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
Problems and conjectures
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
A very brief introduction to permutation patterns
We say a permutation π contains or involves the permutation σ if deleting some of the entries of π gives a permutation that is order isomorphic to σ, and we write σ ≤ π. For example, 534162 contains 321 (delete the values 4, 6, and 2). A permutation avoids a permutation if it does not contain it.
This notion of containment defines a partial order on the set of all finite permutations, and the downsets of this order are called permutation classes. For a set of permutations B define Av(B) to be the set of permutations that avoid all of the permutations in B. Clearly Av(B) is a permutation class for every set B, and conversely, every permutation class can be expressed in the form Av(B).
For the problems we need one more bit of notation. Given permutations π and σ of lengths m and n, respectively, their direct sum, π ⊕ σ, is the permutation of length m + n in which the first m entries are equal to π and the last n entries are order isomorphic to σ while their skew sum, π ⊖ σ, is the permutation of length m + n in which the first m entries are order isomorphic to π while the last n entries are equal to π.
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- Permutation Patterns , pp. 339 - 345Publisher: Cambridge University PressPrint publication year: 2010
References
[1] , , , , , , and . Longest subsequences in permutations. Australas. J. Combin., 28:225–238, 2003.Google Scholar
[2] , , , , , and . Growth rates for subclasses of Av(321). arXiv:0903.1999 [math.CO].
[3] , , , , 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
[5] . On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern. Electron. J. Combin., 6:Note, N1, 4 pp., 1999.Google Scholar
[6] , , and . Sorting with two ordered stacks in series. Theoret. Comput. Sci., 289(1):205–223, 2002.Google Scholar
[7] and . Generalized permutation patterns and a classification of the Mahonian statistics. Sém. Lothar. Combin., 44:Article B44b, 18 pp., 2000.Google Scholar
[8] and . The permutations 123p4 … pm and 321p4 … pm are Wilf-equivalent. Graphs Combin., 16(4):373–380, 2000.Google Scholar
[9] , , and . Wilf-equivalence for singleton classes. Adv. in Appl. Math., 38(2):133–148, 2007.Google Scholar
[10] . 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
[11] . A survey of stack-sorting disciplines. Electron. J. Combin., 9(2):Article 1, 16 pp., 2003.Google Scholar
[12] . The limit of a Stanley-Wilf sequence is not always rational, and layered patterns beat monotone patterns. J. Combin. Theory Ser. A, 110(2):223–235, 2005.Google Scholar
[13] and . Dumont permutations of the third kind. Extended abstract, FPSAC 2007.
[14] and . Problems and conjectures presented at the Third International Conference on Permutation Patterns, University of Florida, March 7–11, 2005. arXiv:0505504 [math.CO].
[15] . Avoidance of partially ordered generalized patterns of the form κ-σ-κ. arXiv:0805.1872v1 [math.CO].
[16] and . Excluded permutation matrices and the Stanley-Wilf conjecture. J. Combin. Theory Ser. A, 107(1):153–160, 2004.Google Scholar
[17] . Computing permutations with double-ended queues, parallel stacks and parallel queues. In STOC '73: Proceedings of the fifth annual ACM symposium on Theory of computing, pages 268–277, New York, NY, USA, 1973.Google Scholar
[18] and . Posets of matrices and permutations with forbidden subsequences. Ann. Comb., 7(1):55–88, 2003.Google Scholar
[19] . Asymptotic values for degrees associated with strips of Young diagrams. Adv. in Math., 41(2):115–136, 1981.Google Scholar
[20] . Longest increasing and decreasing subsequences. Canad. J. Math., 13:179–191, 1961.Google Scholar
[21] and . A new class of Wilf-equivalent permutations. J. Algebraic Combin., 15(3):271–290, 2002.Google Scholar
[23] . Permutation classes of every growth rate above 2.48188. Mathematika, 56:182–192, 2010.Google Scholar
[24] . Permutations with forbidden subsequences and stack-sortable permutations. PhD thesis, M.I.T., 1990.
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