Pattern Popularity in 132-Avoiding Permutations

Source abstract

The popularity of a pattern $p$ is the total number of copies of $p$ within all permutations of a set. We address popularity in the set of $132$-avoidng permutations. Bóna showed that in this set, all other non-monotone length-$3$ patterns are equipopular, and proved equipopularity relations between some length-$k$ patterns of a specific form. We prove equipopularity relations between general length-$k$ patterns, based on the structure of their corresponding binary plane trees. Our result explains all equipopularity relations for patterns of length up to $7$, and we conjecture that it provides a complete classification of equipopularity in $132$-avoiding permutations.