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A survey of simple permutations

Published online by Cambridge University Press:  05 October 2010

Robert Brignall
Department of Mathematics University of Bristol Bristol, BS8 1UJ England
Steve Linton
University of St Andrews, Scotland
Nik Ruškuc
University of St Andrews, Scotland
Vincent Vatter
Dartmouth College, New Hampshire
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We survey the known results about simple permutations. In particular, we present a number of recent enumerative and structural results pertaining to simple permutations, and show how simple permutations play an important role in the study of permutation classes. We demonstrate how classes containing only finitely many simple permutations satisfy a number of special properties relating to enumeration, partial well-order and the property of being finitely based.


An interval of a permutation π corresponds to a set of contiguous indices I = [a, b] such that the set of values π(I) = {π(i) : iI} is also contiguous. Every permutation of length n has intervals of lengths 0, 1 and n. If a permutation π has no other intervals, then π is said to be simple. For example, the permutation π = 28146357 is not simple as witnessed by the non-trivial interval 4635 (= π(4)π(5)π(6)π(7)), while σ = 51742683 is simple.

While intervals of permutations have applications in biomathematics, particularly to genetic algorithms and the matching of gene sequences (see Corteel, Louchard, and Pemantle for extensive references), simple permutations form the “building blocks” of permutation classes and have thus received intensive study in recent years. We will see in Section 3 the various ways in which simplicity plays a role in the study of permutation classes, but we begin this short survey by introducing the substitution decomposition in Subsection 1.1, and thence by reviewing the structural and enumerative results of simple permutations themselves in Section 2.

Permutation Patterns , pp. 41 - 66
Publisher: Cambridge University Press
Print publication year: 2010

