Book contents
- Frontmatter
- Contents
- Preface
- Some general results in combinatorial enumeration
- A survey of simple permutations
- Permuting machines and permutation patterns
- On three different notions of monotone subsequences
- A survey on partially ordered patterns
- Generalized permutation patterns – a short survey
- An introduction to structural methods in permutation patterns
- Combinatorial properties of permutation tableaux
- Enumeration schemes for words avoiding permutations
- The lexicographic first occurrence of a I-II-III-pattern
- Enumeration of partitions by rises, levels and descents
- Restricted patience sorting and barred pattern avoidance
- Permutations with k-regular descent patterns
- Packing rates of measures and a conjecture for the packing density of 2413
- On the permutational power of token passing networks
- Problems and conjectures
- References
Generalized permutation patterns – a short survey
Published online by Cambridge University Press: 05 October 2010
Book contents
- Frontmatter
- Contents
- Preface
- Some general results in combinatorial enumeration
- A survey of simple permutations
- Permuting machines and permutation patterns
- On three different notions of monotone subsequences
- A survey on partially ordered patterns
- Generalized permutation patterns – a short survey
- An introduction to structural methods in permutation patterns
- Combinatorial properties of permutation tableaux
- Enumeration schemes for words avoiding permutations
- The lexicographic first occurrence of a I-II-III-pattern
- Enumeration of partitions by rises, levels and descents
- Restricted patience sorting and barred pattern avoidance
- Permutations with k-regular descent patterns
- Packing rates of measures and a conjecture for the packing density of 2413
- On the permutational power of token passing networks
- Problems and conjectures
- References
Summary
Abstract
An occurrence of a classical pattern p in a permutation π is a subsequence of π whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidance–or the prescribed number of occurrences–of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns.
Introduction
Patterns in permutations have been studied sporadically, often implicitly, for over a century, but in the last two decades this area has grown explosively, with several hundred published papers. As seems to be the case with most things in enumerative combinatorics, some instances of permutation patterns can be found already in MacMahon's classical book from 1915, Combinatory Analysis. In the seminal paper Restricted permutations of Simion and Schmidt from 1985 the systematic study of permutation patterns was launched, and it now seems clear that this field will continue growing for a long time to come, due to its plethora of problems that range from the easy to the seemingly impossible, with a rich middle ground of challenging but solvable problems.
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- Permutation Patterns , pp. 137 - 152Publisher: Cambridge University PressPrint publication year: 2010
References
[1] , , , , and . On the Wilf-Stanley limit of 4231-avoiding permutations and a conjecture of Arratia. Adv. in Appl. Math., 36(2):95–105, 2006.Google Scholar
[2] . Développements de sec x et de tang x. C. R. Math. Acad. Sci. Paris, 88:965–967, 1879.Google Scholar
[4] and . Generalized permutation patterns and a classification of the Mahonian statistics. Sém. Lothar. Combin., 44:Article B44b, 18 pp., 2000.Google Scholar
[5] , , and . Enumeration of some classes of words avoiding two generalized patterns of length three. arXiv:0711.3387v1 [math.CO].
[6] , , and . Enumerating permutations avoiding three Babson-Steingrímsson patterns. Ann. Comb., 9(2):137–162, 2005.Google Scholar
[7] and . Enumerating permutations avoiding more than three Babson-Steingrímsson patterns. J. Integer Seq., 10(6):Article 07.6.4, 21 pp., 2007.Google Scholar
[8] . Exact enumeration of 1342-avoiding permutations: a close link with labeled trees and planar maps. J. Combin. Theory Ser. A, 80(2):257–272, 1997.Google Scholar
[10] and . On the enumeration of rooted non-separable planar maps. Canad. J. Math., 16:572–577, 1964.Google Scholar
[11] and . Restricted patience sorting and barred pattern avoidance. In this volume, 233–257.
[12] and . Words restricted by 3-letter generalized multipermutation patterns. Ann. Comb., 7(1):1–14, 2003.Google Scholar
[13] . A Wilf equivalence related to two stack sortable permutations. arXiv:math/0510211v1 [math.CO].
[14] . A combinatorial interpretation of the eigensequence for composition. J. Integer Seq., 9(1):Article 06.1.4, 12 pp., 2006.Google Scholar
[15] , , , Jr., and . The number of Baxter permutations. J. Combin. Theory Ser. A, 24(3):382–394, 1978.Google Scholar
[17] , , and . Stack sorting, trees, and pattern avoidance. arXiv:0801.4037v1 [math.CO].
