In the study of permutation patterns, there has been considerable interest in enumerating specific permutation classes, especially those with relatively few basis elements. This area of study has turned up unexpected instances of Wilf equivalence, where two seemingly-unrelated permutation classes have the same numbers 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 Marinov & Radoičić (2003). A more efficient algorithm using functional equations was given by Johansson & Nakamura (2014), which was enhanced by Conway & Guttmann (2015), and then further enhanced by Conway, Guttmann & Zinn-Justin (2018) who give the first 50 terms of the enumeration. Bevan et al. (2017) have provided lower and upper bounds for the growth of this class.

Classes avoiding two patterns of length 3

There are five symmetry classes and three Wilf classes, all of which were enumerated in Simion & Schmidt (1985).

B	sequence enumerating Avn(B)	OEIS	type of sequence
123, 321	1, 2, 4, 4, 0, 0, 0, 0, ...	n/a	finite
213, 321	1, 2, 4, 7, 11, 16, 22, 29, ...	A000124	polynomial, ${\displaystyle {n \choose 2}+1}$
231, 321 132, 312 231, 312	1, 2, 4, 8, 16, 32, 64, 128, ...	A000079	rational g.f., ${\displaystyle 2^{n-1}}$

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 Atkinson (1999) or West (1996).

Classes avoiding two patterns of length 4

Heatmaps of classes avoiding two patterns of length 4.

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 Albert et al. (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 ${\displaystyle (i,j)}$ represents how many permutations have value ${\displaystyle j}$ at index ${\displaystyle i}$. Higher resolution versions can be obtained at PermPal

See also

• Baxter permutation
• Riffle shuffle permutation

References

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External links

The Database of Permutation Pattern Avoidance, maintained by Bridget Tenner, contains details of the enumeration of many other permutation classes with relatively few basis elements.