Counting pattern avoiding permutations by number of movable letters

By a movable letter within a pattern avoiding permutation, we mean one that may be transposed with its predecessor while still avoiding the pattern. In this paper, we enumerate permutations avoiding a single pattern of length three according to the number of movable letters, thereby obtaining new q-analogues of the Catalan number sequence. Indeed, we consider the joint distribution with the statistics recording the number of descents and occurrences of certain vincular patterns. To establish several of our results, we make use of the kernel method to solve the functional equations that arise.