Missouri Journal of Mathematical Sciences

Permutation Pattern Avoidance and the Catalan Triangle

Derek Desantis, Rebecca Field, Wesley Hough, Brant Jones, Rebecca Meissen, and Jacob Ziefle

Full-text: Open access


In the study of various objects indexed by permutations, a natural notion of minimal excluded structure, now known as a permutation pattern, has emerged and found diverse applications. One of the earliest results from the study of permutation pattern avoidance in enumerative combinatorics is that the Catalan numbers $c_n$ count the permutations of size $n$ that avoid any fixed pattern of size three. We refine this result by enumerating the permutations that avoid a given pattern of size three and have a given letter in the first position of their one-line notation. Since there are two parameters, we obtain triangles of numbers rather than sequences. Our main result is that there are two essentially different triangles for any of the patterns of size three, and each of these triangles generalizes the Catalan sequence in a natural way. All of our proofs are bijective, and relate the permutations being counted to recursive formulas for the triangles.

Article information

Missouri J. Math. Sci., Volume 25, Issue 1 (2013), 50-60.

First available in Project Euclid: 28 May 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05A05: Permutations, words, matrices
Secondary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx]

permutation patterns Catalan numbers Wilf equivalence


Desantis, Derek; Field, Rebecca; Hough, Wesley; Jones, Brant; Meissen, Rebecca; Ziefle, Jacob. Permutation Pattern Avoidance and the Catalan Triangle. Missouri J. Math. Sci. 25 (2013), no. 1, 50--60. doi:10.35834/mjms/1369746397. https://projecteuclid.org/euclid.mjms/1369746397

Export citation


  • S. Billey and V. Lakshmibai, Singular loci of Schubert varieties, Volume 182 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, 2000.
  • M. Bóna, Combinatorics of Permutations, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2004.
  • S. Billey and A. Postnikov, Smoothness of Schubert varieties via patterns in root subsystems, Adv. in Appl. Math., 34.3 (2005), 447–466.
  • E. Babson and E. Steingrímsson, Generalized permutation patterns and a classification of the Mahonian statistics, Sém. Lothar. Combin., 44 (2000), Art. B44b, 18 pp. (electronic).
  • S. C. Billey and G. S. Warrington, Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations, J. Algebraic Combin., 13.2 (2001), 111–136.
  • D. E. Knuth, The Art of Computer Programming Volume 3, Sorting and Searching, Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1973.
  • OEIS Foundation Inc. (2011), The On-Line Encyclopedia of Integer Sequences, \tthttp://oeis.org.
  • R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6.4 (1985), 383–406.
  • R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999.
  • R. Tarjan, Sorting using networks of queues and stacks, J. Assoc. Comput. Mach., 19 (1972), 341–346.
  • H. N. V. Temperley and E. H. Lieb, Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem, Proc. Roy. Soc. London Ser. A, 322.1549 (1971), 251–280.
  • V. Vatter, Enumeration schemes for restricted permutations, Combin. Probab. Comput., 17.1 (2008), 137–159.
  • J. West, Generating trees and the Catalan and Schröder numbers, Discrete Math., 146.1–3 (1995), 247–262.
  • B. W. Westbury. The representation theory of the Temperley-Lieb algebras, Math. Z., 219.4 (1995), 539–565.
  • A. Woo and A. Yong, Governing singularities of Schubert varieties, J. Algebra, 320.2 (2008), 495–520.