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Generalized permutation patterns – a short survey

Published online by Cambridge University Press:  05 October 2010

Einar Steingrímsson
The Mathematics Institute Reykjavík University IS-103 Reykjavík, Iceland
Steve Linton
University of St Andrews, Scotland
Nik Ruškuc
University of St Andrews, Scotland
Vincent Vatter
Dartmouth College, New Hampshire
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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.


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.

Permutation Patterns , pp. 137 - 152
Publisher: Cambridge University Press
Print publication year: 2010

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