In the study of

permutation pattern In combinatorial mathematics and theoretical computer science, a (classical) permutation pattern is a sub-permutation of a longer permutation. Any permutation may be written in one-line notation as a sequence of entries representing the result of a ...

s, there has been considerable interest in enumerating specific

permutation class In the study of permutations and permutation patterns, a permutation class is a set C of permutations for which every pattern within a permutation in C is also in C. In other words, a permutation class is a hereditary property of permutations, or a ...

es, especially those with relatively few basis elements. This area of study has turned up unexpected instances of

Wilf equivalence In the study of permutations and permutation patterns, Wilf equivalence is an equivalence relation on permutation classes. Two permutation classes are Wilf equivalent when they have the same numbers of permutations of each possible length, or equiv ...

, where two seemingly-unrelated permutation classes have the same number of permutations of each length.

Classes avoiding one pattern of length 3

There are two symmetry classes and a single Wilf class for single permutations of length three.

Classes avoiding one pattern of length 4

There are seven symmetry classes and three Wilf classes for single permutations of length four. No non-recursive formula counting 1324-avoiding permutations is known. A recursive formula was given by . A more efficient algorithm using functional equations was given by , which was enhanced by , and then further enhanced by who give the first 50 terms of the enumeration. currently have the best rigorously established lower and upper bounds for the growth rate of this class, having established that this growth rate lies in the interval 0.271, 13.5

Classes avoiding two patterns of length 3

There are five symmetry classes and three Wilf classes, all of which were enumerated in .

Classes avoiding one pattern of length 3 and one of length 4

There are eighteen symmetry classes and nine Wilf classes, all of which have been enumerated. For these results, see or .

Classes avoiding two patterns of length 4

Heatmaps of all non-finite permutation classes avoiding two length four patterns 11 by 5

There are 56 symmetry classes and 38 Wilf equivalence classes. Only 3 of these remain unenumerated, and their

generating function In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...

s are conjectured not to satisfy any

algebraic differential equation In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several such notions, according to the concept of differential algebra used. The intention is to i ...

(ADE) by ; in particular, their conjecture would imply that these generating functions are not D-finite. Heatmaps of each of the non-finite classes are shown on the right. The lexicographically minimal symmetry is used for each class, and the classes are ordered in lexicographical order. To create each heatmap, one million permutations of length 300 were sampled uniformly at random from the class. The color of the point

(i,j)

represents how many permutations have value

j

at index

i

. Higher resolution versions can be obtained a
PermPal

References

*. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. *. * . *. *. *. *. * . *. *. *. *. *. *. *. *{{Citation , last1=West , first1=Julian , title=Generating trees and forbidden subsequences , year=1996 , journal=

Discrete Mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...

, volume=157 , issue=1–3 , pages=363–374 , mr=1417303 , doi=10.1016/S0012-365X(96)83023-8, doi-access= .

External links

Th
Database of Permutation Pattern Avoidance
maintained by

Bridget Tenner Bridget Eileen Tenner is a professor of mathematics at DePaul University in Chicago. Her research focuses on permutation patterns, and has also included work in algebraic combinatorics, discrete geometry, Coxeter groups, and electoral geography. ...

, contains details of the enumeration of many other permutation classes with relatively few basis elements. Enumerative combinatorics Permutation patterns