- Print publication year: 2010
- Online publication date: October 2010

- Publisher: Cambridge University Press
- DOI: https://doi.org/10.1017/CBO9780511902499.008
- pp 153-170

Abstract

Structural methods as applied to the study of classical permutation pattern avoidance are introduced and described. These methods allow for more detailed study of pattern classes, answering questions beyond basic enumeration. Additionally, they frequently can be applied wholesale, producing results valid for a wide collection of pattern classes, rather than simply ad hoc application to individual classes.

Introduction

In the study of permutation patterns, the important aspects of permutations of [n] = {1, 2, …, n} are considered to be the relative order of both the argument and the value. Specifically, we study a partial order, denoted ≼ and called involvement, on the set of such permutations where π ∈ Sk is involved in σ ∈ Sn, i.e. π ≼ σ if, for some increasing function f : [k] → [n] and all 1 ≤ i < j ≤ k, σ(i) < σ(j) if and only if π(f(i)) < π(f(j)). This dry and uninformative definition is necessary to get us started, but the reader should certainly be aware that another definition of involvement is that some of the points in the graph of π can be erased so that what remains is the graph of σ (possibly with a non-uniform scale on both axes) – in other words the pattern of σ (its graph) occurs as part of the pattern of π.

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[1] 2007. . Aspects of separability. Abstract for Permutation Patterns

[2] Simple permutations and pattern restricted permutations. Discrete Math., 300(1-3):1–15, 2005. and .

[3] The enumeration of simple permutations. J. Integer Seq., 6(4):Article 03.4.4, 18 pp., 2003. , , and .

[4] Regular closed sets of permutations. Theoret. Comput. Sci., 306(1-3):85–100, 2003. , , and .

[5] On the Wilf-Stanley limit of 4231-avoiding permutations and a conjecture of Arratia. Adv. in Appl. Math., 36(2):95–105, 2006. , , , , and .

[6] Growing at a perfect speed. Combin. Probab. Comput., 18:301–308, 2009. and .

[7] The insertion encoding of permutations. Electron. J. Combin., 12(1):Research paper 47, 31 pp., 2005. , , and .

[8] On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern. Electron. J. Combin., 6:Note, N1, 4 pp., 1999. .

[9] Permutations generated by token passing in graphs. Theoret. Comput. Sci., 178(1-2):103–118, 1997. , , and .

[10] Partially well-ordered closed sets of permutations. Order, 19(2):101–113, 2002. , , and .

[11] Restricted permutations and the wreath product. Discrete Math., 259(1-3):19–36, 2002. and .

[12] Exhaustive generation of combinatorial objects by ECO. Acta Inform., 40(8):585–602, 2004. , , , and .

[13] Generating functions for generating trees. Discrete Math., 246(1-3):29–55, 2002. , , , , , and .

[14] From Motzkin to Catalan permutations. Discrete Math., 217(1-3):33–49, 2000. , , , and .

[15] Permutations avoiding an increasing number of length-increasing forbidden subsequences. Discrete Math. Theor. Comput. Sci., 4(1):31–44, 2000. , , , and .

[16] Four classes of pattern-avoiding permutations under one roof: generating trees with two labels. Electron. J. Combin., 9(2):Research paper 19, 31 pp., 2003. .

[17] Wreath products of permutation classes. Electron. J. Combin., 14(1):Research paper 46, 15 pp., 2007. .

[18] Decomposing simple permutations, with enumerative consequences. Combinatorica, 28:385–400, 2008. , , and .

[19] Simple permutations and algebraic generating functions. J. Combin. Theory Ser. A, 115(3):423–441, 2008. , , and .

[20] Simple permutations: decidability and unavoidable substructures. Theoret. Comput. Sci., 391(1–2):150–163, 2008. , , and .

[21] Computer programming and formal systems, pages 118–161. North-Holland, Amsterdam, 1963. and . The algebraic theory of context-free languages. In

[22] Forbidden subsequences and Chebyshev polynomials. Discrete Math., 204(1-3):119–128, 1999. and .

[23] From object grammars to ECO systems. Theoret. Comput. Sci., 314(1-2):57–95, 2004. , , and .

[24] A combinatorial problem in geometry. Compos. Math., 2:463–470, 1935. and .

[25] Analytic combinatorics. Cambridge University Press, Cambridge, 2009. and .

[26] Combinatorial enumeration. Dover Publications Inc., Mineola, NY, 2004. and .

[27] Vexillary involutions are enumerated by Motzkin numbers. Ann. Comb., 5(2):153–147, 2001. , , and .

[28] Introduction to automata theory, languages, and computation. Addison-Wesley Publishing Co., Reading, Mass., 1979. Addison-Wesley Series in Computer Science. and .

[29] Grid classes and the Fibonacci dichotomy for restricted permutations. Electron. J. Combin., 13:Research paper 54, 14 pp., 2006. and .

[30] On growth rates of closed permutation classes. Electron. J. Combin., 9(2):Research paper 10, 20 pp., 2003. and .

[31] The art of computer programming. Vol. 1: Fundamental algorithms. Addison-Wesley Publishing Co., Reading, Mass., 1969. .

[32] Finite transition matrices for permutations avoiding pairs of length four patterns. Discrete Math., 268(1-3):171–183, 2003. and .

[33] Completion of the Wilf-classification of 3-5 pairs using generating trees. Electron. J. Combin., 13(1):Research paper 31, 19 pp., 2006. .

[34] Restricted 132-avoiding permutations. Adv. in Appl. Math., 26(3):258–269, 2001. and .

[35] Excluded permutation matrices and the Stanley-Wilf conjecture. J. Combin. Theory Ser. A, 107(1):153–160, 2004. and .

[36] Permutation classes of every growth rate above 2.48188. Mathematika, 56:182–192, 2010. .

[37] . Small permutation classes. arXiv:0712.4006v2 [math.CO].

[38] Finitely labeled generating trees and restricted permutations. J. Symbolic Comput., 41(5):559–572, 2006. .

[39] Generating trees and the Catalan and Schröder numbers. Discrete Math., 146(1-3):247–262, 1995. .

[40] Generating trees and forbidden subsequences. Discrete Math., 157(1-3):363–374, 1996. .