Mapping Uncharted Symmetries: Machine Discovery in Combinatorics

Alessandro Iraci; Eugenio Cainelli; Giovanni Paolini; Lorenzo Luccioli; Michele D'Adderio

arxiv: 2605.19063 · v1 · pith:5OPRPFEMnew · submitted 2026-05-18 · 💻 cs.LG

Mapping Uncharted Symmetries: Machine Discovery in Combinatorics

Eugenio Cainelli , Lorenzo Luccioli , Alessandro Iraci , Michele D'Adderio , Giovanni Paolini This is my paper

Pith reviewed 2026-05-20 11:45 UTC · model grok-4.3

classification 💻 cs.LG

keywords q,t-Narayana polynomialsnoncrossing partitionscombinatorial interpretationmachine discoveryalgebraic combinatoricssymmetry proofSLURPLean formalization

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The pith

Machine learning discovers a new combinatorial interpretation of the q,t-Narayana polynomials based on noncrossing partitions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors formalize a machine learning task called Simple Learning Under Rigid Proportions to search for exact mathematical functions that satisfy strict distributional constraints. They develop two search procedures, MapSeek-Functional and MapSeek-Symbolic, and apply them to an open question in algebraic combinatorics. The procedures surface statistics on noncrossing partitions that generate the q,t-Narayana polynomials arising from representation theory. This yields the first such combinatorial interpretation using noncrossing partitions. One of the surfaced statistics supplies a combinatorial proof of symmetry for these polynomials in a case that had remained open.

Core claim

The central discovery is a new combinatorial interpretation of the q,t-Narayana polynomials that arises from representation theory and is realized by statistics on noncrossing partitions. Machine learning under exact proportion constraints identifies these statistics, and one of them directly yields a combinatorial proof of the polynomials' symmetry in a previously unsolved case. All findings are formalized in Lean 4.

What carries the argument

Statistics on noncrossing partitions discovered by MapSeek-Functional and MapSeek-Symbolic that generate the q,t-Narayana polynomials and establish their symmetry.

If this is right

The interpretation supplies the first combinatorial model of these polynomials using noncrossing partitions.
One discovered statistic gives a direct combinatorial proof of symmetry for a case without prior proof.
The SLURP framework and MapSeek procedures are shown to produce verifiable mathematical objects in algebraic combinatorics.
Full Lean 4 formalization and released code make the discoveries immediately checkable by others.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same constrained search approach could be tried on other families of polynomials whose combinatorial interpretations remain incomplete.
Success here indicates that rigid-proportion constraints can guide machine search toward objects that admit human-readable proofs.
Combining the methods with proof assistants may create a repeatable workflow for generating candidate combinatorial identities.

Load-bearing premise

The statistics produced by the search procedures must reflect genuine combinatorial structure rather than artifacts of the training distribution or search heuristic.

What would settle it

Explicit enumeration of noncrossing partitions for small n and k must show that the proposed statistic reproduces the exact coefficients of the q,t-Narayana polynomial.

Figures

Figures reproduced from arXiv: 2605.19063 by Alessandro Iraci, Eugenio Cainelli, Giovanni Paolini, Lorenzo Luccioli, Michele D'Adderio.

**Figure 2.** Figure 2: Noncrossing partitions NC(4, 3). The corresponding q, t-Narayana is q 2+qt+t 2+q+t+1. 4 q, t-Combinatorics Setup Within algebraic combinatorics, the area of q, t-combinatorics studies families of bivariate polynomials F(q, t) with nonnegative integer coefficients arising in a variety of mathematical domains such as symmetric functions, representation theory, algebraic geometry, knot theory, probability, a… view at source ↗

**Figure 3.** Figure 3: Left: PCA scatter plot of the MapSeek-Functional successful runs (refined [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: A 12 × 7 polyomino with area 30 − 18 = 12 (left) and its bounce path (right); the bounce statistic is equal to 41 − 18 = 23. Definition A.1. The area statistic on PP(m, n) is the number of whole squares between the two paths, normalized by subtracting m + n − 1 (so that the minimum possible area is 0). Parallelogram polyominoes have a bounce path. To compute it, start from (0, 0), draw a single east step, … view at source ↗

**Figure 5.** Figure 5: The embedding used in Proposition A.3. A new bottom row is added, but only its leftmost [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗

**Figure 6.** Figure 6: An example of the bijection η. This is closely related to the known inv statistic on ordered set partitions [24]. The skip statistic from (2) can also be read directly from the drawing of a noncrossing partition. If β = {b1 < · · · < br} is a block, draw the edges joining consecutive elements in increasing order, namely b1b2, b2b3, . . . , br−1br, and ignore only the closing edge brb1. Each counted edge bi… view at source ↗

**Figure 7.** Figure 7: Computing the skip statistic from a noncrossing partition. The contribution of each counted [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗

**Figure 8.** Figure 8: Exchanging bijection for n = 4 and k = 3. A.4 The Min Gap Arc Statistic The min gap arc statistic (mingarc) is a human-defined extension of the flipped skew statistic discovered by our ML methods on NC(n, 3). It is defined on NC(n, k) for arbitrary k through the arc-selection procedure below and agrees with flipped skew when k = 3. Together with skip, it gives a combinatorial interpretation of Nn,k(q, t). … view at source ↗

**Figure 9.** Figure 9: Min gap arc construction on two noncrossing partitions. Solid red arcs are selected; dashed [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗

