Multi-level Loop-less Algorithm for Multi-set Permutations

TL;DR

This paper introduces a new, efficient loop-less algorithm for generating multiset permutations in a unique order, combining existing methods and applying it to parking functions, with improvements in simplicity and speed.

Contribution

It presents a novel loop-less algorithm for multiset permutations that is simpler, faster, and generates permutations in a new order, also extending to parking functions.

Findings

Generates multiset permutations in O(1) time per permutation

Combines two existing loop-less algorithms for improved performance

Applies the algorithm to parking functions

Abstract

We present an algorithm that generates multiset permutations in O(1) time for each permutation, that is, by a loop-less algorithm with O(n) extra memory requirement. There already exist several such algorithms that generate multiset permutations in various orders. For multiset permutations, we combine two loop-less algorithms that are designed in the same principle of tree traversal. Our order of generation is different from any existing order, and the algorithm is simpler and faster than the previous ones. We also apply the new algorithm to parking functions.