The equidistribution of some length three vincular patterns on $S_n(132)$

Abstract: In 2012 B\'ona showed the rather surprising fact that the cumulative number of occurrences of the classical patterns $231$ and $213$ are the same on the set of permutations avoiding $132$, beside 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 the following more general results: the statistics based on the patterns $b-ca$, $b-ac$ and $ba-c$, together with other statistics, have the same joint distribution on $S_n(132)$, and so do the patterns $bc-a$ and $c-ab$; and up to trivial transformations, these statistics are the only based on length three proper (not classical nor adjacent) vincular patterns which are equidistributed on a set of permutations avoiding a classical length three pattern.