A Versatile Algorithm to Generate Various Combinatorial Structures

Abstract: Algorithms to generate various combinatorial structures find tremendous importance in computer science. In this paper, we begin by reviewing an algorithm proposed by Rohl that generates all unique permutations of a list of elements which possibly contains repetitions, taking some or all of the elements at a time, in any imposed order. The algorithm uses an auxiliary array that maintains the number of occurrences of each unique element in the input list. We provide a proof of correctness of the algorithm. We then show how one can efficiently generate other combinatorial structures like combinations, subsets, n-Parenthesizations, derangements and integer partitions & compositions with minor changes to the same algorithm.