enumerations of specific permutation classes

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

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 {{harvtxt|Marinov|Radoičić|2003}}.

A more efficient algorithm using functional equations was given by {{harvtxt|Johansson|Nakamura|2014}}, which was enhanced by {{harvtxt|Conway|Guttmann|2015}}, and then further enhanced by {{harvtxt|Conway|Guttmann|Zinn-Justin|2018}} who give the first 50 terms of the enumeration.

{{harvtxt|Bevan|Brignall|Elvey Price|Pantone|2017}} have provided lower and upper bounds for the growth of this class.

There are five symmetry classes and three Wilf classes, all of which were enumerated in {{harvtxt|Simion|Schmidt|1985}}.

There are eighteen symmetry classes and nine Wilf classes, all of which have been enumerated. For these results, see {{harvtxt|Atkinson|1999}} or {{harvtxt|West|1996}}.

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

There are 56 symmetry classes and 38 Wilf equivalence classes. Only 3 of these remain unenumerated, and their generating functions are conjectured not to satisfy any algebraic differential equation (ADE) by {{harvtxt|Albert|Homberger|Pantone|Shar|2018}}; 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 at [https://permpal.com/ PermPal]

• Baxter permutation
• Riffle shuffle permutation

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The [http://math.depaul.edu/~bridget/patterns.html Database of Permutation Pattern Avoidance], maintained by Bridget Tenner, contains details of the enumeration of many other permutation classes with relatively few basis elements.

Category:Enumerative combinatorics

Category:Permutation patterns