# Syntactic View of Sigma-Tau Generation of Permutations

## TL;DR

This paper presents a syntactic perspective on the sigma-tau permutation generation, demonstrating its high compressibility and enabling efficient ranking and unranking algorithms.

## Contribution

It introduces a syntactic framework for the sigma-tau permutation generation sequence, achieving a compact representation and fast algorithms for permutation ranking and unranking.

## Key findings

- Sequence has $O(n^2 \, \log n)$ bit description
- Enables fast permutation ranking and unranking algorithms
- Provides a syntactic view of permutation generation

## Abstract

We give a syntactic view of the Sawada-Williams $(\sigma,\tau)$-generation of permutations. The corresponding sequence of $\sigma-\tau$-operations, of length $n!-1$ is shown to be highly compressible: it has $O(n^2\log n)$ bit description. Using this compact description we design fast algorithms for ranking and unranking permutations.