Abstract
A permutation is simsun if for all k, the subword of the one-line notation consisting of the k smallest entries does not have three consecutive decreasing elements. Simsun permutations were introduced by Simion and Sundaram, who showed that they are counted by the Euler numbers. In this paper we enumerate simsun permutations avoiding a pattern or a set of patterns of length 3. The results involve Motkzin, Fibonacci, and secondary structure numbers. The techniques in the proofs include generating functions, bijections into lattice paths and generating trees.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Babson, E., Steingrímsson, E.: Generalized permutation patterns and a classification of the Mahonian statistics. Sém. Lothar. Combin. 44, Art. B44b (2000)
Callan, D.: Two bijections for Dyck path parameters. Preprint at arXiv:math/0406381 (2004)
Chow C.-O., Shiu W.C.: Counting simsun permutations by descents. Ann. Combin. 15(4), 625–635 (2011)
Deutsch, E., Elizalde, S.: Restricted simsun permutations. \({\tt arXiv:0912.1361}\) (2009)
Deutsch E., Ferrari L., Rinaldi S.: Production matrices. Adv. Appl. Math 34(1), 101–122 (2005)
Donaghey R.: Automorphisms on Catalan trees and bracketings. J. Combin. Theory Ser B 29(1), 75–90 (1980)
Elizalde S., Mansour T.: Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials. Discrete Math 305(1-3), 170–189 (2005)
Elizalde S., Pak I.: Bijections for refined restricted permutations. J. Combin. Theory Ser. A 105(2), 207–219 (2004)
Ferrari L., Pergola E., Pinzani R., Rinaldi S.: An algebraic characterization of the set of succession rules. Theoret. Comput. Sci 281(1-2), 351–367 (2002)
Foata D., Schützenberger M.-P. et al.: Nombres d’Euler et permutations alternantes. In: Srivastava, J.N. (eds) A Survey of Combinatorial Theory (Proc., pp. 173–187. Internat. Sympos., Colorado State Univ. (1973)
Hetyei G.: On the cd-variation polynomials of André and simsun permutations. Discrete Comput. Geom 16(3), 259–275 (1996)
Hetyei G., Reiner E.: Permutation trees and variation statistics. European J. Combin 19(7), 847–866 (1998)
Krattenthaler C.: Permutations with restricted patterns and Dyck paths. Adv. Appl. Math 27(2-3), 510–530 (2001)
Lewis, J.B: Pattern avoidance in alternating permutations and tableaux. Discrete Math. Theor. Comput. Sci. Proc. AN, 391–402 (2010)
Mansour T.: Restricted 132-alternating permutations and Chebyshev polynomials. Ann. Combin 7(2), 201–227 (2003)
Ouchterlony, E.: Pattern avoiding doubly alternating permutations. In: Proc. FPSAC 2006, pp. 652–663. http://garsia.math.yorku.ca/fpsac06/papers/83.pdf
Simion R., Schmidt F.W.: Restricted permutations. European J. Combin 6(4), 383–406 (1985)
Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. Avaiable online at http://www.research.att.com/~njas/sequences
Stanley R.P.: Enumerative Combinatorics Vol II. Cambridge University Press, Cambridge (1999)
Stanley, R.P.: A survey of alternating permutations. In: Brualdi, R.A. et al (eds.) Combinatorics and Graphs, Contemp. Math., Vol. 531, pp. 165–196. Amer. Math. Soc., Providence, RI (2010)
Stein, P.R., Waterman, M.S.: On some new sequences generalizing the Catalan and Motzkin numbers. Discrete Math. 26(3), 261–272 (1979)
Sundaram, S.: The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice. Adv. Math. 104(2), 225–296 (1994)
West, J.: Generating trees and forbidden subsequences. Discrete Math. 157(1-3), 363–374 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
To the memory of Rodica Simion
Rights and permissions
About this article
Cite this article
Deutsch, E., Elizalde, S. Restricted Simsun Permutations. Ann. Comb. 16, 253–269 (2012). https://doi.org/10.1007/s00026-012-0129-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00026-012-0129-6