Enumerations of specific permutation classes

In the study of permutation patterns, there has been considerable interest in enumerating specific permutation classes, especially those with relatively few basis elements.

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

123
231

There are seven symmetry classes and three Wilf classes for single permutations of length four.

1342
2413

1234
1243
1432
2143

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). Bevan (2015) has provided a lower bound and Bóna (2015) an upper bound for the growth of this class.

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

123, 132
132, 312
231, 312

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

132, 4312
132, 4231

321, 2341
321, 3412
321, 3142
132, 1234
132, 4213
132, 4123
132, 3124
132, 2134
132, 3412

There are 56 symmetry classes and 38 Wilf equivalence classes. Only 5 of these remain unenumerated, and the generating functions for 3 of those 5 classes are conjectured not to satisfy any algebraic differential equation (ADE) by Albert et al.; in particular, their conjecture would imply that the generating functions are not D-finite.

4321, 4123
4321, 3412
4123, 3214
4123, 2143

4312, 3421
4213, 2431

4231, 3412
4231, 3142
4213, 3241
4213, 3124
4213, 2314

4213, 3412
4123, 3142

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