
Lynn Chua

Krishanu Roy Sankar
Keywords:
Permutations, Pattern avoidance
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 132avoiding permutations. Two patterns are equipopular if, for all $n$, they have the same popularity in the set of length$n$ 132avoiding permutations. There is a wellknown bijection between 132avoiding 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 132avoiding permutations into equipopularity classes.