Update README.md

README.md

@@ -1,25 +1,23 @@
 ---
 license: cc-by-2.0
-pretty_name: Partial orders on lattice paths
+pretty_name: Partial orders on lattice paths from (0,0) to (10,9)
 ---
 
-# Dataset Card for Partial Orders on Lattice Paths, \\((0,0)\\) to \\((10 \times 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 
+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 the respective partial order. Mathematicians would be interested to 
-better understand the relationship between these orders \[2\].
-We provide datasets for lattice paths from \\((0,0)\\) to \\((10, 9)\\),
-\\((11, 10)\\), \\((12, 11)\\), and \\((13, 12)\\).
+numbers each define a partial order. Mathematicians are interested in 
+better understanding the relationship between these orders \[2\].
 
 ## Background on Posets
 
@@ -41,22 +39,13 @@ example, \\(\{1, 2, 4\}\\) covers \\(\{1, 2\}\\), \\(\{2, 4\}\\), and \\(\{1, 4\
 
 ## Dataset Details
 
-This dataset contains pairs of lattice paths starting at \\((0, 0)\\) and ending at \\((n, n − 1)\\) 
-that are only allowed to take one unit steps either north or east, and must stay below the line \\(y =
-\frac{n}{n−1}x\\). They are thus encoded by a sequence of \\(1\\)’s (for steps
-east) and \\(0\\)’s (for steps north) of length \\((n + 1) + n = 2n + 1\\). Each pair of lattice paths 
+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.
 
-Each line in a file is the concatenation of two \\(0\\)-\\(1\\) sequences (one for each path) for a length \\(4n + 2\\) row of \\(0\\)’s and \\(1\\)’s. The
-lattice paths are separated by ‘;’. For an 3 × 2 grid, the sequence:
-`1, 1, 1, 0, 0; 1, 1, 0, 1, 0`
-corresponds to the two lattice paths. The first moves right three steps and then up two. 
-The second moves right two steps, up one, right one, and up one. 
-
-The datasets can also be found [here](https://drive.google.com/file/d/1Wm9mtZQjXXQ4rl0TU9KtJ1T4RQaGsJNz/view?usp=sharing).
-Data loaders can be found [here](https://github.com/pnnl/ML4AlgComb/tree/master/lattice_path_posets).
+Data loaders can be found [here](https://github.com/pnnl/ML4AlgComb/tree/master).
 
 **Statistics**
 |  | Lagrange order | Matching order |
@@ -64,17 +53,13 @@ Data loaders can be found [here](https://github.com/pnnl/ML4AlgComb/tree/master/
 | Train | 7,589 |  3,875 |
 | Test  | 1,895 | 968 |
 
-**Math question:** Find an algorithm or set of rules that can efficiently distinguish between 
-weaving pattern matrices and non-weaving pattern matrices. This should be more efficient than the 
-\\(O(n^2\\) algorithm that can be found in the references above.
+**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.
 
-If a successful model is trained, it would be interesting to understand whether the model has 
-learned an existing algorithm or whether it has discovered something new.
-
 ## Small model performance
 
 We provide some basic baselines for this task. Benchmarking details can be found in the associated paper.
@@ -95,47 +80,6 @@ The \\(\pm\\) signs indicate 95% confidence intervals from random weight initial
 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)
-  
-## Uses
-
-This dataset was generated to study ML model's ability yield insight on an open 
-problem in algebraic combinatorics, specifically, a characterization of Lagrange and partial orders
-of lattice paths.
-
-### Direct Use
-
-We use this dataset for a classification task distinguishing between covers in the Lagrange and partial orders.
-
-### Out-of-Scope Use
-
-None.
-
-## Dataset Structure
-
-Each line in a file is the concatenation of two \\(0\\)-\\(1\\) sequences (one for each path) for a length \\(4n + 2\\) row of \\(0\\)’s and \\(1\\)’s. The
-lattice paths are separated by ‘;’. For an 3 × 2 grid, the sequence:
-`1, 1, 1, 0, 0; 1, 1, 0, 1, 0`
-corresponds to the two lattice paths. The first moves right three steps and then up two. 
-The second moves right two steps, up one, right one, and up one. 
-
-## Dataset Creation
-
-Data generation scripts can be found 
-[here](https://github.com/pnnl/ML4AlgComb/tree/master/lattice_path_posets).
-
-### Curation Rationale
-
-This dataset was generated as a test of current AI system's ability to advance 
-research mathematics.
-
-#### Who are the source data producers?
-
-Helen Jenne wrote code to generate this dataset using [SageMath](https://www.sagemath.org/).
-
-## Bias, Risks, and Limitations
-
-We only provide data for lattice paths from \\(10 \times 9\\), \\(11 \times 10\\), and \\(12 \times 11\\) in this repository. 
-We are happy to generate (subsets) of datasets for larger values of \\(n \times n-1\)).
 
 ## Citation