Merging-Free Partitions and Run-Sorted Permutations

TL;DR

This paper investigates merging-free partitions and run-sorted permutations, providing combinatorial proofs, distribution analyses of permutation statistics, and algorithms for permutation generation, advancing understanding of their combinatorial structures.

Contribution

It offers a combinatorial proof of a conjecture, characterizes distributions of permutation statistics, and develops an algorithm for generating run-sorted permutations.

Findings

Number of right-to-left minima follows a shifted Stirling distribution.

Non-crossing merging-free partitions are counted by powers of 2.

An algorithm for exhaustive generation of run-sorted permutations is presented.

Abstract

In this paper, we study merging-free partitions with their canonical forms and run-sorted permutations. We give a combinatorial proof of the conjecture made by Nabawanda et al. We describe the distribution of the statistics of runs and right-to-left minima over the set of run-sorted permutations and we give the exponential generating function for their joint distribution. We show the number of right-to-left minima is given by the shifted distribution of the Stirling number of the second kind. We also prove that the non-crossing merging-free partitions are enumerated by powers of 2. We use one of the constructive proofs given in the paper to implement an algorithm for the exhaustive generation of run-sorted permutations by number of runs.