Equipopularity Classes in the Separable Permutations

TL;DR

This paper classifies patterns in separable permutations that occur with equal frequency, revealing a connection between equipopularity classes and partitions of integers.

Contribution

It provides a complete classification of equipopular patterns in separable permutations and links these classes to integer partitions.

Findings

Number of equipopularity classes equals the number of partitions of n-1.

Structural and analytic tools used for classification.

Equipopularity classes correspond to partitions of n-1.

Abstract

When two patterns occur equally often in a set of permutations, we say that these patterns are equipopular. Using both structural and analytic tools, we classify the equipopular patterns in the set of separable permutations. In particular, we show that the number of equipopularity classes for length $n$ patterns in the separable permutations is equal to the number of partitions of $n-1$.