[18] and . Counting occurrences of a pattern of type (1, 2) or (2, 1) in permutations. Adv. in Appl. Math., 29(2):293–310, 2002.Google Scholar
[19] and . Enumerating permutations avoiding a pair of Babson-Steingrímsson patterns. Ars Combin., 77:17–31, 2005.Google Scholar
[20] , , and . New Euler-Mahonian statistics on permutations and words. Adv. in Appl. Math., 18(3):237–270, 1997.Google Scholar
[21] . Crossings and alignments of permutations. Adv. in Appl. Math., 38(2):149–163, 2007.Google Scholar
[22] and . Bijections for permutation tableaux. European J. Combin., 30(1):295–310, 2009.Google Scholar
[23] and . A Markov chain on permutations which projects to the PASEP. Int. Math. Res. Not. IMRN, 2007(17):Art. ID rnm055, 27, 2007.Google Scholar
[24] and . Tableaux combinatorics for the asymmetric exclusion process. Adv. in Appl. Math., 39(3):293–310, 2007.Google Scholar
[25] , , and . Permutations with forbidden subsequences and nonseparable planar maps. Discrete Math., 153(1-3):85–103, 1996.Google Scholar
[26] . Asymptotic enumeration of permutations avoiding generalized patterns. Adv. in Appl. Math., 36(2):138–155, 2006.Google Scholar
[27] and . Consecutive patterns in permutations. Adv. in Appl. Math., 30(1-2):110–125, 2003.Google Scholar
[28] . Combinatorial aspects of continued fractions. Discrete Math., 32(2):125–161, 1980.Google Scholar
[29] and . Two oiseau decompositions of permutations and their application to Eulerian calculus. European J. Combin., 27(3):342–363, 2006.Google Scholar
[30] and . Denert's permutation statistic is indeed Euler-Mahonian. Stud. Appl. Math., 83(1):31–59, 1990.Google Scholar
[31] and . Babson–Steingrímsson statistics are indeed Mahonian (and sometimes even Euler–Mahonian). Adv. in Appl. Math., 27(2–3):390–404, 2001.Google Scholar
[32] and . Permutations selon leurs pics, creux, doubles montées et double descentes, nombres d'Euler et nombres de Genocchi. Discrete Math., 28(1):21–35, 1979.Google Scholar
[33] . Symmetric functions and P-recursiveness. J. Combin. Theory Ser. A, 53(2):257–285, 1990.Google Scholar
[34] . Arbres, permutations à motifs exclus et cartes planaires: quelques problèmes algorithmique et combinatoire. PhD thesis, Université Bordeaux I, 1993.
[36] . q-rook polynomials and matrices over finite fields. Adv. in Appl. Math., 20(4):450–487, 1998.Google Scholar
[37] . personal communication. 2008.
[38] . Avoidance of partially ordered generalized patterns of the form κ-σ-κ. arXiv:0805.1872v1 [math.CO].
[39] and . A bijective census of nonseparable planar maps. J. Combin. Theory Ser. A, 83(1):1–20, 1998.Google Scholar
[40] . Generalized pattern avoidance with additional restrictions. Sém. Lothar. Combin., 48:Article B48e, 19 pp., 2002.Google Scholar
[41] . Multi-avoidance of generalised patterns. Discrete Math., 260(1-3):89–100, 2003.Google Scholar
[42] . Partially ordered generalized patterns. Discrete Math., 298(1-3):212–229, 2005.Google Scholar
[44] and . Simultaneous avoidance of generalized patterns. Ars Combin., 75:267–288, 2005.Google Scholar
[46] and . Excluded permutation matrices and the Stanley-Wilf conjecture. J. Combin. Theory Ser. A, 107(1):153–160, 2004.Google Scholar
[47] . Building generating functions brick by brick. PhD thesis, University of California, San Diego, 2004.
[48] and . Permutations and words counted by consecutive patterns. Adv. in Appl. Math., 37(4):443–480, 2006.Google Scholar
[49] and . The enumeration of permutations with a prescribed number of “forbidden” patterns. Adv. in Appl. Math., 17(4):381–407, 1996.Google Scholar
[50] . On Young tableaux involutions and patterns in permutations. PhD thesis, Matematiska institutionen Linköpings Universitet, Linköping, Sweden, 2005.
[51] . Lattice path enumeration of permutations with k occurrences of the pattern 2-13. J. Integer Seq., 9(3):Article 06.3.2, 8 pp., 2006.Google Scholar
[53] and . Octabasic Laguerre polynomials and permutation statistics. J. Comput. Appl. Math., 68(1-2):297–329, 1996.Google Scholar
[54] and . Permutation tableaux and permutation patterns. J. Combin. Theory Ser. A, 114(2):211–234, 2007.Google Scholar
[55] . Permutations with forbidden subsequences and stack-sortable permutations. PhD thesis, M.I.T., 1990.
[56] . Enumeration of totally positive Grassmann cells. Adv. Math., 190(2):319–342, 2005.Google Scholar
[58] . The umbral transfer-matrix method. I. Foundations. J. Combin. Theory Ser. A, 91(1-2):451–463, 2000.Google Scholar
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