The equidistribution of some vincular patterns on 132-avoiding permutations

A pattern in a permutation π is a sub-permutation of π, and this paper deals mainly with length three patterns. In 2012 Bóna showed the rather surprising fact that the cumulative number of occurrences of the patterns 231 and 213 are the same on the set of permutations avoiding 132, even though the pattern based statistics 231 and 213 do not have the same distribution on this set. Here we show that if it is required for the symbols playing the role of 1 and 3 in the occurrences of 231 and 213 to be adjacent, then the obtained statistics are equidistributed on the set of 132-avoiding permutations. Actually, expressed in terms of vincular patterns, we prove bijectively the following more general results: the statistics based on the patterns 231, 213 and 213, together with other statistics, have the same joint distribution on Sn(132) of length n permutations avoiding 132, and so do the patterns 231 and 312; and up to trivial transformations, these statistics are the only based on lengththree proper (not classical nor consecutive) vincular patterns which are equidistributed on a set of permutations avoiding a classical length-three pattern.