ACDRepo/partial_orders_on_lattice_paths_10x9

Dataset Card for Partial Orders on Lattice Paths from $(0,0)(0,0)$ to $(10,9)(10,9)$

Consider northeast lattice paths that travel along the edges of a grid from $(0,0)(0,\; 0)$ to $(a,b)(a,\; b)$, only taking steps north and east and never passing through the diagonal $y=\frac{b}{a}xy\; =\; \backslash frac\{b\}\{a\}x$, where $aa$ and $bb$ are relatively prime. [1] defines two order relations on such paths called the matching ordering $({\le }_M)(\backslash leq\_M)$ and the Lagrange ordering $({\le }_L)(\backslash leq\_L)$, motivated by questions in number theory. The matching ordering assigns a number to each lattice path based on the number of perfect matchings of an associated snake graph, while the Lagrange ordering assigns a number to each lattice path equal to the Lagrange number of a certain continued fraction. These numbers each define a partial order. Mathematicians are interested in better understanding the relationship between these orders [2].

Background on Posets

A partially ordered set (poset) is a set $PP$ of objects equipped with a binary relation, typically denoted $\le \backslash leq$, that is reflexive, antisymmetric, and transitive. This means that for all elements $a,b,c\in Pa,b,c\; \backslash in\; P$: (1) $a\le aa\; \backslash leq\; a$, (2) if $a\le ba\; \backslash leq\; b$ and $b\le ab\; \backslash leq\; a$, then $b=ab\; =\; a$, and (3) if $a\le ba\; \backslash leq\; b$ and $b\le cb\; \backslash leq\; c$, then $a\le ca\; \backslash leq\; c$. Unlike total orders which are more familiar (e.g., $\mathbb{Z}\backslash mathbb\{Z\}$), in a partial order some pairs of elements may be incomparable. An example of a partially ordered set is the set of all subsets of $\{1,2,3,4\}\backslash \{1,\; 2,\; 3,\; 4\backslash \}$, ordered by inclusion. This is a partial order and not a total order because $\{1,2\}\backslash \{1,\; 2\backslash \}$ is not comparable to $\{2,3\}\backslash \{2,\; 3\backslash \}$ or to $\{2,3,4\}\backslash \{2,\; 3,\; 4\backslash \}$, for example. In a poset, $yy$ covers $xx$ if $yy$ is greater than $xx$ with respect to the ordering, and for any $zz$ such that $x\le z\le yx\; \backslash leq\; z\; \backslash leq\; y$, either $z=xz\; =\; x$ or $z=yz\; =\; y$. In this example, $\{1,2,4\}\backslash \{1,\; 2,\; 4\backslash \}$ covers $\{1,2\}\backslash \{1,\; 2\backslash \}$, $\{2,4\}\backslash \{2,\; 4\backslash \}$, and $\{1,4\}\backslash \{1,\; 4\backslash \}$, but not $\{1\}\backslash \{1\backslash \}$, $\{2\}\backslash \{2\backslash \}$, or $\{4\}\backslash \{4\backslash \}$.

Dataset Details

This dataset contains pairs of lattice paths encoded by a sequence of $11$’s (for steps east) and $00$’s (for steps north). Each pair of lattice paths is a covering pair in exactly one of the two partial orders, the Lagrange order or the matching order (pairs that are covers in both are few and were removed). The task is to predict which partial order a covering pair belongs to.

Statistics

	Lagrange order	Matching order	Total instances
Train	7,558	3,875	11,433
Test	1,895	968	2,863

Math question: Characterize the relationship between the matching and Lagrange orders.

ML task: Given a pair of lattice paths $(w,w\text{′})(w,\; w\prime )$, train a model that can predict whether $w\text{′}w\prime$ covers $ww$ in either the matching or Lagrange order.

Small model performance

We provide some basic baselines for this task. Benchmarking details can be found in the associated paper.

Size	Logistic regression	MLP	Transformer	Guessing largest class
Paths from $(0,0)(0,0)$ to $(10,9)(10,9)$	$66.2\mathrm{\%}66.2\backslash \%$	$90.6\mathrm{\%}\pm 0.8\mathrm{\%}90.6\backslash \%\; \backslash pm\; 0.8\backslash \%$	$65.3\mathrm{\%}\pm 0.0\mathrm{\%}65.3\backslash \%\; \backslash pm\; 0.0\backslash \%$	$66.2\mathrm{\%}66.2\backslash \%$

The $\pm \backslash pm$ signs indicate 95% confidence intervals from random weight initialization and training.

Further information

• Curated by: Helen Jenne
• Funded by: Pacific Northwest National Laboratory
• Language(s) (NLP): NA
• License: CC-by-2.0

Dataset Sources

Data generation scripts can be found here.

• Repository: ACD Repo

Citation

BibTeX:

@article{chau2025machine,
    title={Machine learning meets algebraic combinatorics: A suite of datasets capturing research-level conjecturing ability in pure mathematics},
    author={Chau, Herman and Jenne, Helen and Brown, Davis and He, Jesse and Raugas, Mark and Billey, Sara and Kvinge, Henry},
    journal={arXiv preprint arXiv:2503.06366},
    year={2025}
}

APA:

Chau, H., Jenne, H., Brown, D., He, J., Raugas, M., Billey, S., & Kvinge, H. (2025). Machine learning meets algebraic combinatorics: A suite of datasets capturing research-level conjecturing ability in pure mathematics. arXiv preprint arXiv:2503.06366.

Dataset Card Contact

Henry Kvinge, acdbenchdataset@gmail.com

References

[1] Schiffler, Ralf. "Perfect matching problems in cluster algebras and number theory." Open Problems in Algebraic Combinatorics 110 (2024): 361.
[2] Apruzzese, P. J., and Kevin Cong. "On two orderings of lattice paths." arXiv preprint arXiv:2310.16963 (2023).