, and bracket-vector encodings—which support explicit structural and enumerative analysis. A later generalization to ornamentation lattices shows that when the underlying rooted plane tree is an $M$3-element chain, the ornamentation lattice is the $M$4-th Tamari lattice, so the Tamari case is the basic linear instance of a larger family (Ajran et al., 17 Jan 2025).

2. Enumeration of $M$5-$M$6-sortable elements

The principal enumerative theorem gives an explicit rational generating function: $M$7 where $M$8 is the $M$9-th Catalan number. This verifies Defant’s conjecture that for fixed $$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}),$$0, the generating function is rational (Hong, 2022).

The significance of this formula is twofold. First, it identifies a closed-form rational structure in an iterated lattice-dynamical process that a priori need not have any such regularity. Second, it shows that the dependence on the iteration depth $$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}),$$1 is controlled by Catalan coefficients, reinforcing that the Tamari case is governed not merely by generic lattice theory but by the specific algebra of Catalan combinatorics. The theorem is not only an existence result: it gives a concrete series for the full family of counting sequences $$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}),$$2.

A further structural ingredient is the decomposition of $$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}),$$3-$$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}),$$4-sortable bracket vectors into irreducible components. If $$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}),$$5 is the generating function for all $$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}),$$6-$$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}),$$7-sortable elements and $$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}),$$8 the generating function for irreducible ones, then

$$\mathsf{Pop}_M(x)=\bigwedge(\{y\in M: y\lessdot x\}\cup \{x\}),$$9

This relation exposes the enumerative mechanism behind the rational formula: irreducibles behave as the primitive combinatorial units, and general elements assemble from them by a free-sequence type construction (Hong, 2022).

3. Image of the operator and Motzkin enumeration

A second main theorem concerns the image $\lessdot$0. The refined generating function is

$\lessdot$1

At $\lessdot$2, the image size is a Motzkin number: $\lessdot$3 This settles a conjecture of Defant and Williams identifying the image size with a fundamental integer sequence (Hong, 2022).

The structural characterization of the image is equally important. Under the isomorphism between $\lessdot$4 and the lattice of 312-avoiding permutations under the right weak order, the image consists exactly of 312-avoiding permutations ending with $\lessdot$5 and with no double descents. The counting argument then passes through a reflection to 231-avoiding permutations and uses a known formula, attributed in the summary to Petersen, for permutations with $\lessdot$6 descents and $\lessdot$7 peaks: $\lessdot$8

This result clarifies that the image is not an opaque subset produced by an iterative meet operation. It has a direct pattern-avoidance characterization and a classical combinatorial enumeration. In later work on ornamentation lattices, the image of $\lessdot$9 and even the images of $\mathrm{Tam}_n$0 were further analyzed; for chains, this yields a complete characterization on Tamari lattices, and for $\mathrm{Tam}_n$1 the resulting generating function gives Motzkin numbers, recovering and generalizing Hong’s Tamari result (Ajran et al., 17 Jan 2025).

4. Proof architecture and combinatorial models

The original Tamari analysis proceeds by encoding Dyck paths as $\mathrm{Tam}_n$2-bracket vectors and describing the pop-stack operator explicitly through updates to these vectors. The argument then decomposes bracket vectors into irreducible components, derives generating-function recurrences for irreducible and general cases, and solves the resulting functional equations to obtain the rational formula for $\mathrm{Tam}_n$3 (Hong, 2022).

For the image theorem, the proof uses the weak-order model of Tamari lattices together with lattice congruence machinery. The summary specifically invokes results of Reading and Björner–Wachs: 312-avoiding permutations form sublattices and can be described as minimal representatives of sylvester-congruence classes. This transfer from Tamari-lattice elements to pattern-avoiding permutations is what makes the image characterization tractable.

These methods illustrate a recurring feature of T-POP. Although the operator is defined purely in terms of meets and covers, the effective analysis is carried out in auxiliary combinatorial languages: bracket vectors for iteration, weak-order embeddings for image structure, and classical pattern-avoidance statistics for enumeration. The Tamari case is therefore a meeting point of semilattice dynamics and Catalan combinatorics rather than a purely order-theoretic curiosity.

5. Extensions beyond the original Tamari analysis

Subsequent work substantially broadened the T-POP perspective. The paper on ornamentation lattices studies the pop-stack operator $\mathrm{Tam}_n$4 for a rooted plane tree $\mathrm{Tam}_n$5, where the ornamentation lattice of an $\mathrm{Tam}_n$6-element chain is the $\mathrm{Tam}_n$7-th Tamari lattice. It computes the maximum size of a forward orbit, characterizes the image of $\mathrm{Tam}_n$8, provides necessary conditions for membership in the image of $\mathrm{Tam}_n$9, and completely characterizes the image of $n$0 on a Tamari lattice (Ajran et al., 17 Jan 2025).

In that framework, the Tamari case appears as the linear prototype. The summary emphasizes several structural notions—minimal reductions, sections, beads, and hugs. A “no-hug” criterion generalizes Hong’s characterization of the image for Tamari lattices, while a beads condition controls iterated images. For chains, the paper gives a complete characterization and an explicit generating function for $n$1; for $n$2 this is the Catalan generating function, and for $n$3 it gives Motzkin numbers (Ajran et al., 17 Jan 2025).

A complementary line of work studies the image of $n$4 on other lattices, including the weak order of type $n$5, the Tamari lattice of type $n$6, and the lattices of order ideals of the root posets of types $n$7 and $n$8. In each case the focus is the generating function

$n$9

and the paper settles four conjectures of Defant and Williams on these images (Choi et al., 2022). This suggests that the Tamari case is the most explicit member of a larger image-enumeration program for pop operators on structured lattices.

6. Related operators and terminological ambiguity

T-POP in the Tamari sense should be distinguished from the Coxeter pop-tsack torsing operator, denoted $t$0, defined for a finite irreducible Coxeter group $t$1 with fixed Coxeter element $t$2 by

$t$3

where $t$4 is the join in the noncrossing partition lattice of the reflections lying weakly below $t$5 in absolute order. This operator is a “Bessis dual” version of the Coxeter pop-stack sorting map rather than the Tamari-lattice operator itself (Defant et al., 2021).

Its dynamics differ sharply from those of T-POP on Tamari lattices. In coincidental types and type $t$6, the identity element is the unique periodic point of $t$7, the maximum size of a forward orbit is the Coxeter number $t$8, and the forward orbit of $t$9 has size $\mathsf{Pop}$0 and is isolated in the sense that none of the non-identity elements of the orbit have preimages lying outside of the orbit (Defant et al., 2021). A later paper resolves conjectures on enumerating elements with near-maximal orbit length and gives complete classifications in types $\mathsf{Pop}$1, $\mathsf{Pop}$2, and $\mathsf{Pop}$3 (Li, 2022).

There is also an unrelated modern use of the acronym in machine learning: “T-POP: Test-Time Personalization with Online Preference Feedback,” an algorithm for LLM personalization via test-time alignment and dueling bandits (Qu et al., 29 Sep 2025). That usage is entirely separate from the Tamari-lattice literature. In combinatorics and lattice dynamics, T-POP is best understood as shorthand for the Tamari specialization of Defant’s pop-stack operator, centered on rational $\mathsf{Pop}$4-sortable enumeration, Motzkin-characterized images, and a growing web of generalizations across Catalan and Coxeter structures.