Ranking And Unranking Permutations WithApplications

TLDR: This paper examines and implements alternative algorithms to the Rank permutations and NthPermutations already contained in the standard add-on Mathematica packages, and provides some applications of these ranking procedures in different topics such as probability, statistics and elementary calculus.

Abstract: Permutation theory has many applications in several fields of science and technology and it also has a charm in itself. Mathematica is particularly suitable for writing combinatorial algorithms because it provides many easy-to-use tools for handling lists. Several combinatorial built-in functions which involve permutations and combinations are available as standard add-on Mathematica packages. They are grouped under the name of DiscreteMath and are described by their author Steven Skiena in his book [7]. In this paper we focus on ranking and unranking procedures and we examine and implement alternative algorithms to the RankPermutations and NthPermutations already contained in the above mentioned add-on packages. Moreover we provide some applications of these ranking procedures in different topics such as probability, statistics and elementary calculus. Let /„={(), 1, ..., n-\\ } be the set of the nonnegative integers smaller than n and let A be a rc-set. We denote the set of all permutations of A by the n!-set: