On computing the total displacement number via weighted Motzkin paths

TL;DR

This paper introduces efficient algorithms for counting and sampling permutations with a given total displacement by leveraging the connection to weighted Motzkin paths, advancing understanding of this open combinatorial problem.

Contribution

It presents novel algorithms for counting and sampling permutations with specified displacement using weighted Motzkin paths, offering new computational tools.

Findings

Algorithms enable efficient counting and sampling of permutations.

New counting method based on building sequences for Motzkin paths.

Provides insights into the open problem of permutation displacement enumeration.

Abstract

Counting the number of permutations of a given total displacement is equivalent to counting weighted Motzkin paths of a given area (Guay-Paquet and Petersen, 2014). The former combinatorial problem is still open. In this work, we show that this connection allows to construct efficient algorithms for counting and for sampling such permutations. These algorithms provide a tool to better understand the original combinatorial problem. A by-product of our approach is a different way of counting based on certain building sequences for Motzkin paths, which may be of independent interest.