Ranking and Unranking Restricted Permutations

TL;DR

This paper presents efficient algorithms for ranking and unranking restricted permutations like derangements and me9nage permutations, using rook theory to count permutations with given prefixes, with broad applications.

Contribution

It introduces a novel method to efficiently unrank restricted permutations by reducing the problem to counting permutations with specific prefixes using rook theory.

Findings

Algorithms for unranking derangements and me9nage permutations

Reduction of unranking problem to prefix counting via rook theory

Applications in combinatorics, probability, and statistics

Abstract

We discuss efficient methods for unranking derangements and m\'enage permutations. That is, we will provide an algorithm to efficiently extract the $k$-th earliest such permutation under the lexicographic ordering. We will show that this problem can be reduced to the problem of computing the number of restricted permutations with a given prefix, and then we will use rook theory to solve this counting problem. This has applications to combinatorics, probability, statistics, and modeling.