Equipopularity classes of 123-pattern occurring and 213-pattern avoiding of Aunu permutations

Abstract

The popularity of a pattern  in a set of permutations is the sum of the number of copies/occurrences of  in each permutation of the set. We study pattern popularity in the set of 213-avoiding Aunu permutations. Two patterns are equipopular if, for all , they have the same popularity (number of occurrences) in the set of length- 213-avoiding permutations. There is a bijection between the set of  Aunu permutations and set of Aunu-permutations with exactly one occurrence of .  Reseach has shown 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  Aunu- permutations into equipopularity classes..