A Recursive Algorithmic Approach to the Finding of Permutations for the Combination of Any Two Sets

TL;DR

This paper introduces a recursive algorithm for generating permutations of the union of two sets efficiently, representing permutations as binary integers and exploring their structure through a fractal-like tree graph.

Contribution

It presents a novel recursive algorithm for permutation generation of set unions, with potential applications in various scientific fields.

Findings

Algorithm efficiently finds all permutations of two set unions.

Permutations are represented as binary integers manipulated mathematically.

The permutation routes form a fractal-like tree structure.

Abstract

In this paper I present a conjecture for a recursive algorithm that finds each permutation of combining two sets of objects (AKA the Shuffle Product). This algorithm provides an efficient way to navigate this problem, as each atomic operation yields a permutation of the union. The permutations of the union of the two sets are represented as binary integers which are then manipulated mathematically to find the next permutation. The routes taken to find each of the permutations then form a series of associations or adjacencies which can be represented in a tree graph which appears to possess some properties of a fractal. This algorithm was discovered while attempting to identify every possible end-state of a Tic-Tac-Toe (Naughts and Crosses) board. It was found to be a viable and efficient solution to the problem, and now---in its more generalized state---it is my belief that it may find applications among a wide range of theoretical and applied sciences. I hypothesize that, due to the fractal-like nature of the tree it traverses, this algorithm sheds light on a more generic principle of combinatorics and as such could be further generalized to perhaps be applied to the union of any number of sets.