Pattern popularity in 132-avoiding permutations (Q1953384)

Summary: 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. \textit{M. Bóna} [ibid. 19, No. 1, Research Paper P62, 11 p., electronic only (2012; Zbl 1243.05006)] 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.