**Figure 10.** Figure 10: Overview of the objects and statistics covered by the Lean code. Thick edges indicate [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗

read the original abstract

Inspired by long-standing open problems in algebraic combinatorics, we show that modern machine learning can meaningfully contribute to verifiable mathematical discoveries. In particular, we focus on the construction of simple mathematical functions under exact distributional constraints, a setting we formalize as Simple Learning Under Rigid Proportions (SLURP). We tackle this problem by introducing two methods: MapSeek-Functional, which models the desired function alternating pseudo-labeling and supervised training steps; and MapSeek-Symbolic, designed to directly produce symbolic formulas. We successfully apply both methods to a research problem in algebraic combinatorics, discovering a new combinatorial interpretation of the $q,t$-Narayana polynomials arising from representation theory. To our knowledge, this is the first such interpretation based on noncrossing partitions. Using one discovered statistic, we find a combinatorial proof of the symmetry of these polynomials in a previously unsolved case. To streamline verification and reproducibility, we release all code, including a formalization of all the mathematical discoveries of this paper in Lean 4.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They found a new noncrossing partition model for the q,t-Narayana polynomials and proved symmetry in an open case, all formalized in Lean.

read the letter

The paper's main takeaway is straightforward: machine search turned up a combinatorial interpretation of the q,t-Narayana polynomials using noncrossing partitions, and one of the statistics gave a proof for a symmetry that had been open. Both results are stated explicitly and checked in Lean 4, with code released. That part is solid and independent of how the search was run once the objects are in hand. The authors frame this as the first such interpretation based on noncrossing partitions, which matches what they cite from prior work. The Lean formalization is the strongest part here because it lets anyone verify the identities without trusting the discovery step. The two methods, MapSeek-Functional and MapSeek-Symbolic, are presented as practical ways to handle exact distributional constraints under their SLURP setup, and they applied them successfully to this combinatorics problem. The symmetry proof is a direct combinatorial argument rather than something derived from fitted parameters. On the softer side, the description of the training distributions used to generate the SLURP instances is light on specifics, so it is hard to judge how much the search space was shaped by those choices. That is a minor gap rather than a flaw in the final claims, since the math stands on its own. The methods are adaptations of existing supervised and symbolic techniques rather than brand-new machinery. This paper is for algebraic combinatorists who work with q,t-analogs or noncrossing partitions and are curious about whether targeted search can surface useful objects in open cases. Readers who value machine-checked proofs will get the most out of it. It deserves a serious referee because the central objects and the proof are concrete and independently checkable. I would send it to review.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces the SLURP setting for discovering simple functions under exact distributional constraints and develops two methods (MapSeek-Functional, which alternates pseudo-labeling with supervised training, and MapSeek-Symbolic) to address it. These methods are applied to an open problem in algebraic combinatorics, yielding a new combinatorial interpretation of the q,t-Narayana polynomials in terms of noncrossing partitions (the first such interpretation) together with a combinatorial proof of symmetry in a previously unsolved case; all identities and the proof are formalized and machine-checked in Lean 4, with full code released.

Significance. If the results hold, the work demonstrates that modern machine-learning techniques can produce verifiable, parameter-free contributions to algebraic combinatorics by supplying the first noncrossing-partition interpretation of the q,t-Narayana polynomials and resolving an open symmetry question combinatorially. The explicit Lean 4 formalization of the discovered statistics, identities, and proof, together with the public release of all code, constitutes a major strength: the mathematical claims become independently verifiable and decoupled from the training procedure once found.

minor comments (3)

The precise proportions and sampling procedure used to generate the SLURP training instances are described only at a high level in the application section; adding an explicit table or pseudocode listing the exact distributional constraints would improve reproducibility of the discovery process.
Notation for the newly discovered statistics on noncrossing partitions is introduced inline; a dedicated subsection collecting the definitions, together with their q,t-generating functions, would aid readability.
The Lean 4 formalization is referenced in the abstract and conclusion, but the main text should include a direct pointer to the repository or a short summary of which theorems have been checked (e.g., the symmetry identity in the open case).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including recognition of the SLURP framework, the application to the q,t-Narayana polynomials, the first noncrossing-partition interpretation, the combinatorial symmetry proof, and the Lean 4 formalization with public code release. We note the recommendation for minor revision and will address any editorial or presentational suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper employs MapSeek-Functional and MapSeek-Symbolic to surface candidate statistics on noncrossing partitions, then extracts an explicit combinatorial interpretation of the q,t-Narayana polynomials together with a symmetry proof; both are stated as ordinary combinatorial objects and are independently formalized and machine-checked in Lean 4 with released code. Because the final mathematical statements are parameter-free, externally verifiable, and do not redefine any quantity in terms of the search procedure or fitted values, the derivation chain does not reduce to its inputs by construction. No self-definitional equations, fitted-input predictions, or load-bearing self-citations appear in the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard facts from algebraic combinatorics and representation theory (q,t-analogs, noncrossing partitions) plus the correctness of the Lean 4 kernel. No new free parameters are introduced to define the final statistic or proof; the machine-learning stage is used only for discovery and is not part of the mathematical claim.

axioms (1)

domain assumption Standard properties of q,t-Narayana polynomials and noncrossing partitions as defined in the algebraic combinatorics literature
Invoked when stating that the discovered statistic matches the known polynomials and when constructing the symmetry proof.

pith-pipeline@v0.9.0 · 5716 in / 1365 out tokens · 44761 ms · 2026-05-20T11:45:33.760999+00:00 · methodology

discussion (0)

Reference graph

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