---
license: mit
task_categories:
- text-generation
language:
- en
tags:
- mathematics
- group-theory
- permutations
- symbolic-reasoning
- algebra
- sequence-modeling
pretty_name: Permutation Groups Composition Dataset
size_categories:
- 10M<n<100M
---

# Permutation Groups Composition Dataset

A comprehensive collection of permutation composition datasets for various mathematical groups including symmetric, alternating, cyclic, dihedral, and special groups, with multiple sequence length variants.

## Dataset Description

This dataset contains permutation composition problems across 30 different mathematical groups with 8 different sequence length variants each, totaling 270 distinct configurations.

### Supported Groups

#### Symmetric Groups (Sn)
- **S3** to **S7**: All permutations of n elements (orders: 6, 24, 120, 720, 5040)

#### Alternating Groups (An)
- **A3** to **A7**: Even permutations of n elements (orders: 3, 12, 60, 360, 2520)

#### Cyclic Groups (Cn/Zn)
- **C3** to **C12**: Cyclic groups of order n
- **Z3** to **Z6**: Alternative notation for cyclic groups
- Orders: 3, 4, 5, 6, 7, 8, 10, 12

#### Dihedral Groups (Dn)
- **D3** to **D8**: Symmetries of regular n-gons (orders: 6, 8, 10, 12, 14, 16)

#### Special Groups
- **PSL(2,5)**: Projective special linear group (order 60)
- **F20**: Frobenius group F(5,4) (order 20)

### Length Variants

Each group is available with 8 different maximum sequence lengths:
- 2², 2³, 2⁴, 2⁵, 2⁶, 2⁷, 2⁸, 2⁹ (4, 8, 16, 32, 64, 128, 256, 512)

Each dataset consists of sequences of permutations that need to be composed to produce a target permutation. This is useful for:
- Training models on algebraic reasoning and symbolic computation
- Evaluating mathematical understanding and compositional generalization
- Benchmarking sequence models on structured mathematical tasks
- Studying group theory properties in neural networks
- Research in abstract algebra and computational mathematics

## Usage

```python
from datasets import load_dataset

# NEW: Clean API - just specify group and max_len
s5_data = load_dataset(
    "BeeGass/permutation-groups",
    name="s5",          # Just the group name
    max_len=32,         # Optional: filter sequences ≤ 32 (default: 512)
    trust_remote_code=True
)

# More examples with the clean API
c8_short = load_dataset("BeeGass/permutation-groups", name="c8", max_len=16, trust_remote_code=True)
d4_medium = load_dataset("BeeGass/permutation-groups", name="d4", max_len=64, trust_remote_code=True)
all_short = load_dataset("BeeGass/permutation-groups", name="all", max_len=32, trust_remote_code=True)

# Backwards compatibility - old style still works
s5_data = load_dataset("BeeGass/permutation-groups", name="s5_data", trust_remote_code=True)
s5_len32 = load_dataset("BeeGass/permutation-groups", name="s5_len32", trust_remote_code=True)

# Access the data
train_data = s5_data["train"]
test_data = s5_data["test"]

# Example data point
print(train_data[0])
# {'input_sequence': '23 45 12', 'target': '67'}
```

## Dataset Structure

Each example contains:
- `input_sequence`: A space-separated sequence of permutation IDs to be composed
- `target`: The ID of the resulting permutation after composition

The composition follows standard mathematical convention: for input `[p1, p2, p3]`, the result is `p3 ∘ p2 ∘ p1`.

## Available Configurations

### Efficient Loading (Recommended)
Simply specify the group name and optional `max_len` parameter:

```python
# Load any group with any max sequence length
dataset = load_dataset("BeeGass/permutation-groups", name="s5", max_len=32, trust_remote_code=True)
```

### All Groups (30 total)
| Group | Type | Order | Example Usage |
|-------|------|-------|---------------|
| S3-S7 | Symmetric | 6-5040 | `name="s5", max_len=64` |
| A3-A7 | Alternating | 3-2520 | `name="a4", max_len=32` |
| C3-C12 | Cyclic | 3-12 | `name="c8", max_len=16` |
| Z3-Z6 | Cyclic (alt) | 3-6 | `name="z5", max_len=128` |
| D3-D8 | Dihedral | 6-16 | `name="d4", max_len=256` |
| PSL25 | PSL(2,5) | 60 | `name="psl25", max_len=64` |
| F20 | Frobenius | 20 | `name="f20", max_len=32` |
| all | Combined | - | `name="all", max_len=16` |

### Legacy Configuration Names
For backwards compatibility, old-style names still work:
- `s5_data` (equivalent to `name="s5"`)
- `s5_len32` (equivalent to `name="s5", max_len=32`)
- etc.

## Dataset Features

- **Variable sequence length**: Input sequences range from 3 to maximum configured length
- **Length-specific variants**: 8 different maximum lengths for each group (2² to 2⁹)
- **Consistent formatting**: All permutations use space-separated integer IDs
- **Metadata included**: Each dataset includes a `metadata.json` file mapping IDs to permutation array forms
- **Train/test split**: 80/20 split for all configurations
- **Scaled sample sizes**: Shorter sequences have more samples for efficient training

## Understanding the Data

Each permutation is represented by a unique integer ID. The `metadata.json` file in each dataset folder provides the mapping from IDs to permutation array forms.

For example, in S3:
- ID 0 might map to `[0, 1, 2]` (identity)
- ID 1 might map to `[0, 2, 1]` (transpose elements 1 and 2)
- etc.

## Citation

If you use this dataset in your research, please cite:

```bibtex
@software{permutation_groups_dataset,
  author = {Bryan Gass},
  title = {Permutation Groups Dataset},
  year = {2024},
  publisher = {Hugging Face},
  url = {https://huggingface.co/datasets/BeeGass/permutation-groups}
}
```

## Acknowledgments

This dataset was inspired by the work of [William Merrill](https://github.com/viking-sudo-rm) and his paper ["The Illusion of State in State-Space Models"](https://arxiv.org/abs/2404.08819), which explores the computational properties of state-space models through group theory.

## License

This dataset is released under the MIT License.

## Contact

For questions or issues, please open an issue on the [GitHub repository](https://github.com/BeeGass/permutation-groups).