Enumeration of Permutation Classes and Weighted Labelled Independent Sets

TL;DR

This paper introduces a staircase encoding for permutations, linking permutation classes to weighted independent sets in graphs, enabling enumeration, analysis of Wilf-equivalences, and random sampling methods.

Contribution

It establishes a novel bijection between permutation classes and weighted independent sets, deriving generating functions and enabling enumeration and sampling.

Findings

Derived generating functions for independent sets and their weighted versions.

Enumerated several permutation classes and identified unbalanced Wilf-equivalences.

Outlined methods for random sampling within permutation classes.

Abstract

In this paper, we study the staircase encoding of permutations, which maps a permutation to a staircase grid with cells filled with permutations. We consider many cases, where restricted to a permutation class, the staircase encoding becomes a bijection to its image. We describe the image of those restrictions using independent sets of graphs weighted with permutations. We derive the generating function for the independent sets and then for their weighted counterparts. The bijections we establish provide the enumeration of permutation classes. We use our results to uncover some unbalanced Wilf-equivalences of permutation classes and outline how to do random sampling in the permutation classes. In particular, we cover the classes $\mathrm{Av}(2314,3124)$, $\mathrm{Av}(2413,3142)$, $\mathrm{Av}(2413,3124)$, $\mathrm{Av}(2413,2134)$ and $\mathrm{Av}(2314,2143)$, as well as many subclasses.