Generating the nth Lexicographical Element of a Mathematical k-Permutation using Permutational Number System

Abstract

This paper is about generating the nth lexicographical element of a mathematical k-permutation using a proposed concept called permutational number system and its mapping called Deep code to and from the nth k-permutation. Permutational Number System or Permutadic for short is a number system based on permutations of numbers.

The main application of the permutational number system is to rapidly compute the k-permutation at a given position 'n' (also known as nth k-permutation), in the lexicographical ordering starting from zero, without explicitly computing all the permutations preceding it. This concept is important because the total number of permutations can grow astronomically large. For instance, a number of permutations of 100 elements selected 50 at a time are 100P50 = 3.068518756 x 10^93, which is way beyond the practical limit to be generated sequentially to reach the desired permutation.

Keywords: Combinatorics, Number Theory, Factorial Number System, Factoradic, Combinatorial Number System, Combinadic, Permutational Number System, Permutadic