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[1] M. H., Albert and M. D., Atkinson. Simple permutations and pattern restricted permutations. Discrete Math., 300(1-3):1–15, 2005.Google Scholar
[2] M. H., Albert, M. D., Atkinson, and M., Klazar. The enumeration of simple permutations. J. Integer Seq., 6(4):Article 03.4.4, 18 pp., 2003.Google Scholar
[3] M. D., Atkinson. Restricted permutations. Discrete Math., 195(1-3):27–38, 1999.Google Scholar
[4] M. D., Atkinson, N., Ruškuc, and R., Smith. Substitution-closed pattern classes. Preprint.
[5] M. D., Atkinson and T., Stitt. Restricted permutations and the wreath product. Discrete Math., 259(1-3):19–36, 2002.Google Scholar
[6] J., Balogh, B., Bollobás, and R., Morris. Hereditary properties of partitions, ordered graphs and ordered hypergraphs. European J. Combin., 27(8):1263–1281, 2006.Google Scholar
[7] J., Balogh, B., Bollobás, and R., Morris. Hereditary properties of combinatorial structures: posets and oriented graphs. J. Graph Theory, 56(4):311–332, 2007.Google Scholar
[8] J., Balogh, B., Bollobás, and R., Morris. Hereditary properties of tournaments. Electron. J. Combin., 14(1):Research Paper 60, 25 pp., 2007.Google Scholar
[9] E. A., Bender and L. B., Richmond. An asymptotic expansion for the coefficients of some power series. II. Lagrange inversion. Discrete Math., 50(2–3):135–141, 1984.Google Scholar
[10] A., Bergeron, C., Chauve, F., de Montgolfier, and M., Raffinot. Computing common intervals of κ permutations, with applications to modular decomposition of graphs. In Algorithms – ESA 2005, volume 3669/2005 of Lecture Notes in Computer Science, pages 779–790. Springer Berlin, Heidelberg, 2005.Google Scholar
[11] B., Bollobás. Hereditary properties of graphs: asymptotic enumeration, global structure, and colouring. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), Extra Vol. III, pages 333–342, 1998.Google Scholar
[12] B., Bollobás. Hereditary and monotone properties of combinatorial structures. In A., Hilton and J., Talbot, editors, Surveys in Combinatorics 2007, number 346 in London Mathematical Society Lecture Note Series, pages 1–39. Cambridge University Press, 2007.Google Scholar
[13] M., Bóna. The number of permutations with exactly r 132-subsequences is P-recursive in the size! Adv. in Appl. Math., 18(4):510–522, 1997.Google Scholar
[14] P., Bose, J. F., Buss, and A., Lubiw. Pattern matching for permutations. Inform. Process. Lett., 65(5):277–283, 1998.Google Scholar
[15] M., Bouvel and D., Rossin. Longest common pattern between two permutations. arXiv:math/0611679v1 [math.CO].
[16] R., Brignall, S., Huczynska, and V., Vatter. Decomposing simple permutations, with enumerative consequences. Combinatorica, 28:385–400, 2008.Google Scholar
[17] R., Brignall, S., Huczynska, and V., Vatter. Simple permutations and algebraic generating functions. J. Combin. Theory Ser. A, 115(3):423–441, 2008.Google Scholar
[18] R., Brignall, N., Ruškuc, and V., Vatter. Simple extensions of combinatorial structures. arXiv:math/0911.4378v1 [math.CO].
[19] R., Brignall, N., Ruškuc, and V., Vatter. Simple permutations: decidability and unavoidable substructures. Theoret. Comput. Sci., 391(1–2):150–163, 2008.Google Scholar
[20] L., Comtet. Advanced combinatorics. D. Reidel Publishing Co., Dordrecht, 1974.Google Scholar
[21] S., Corteel, G., Louchard, and R., Pemantle. Common intervals of permutations. Discrete Math. Theor. Comput. Sci., 8(1):189–214, 2006.Google Scholar
[22] A., Cournier and M., Habib. A new linear algorithm for modular decomposition. In Trees in algebra and programming–CAAP '94 (Edinburgh, 1994), volume 787 of Lecture Notes in Comput. Sci., pages 68–84. Springer, Berlin, 1994.Google Scholar
[23] P., Erdős, E., Fried, A., Hajnal, and E. C., Milner. Some remarks on simple tournaments. Algebra Universalis, 2:238–245, 1972.Google Scholar
[24] P., Erdős, A., Hajnal, and E. C., Milner. Simple one-point extensions of tournaments. Mathematika, 19:57–62, 1972.Google Scholar
[25] R., Fraïssé. On a decomposition of relations which generalizes the sum of ordering relations. Bull. Amer. Math. Soc., 59:389, 1953.Google Scholar
[26] T., Gallai. Transitiv orientierbare Graphen. Acta Math. Acad. Sci. Hungar, 18:25–66, 1967.Google Scholar
[27] T., Gallai. A translation of T. Gallai's paper: “Transitiv orientierbare Graphen”, pages 25–66. Wiley-Intersci. Ser. Discrete Math. Optim.Wiley, Chichester, 2001.Google Scholar
[28] V., Giakoumakis. On the closure of graphs under substitution. Discrete Math., 177(1-3):83–97, 1997.Google Scholar
[29] J., Gustedt. Finiteness theorems for graphs and posets obtained by compositions. Order, 15:203–220, 1999.Google Scholar
[30] G., Higman. Ordering by divisibility in abstract algebras. Proc. London Math. Soc. (3), 2:326–336, 1952.Google Scholar
[31] I., Kaplansky. The asymptotic distribution of runs of consecutive elements. Ann. Math. Statistics, 16:200–203, 1945.Google Scholar
[32] T., Mansour and A., Vainshtein. Counting occurrences of 132 in a permutation. Adv. in Appl. Math., 28(2):185–195, 2002.Google Scholar
[33] R. M., McConnell and J. P., Spinrad. Linear-time modular decomposition and efficient transitive orientation of comparability graphs. In Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (Arlington, VA, 1994), pages 536–545, New York, 1994. ACM.Google Scholar
[34] R. H., Möhring. On the distribution of locally undecomposable relations and independence systems. In Colloquium on Mathematical Methods of Operations Research (Aachen, 1980), volume 42 of Methods Oper. Res., pages 33–48. Athenäum/Hain/Hanstein, Königstein/Ts., 1981.Google Scholar
[35] R. H., Möhring. Algorithmic aspects of the substitution decomposition in optimization over relations, sets systems and Boolean functions. Ann. Oper. Res., 4(1-4):195–225, 1985.Google Scholar
[36] R. H., Möhring and F. J., Radermacher. Substitution decomposition for discrete structures and connections with combinatorial optimization. In Algebraic and combinatorial methods in operations research, volume 95 of North-Holland Math. Stud., pages 257–355. North-Holland, Amsterdam, 1984.Google Scholar
[37] M. M., Murphy. Restricted Permutations, Antichains, Atomic Classes, and Stack Sorting. PhD thesis, Univ. of St Andrews, 2002.
[38] C. S. J. A., Nash-Williams. On well-quasi-ordering finite trees. Proc. Cambridge Philos. Soc., 59:833–835, 1963.Google Scholar
[39] J. H., Schmerl and W. T., Trotter. Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures. Discrete Math., 113(1-3):191–205, 1993.Google Scholar
[40] N. J. A., Sloane. The On-line Encyclopedia of Integer Sequences. Available online at∼njas/sequences/.
[41] T., Uno and M., Yagiura. Fast algorithms to enumerate all common intervals of two permutations. Algorithmica, 26(2):290–309, 2000.Google Scholar
[42] J., Wolfowitz. Note on runs of consecutive elements. Ann. Math. Statistics, 15:97–98, 1944.Google Scholar
[43] I., Zverovich. A finiteness theorem for primal extensions. Discrete Math., 296(1):103–116, 2005.Google Scholar
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