Abstract
In [Ferrari, L. and Pinzani, R.: Lattices of lattice paths. J. Stat. Plan. Inference 135 (2005), 77–92] a natural order on Dyck paths of any fixed length inducing a distributive lattice structure is defined. We transfer this order to noncrossing partitions along a well-known bijection [Simion, R.: Noncrossing partitions. Discrete Math. 217 (2000), 367–409], thus showing that noncrossing partitions can be endowed with a distributive lattice structure having some combinatorial relevance. Finally we prove that our lattices are isomorphic to the posets of 312-avoiding permutations with the order induced by the strong Bruhat order of the symmetric group.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aigner, M.: Combinatorial Theory, Springer, Berlin Heidelberg New York, 1979.
Aigner, M.: Catalan and other numbers: A recurrent theme, In: H. Crapo and D. Senato (eds.), Algebraic Combinatorics and Computer Science, Springer, Milano, 2001, pp. 347–390.
Babson, E. and Steingrímsson, E.: Generalized permutation patterns and a classification of the Mahonian statistics, Sém. Lothar. Combin. 44 (2000), Art. B44b, 18 pp.
Banderier, C., Bousquet-Mélou, M., Denise, A., Flajolet, P., Gardy, D. and Gouyou-Beauchamps, D.: Generating functions for generating trees, Discrete Math. 246 (2002), 29–55.
Bandlow, J. and Killpatrick, K.: An area-to-inv bijection between Dyck paths and 312-avoiding permutations, Electron. J. Combin. 8 (2001), #R40, (16 pp.)
Barcucci, E., Del Lungo, A., Pergola, E. and Pinzani, R.: ECO: A methodology for the enumeration of combinatorial objects, J. Differ. Equ. Appl. 5 (1999), 435–490.
Bernini, A., Ferrari, L. and Pinzani, R.: Enumerating permutations avoiding three Babson–Steingrímsson patterns, Ann. Comb. 9 (2005), 137–162.
Bjorner, A. and Wachs, M.: Shellable nonpure complexes and posets. II, Trans. Am. Math. Soc. 349 (1997), 3945–3975.
Cautis, S. and Jackson, D. M.: The matrix of chromatic joins and the Temperley–Lieb algebra, J. Comb. Theory, Ser. B 89 (2003) 109–155.
Claesson, A.: Generalized pattern avoidance, Eur. J. Comb. 22 (2001), 961–971.
Claesson, A. and Mansour, T.: Enumerating permutations avoiding a pair of Babson–Steingrímsson patterns, Ars Comb. (to appear).
Davey, B. A. and Priestley, H. A.: Introduction to Lattices and Order, Cambridge University Press, Cambridge, 1990.
Denise, A. and Simion, R.: Two combinatorial statistics on Dyck paths, Discrete Math. 137 (1995), 155–176.
Deutsch, E.: Dyck path enumeration, Discrete Math. 204 (1999), 167–202.
Drake, B.: The Weak Order on Pattern-Avoiding Permutations, Proceedings of FPSAC 2005.
Elizalde, S. and Pak, I.: Bijections for refined restricted permutations, J. Comb. Theory, Ser. A 105 (2004), 207–219.
Fulmek, M.: Enumeration of permutations containing a prescribed number of occurrences of a pattern of length three, Adv. Appl. Math. 30 (2003), 607–632.
Ferrari, L. and Pinzani, R.: A linear operator approach to succession rules, Linear Algebra Appl. 348 (2002), 231–246.
Ferrari, L. and Pinzani, R.: Lattices of lattice paths, J. Stat. Plan. Inference 135 (2005), 77–92.
Guibert, O., Pergola, E. and Pinzani, R.: Vexillary involutions are enumerated by Motzkin numbers, Ann. Comb. 5 (2001), 153–174.
Krattenthaler, C.: Permutations with restricted patterns and Dyck paths, Adv. Appl. Math. 27 (2001), 510–530.
Kreweras, G.: Sur les partitions non croisées d’un cycle, Discrete Math. 1 (1972), 333–350.
Narayana, T. V.: Lattice Path Combinatorics with Statistical Applications, University of Toronto, Toronto, 1979.
Pergola, E., Pinzani, R. and Rinaldi, S.: ECO-approximation of algebraic functions, In: D. Krob, A. A. Mikhalev, A. V. Mikhalev (eds.), Formal Power Series and Algebraic Combinatorics, Springer, Berlin Heidelberg New York, 2000.
Proctor, R. A.: Bruhat lattices, plane partitions generating functions, and minuscule representations, Eur. J. Comb. 5 (1984), 331–350.
Reifegerste, A.: On the diagram of 132-avoiding permutations, Eur. J. Comb. 24 (2003), 759–776.
Simion, R.: Noncrossing partitions, Discrete Math. 217 (2000), 367–409.
Simion, R. and Schmidt, F. W.: Restricted permutations, Eur. J. Comb. 6 (1985), 383–406.
Sloane, N. J. A.: The On-Line Encyclopedia of Integer Sequences, at http://www.research.att.com/~njas/sequences/index.html.
Stanley, R. P.: Enumerative Combinatorics 1, Cambridge University Press, New York, 1997.
Stanley, R. P.: Enumerative Combinatorics 2, Cambridge University Press, Cambridge, 1999.
West, J.: Generating trees and the Catalan and Schröder numbers, Discrete Math. 146 (1995), 247–262.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barcucci, E., Bernini, A., Ferrari, L. et al. A Distributive Lattice Structure Connecting Dyck Paths, Noncrossing Partitions and 312-avoiding Permutations. Order 22, 311–328 (2005). https://doi.org/10.1007/s11083-005-9021-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-005-9021-x