Equipopularity Classes of 132-Avoiding Permutations

Lynn Chua,Krishanu Roy Sankar

doi:10.37236/3675

Abstract

The popularity of a pattern $p$ in a set of permutations is the sum of the number of copies of $p$ in each permutation of the set. We study pattern popularity in the set of 132-avoiding permutations. Two patterns are equipopular if, for all $n$, they have the same popularity in the set of length-$n$ 132-avoiding permutations. There is a well-known bijection between 132-avoiding permutations and binary plane trees. The spines of a binary plane tree are defined as the connected components when all edges connecting left children to their parents are deleted, and the spine structure is the sorted sequence of lengths of the spines. Rudolph shows that patterns of the same length are equipopular if their associated binary plane trees have the same spine structure. We prove the converse of this result using the method of generating functions, which gives a complete classification of 132-avoiding permutations into equipopularity classes.

Highlights

IntroductionGiven a binary plane tree, we label the root with n, and if the left subtree has i vertices, the values n − i,
Let σ = σ1 · · · σn be a permutation in the symmetric group Sn
The spines of a binary plane tree are defined as the connected components when all edges connecting left children to their parents are deleted, and the spine structure is the sorted sequence of lengths of the spines

Summary

Introduction

Given a binary plane tree, we label the root with n, and if the left subtree has i vertices, the values n − i, . We can label the tree recursively, and the permutation it corresponds to can be recovered by doing an in-order reading of the vertices (left subtree, root, right subtree). Given 132-avoiding permutations p and q, if T (p) and T (q) have the same spine structure, p and q are equipopular. We prove Conjecture 3, by using the method of generating functions This gives a complete classification of 132-avoiding permutations into equipopularity classes. We show that if two patterns have different spine structures, the associated generating functions are different

Preliminaries

Generating function for the decreasing pattern

Arguments using generating functions

Conclusion