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

Authors:
Marilena Barnabei;Flavio Bonetti;Matteo Silimbani
Affiliations:
Dipartimento di Matematica, Universití di Bologna, Piazza di Porta San Donato, 5 - 40126 Bologna, Italy;Dipartimento di Matematica, Universití di Bologna, Piazza di Porta San Donato, 5 - 40126 Bologna, Italy;Dipartimento di Matematica, Universití di Bologna, Piazza di Porta San Donato, 5 - 40126 Bologna, Italy
Venue:
European Journal of Combinatorics
Year:
2010

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Abstract

We exploit Krattenthaler's bijection between the set S"n(3-1-2) of permutations in S"n avoiding the classical pattern 3-1-2 and Dyck n-paths to study the joint distribution over the set S"n(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.