The joint distribution of consecutive patterns and descents in permutations avoiding 3-1-2

We exploit Krattenthaler’s bijection between the set Sn(3-1-2)Sn(3-1-2) of permutations in SnSn avoiding the classical pattern 3-1-23-1-2 and Dyck nn-paths to study the joint distribution over the set Sn(3-1-2)Sn(3-1-2) of a given consecutive pattern of length 3 and of descents. We utilize a involution on Dyck paths due to E. Deutsch to show that these consecutive patterns split into 3 equidistribution classes. In addition, we state equidistribution theorems concerning quadruplets of statistics relative to occurrences of consecutive patterns of length 3 and of descents in a permutation.