README.md

---
license: cc-by-2.0
pretty_name: Partial orders on lattice paths from (0,0) to (10,9)
---

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

Consider northeast lattice paths that travel along the edges
of a grid from \\((0, 0)\\) to \\((a, b)\\), only taking steps north and
east and never passing through the diagonal \\(y = \frac{b}{a}x\\), where
\\(a\\) and \\(b\\) are relatively prime. \[1\] defines two
order relations on such paths called the *matching ordering*
\\((\leq_M)\\) and the *Lagrange ordering* \\((\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 \\(P\\) of objects
equipped with a binary relation, typically denoted \\(\leq\\), that
is reflexive, antisymmetric, and transitive. This means that
for all elements \\(a,b,c \in P\\): (1) \\(a \leq a\\), (2) if \\(a \leq b\\) and
\\(b \leq a\\), then \\(b = a\\), and (3) if \\(a \leq b\\) and \\(b \leq c\\), then \\(a \leq c\\).
Unlike total orders which are more familiar (e.g., \\(\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\}\\), ordered by inclusion. This is a partial order
and not a total order because \\(\{1, 2\}\\) is not comparable to
\\(\{2, 3\}\\) or to \\(\{2, 3, 4\}\\), for example. In a poset, \\(y\\) *covers* \\(x\\)
if \\(y\\) is greater than \\(x\\) with respect to the ordering, and for
any \\(z\\) such that \\(x \leq z \leq y\\), either \\(z = x\\) or \\(z = y\\). In this
example, \\(\{1, 2, 4\}\\) covers \\(\{1, 2\}\\), \\(\{2, 4\}\\), and \\(\{1, 4\}\\), but not
\\(\{1\}\\), \\(\{2\}\\), or \\(\{4\}\\).

## Dataset Details

This dataset contains pairs of lattice paths encoded by a sequence of \\(1\\)’s (for steps
east) and \\(0\\)’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′)\\), train a
model that can predict whether \\(w′\\) covers \\(w\\) 
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)\\) to \\((10,9)\\) | \\(66.2\%\\) | \\(90.6\% \pm 0.8\%\\) | \\(65.3\% \pm 0.0\%\\)| \\(66.2\%\\) |

The \\(\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](https://github.com/pnnl/ML4AlgComb/tree/master/lattice_path_posets).

- **Repository:** [ACD Repo](https://github.com/pnnl/ML4AlgComb/tree/master/lattice_path_posets)

## 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, [email protected]

## 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).