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 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.

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., ${\displaystyle 2^{n-1}}$

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

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.

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.

• Baxter permutation
• Riffle shuffle permutation

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