Ranking and unranking restricted permutations

Abstract

We provide computationally efficient methods for unranking derangements and ménage permutations. That is, we provide a polynomial-time algorithm to extract the $k$th such permutation under the lexicographic ordering. More generally, we show that there exists a polynomial-time algorithm for unranking words in lexicographic order whenever there exists a polynomial-time algorithm for counting the number of such words with a given prefix. We use rook theory to give a polynomial-time algorithm for counting the number of derangements and ménage permutations with a given prefix and in turn use this to give an unranking algorithm. This has applications to combinatorics, probability, modeling, and data compression.