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.
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.
| β | sequence enumerating Avn(β) | type of sequence | exact enumeration reference | ||
|---|---|---|---|---|---|
| 1342 2413 | 1, 2, 6, 23, 103, 512, 2740, 15485, ... | ||||
| 1234 1243 1432 2143 | 1, 2, 6, 23, 103, 513, 2761, 15767, ... | ||||
| 1324 | 1, 2, 6, 23, 103, 513, 2762, 15793, ... |
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. 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 .
| B | sequence enumerating Avn(B) | 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, ... | polynomial, {n\choose2}+1 | ||
| 231, 321 132, 312 231, 312 | 1, 2, 4, 8, 16, 32, 64, 128, ... | rational g.f., 2n-1 |
There are eighteen symmetry classes and nine Wilf classes, all of which have been enumerated. For these results, see or .
| B | sequence enumerating Avn(B) | type of sequence | ||
|---|---|---|---|---|
| 321, 1234 | 1, 2, 5, 13, 25, 25, 0, 0, ... | n/a | finite | |
| 321, 2134 | 1, 2, 5, 13, 30, 61, 112, 190, ... | polynomial | ||
| 132, 4321 | 1, 2, 5, 13, 31, 66, 127, 225, ... | polynomial | ||
| 321, 1324 | 1, 2, 5, 13, 32, 72, 148, 281, ... | polynomial | ||
| 321, 1342 | 1, 2, 5, 13, 32, 74, 163, 347, ... | rational g.f. | ||
| 321, 2143 | 1, 2, 5, 13, 33, 80, 185, 411, ... | rational g.f. | ||
| 132, 4312 132, 4231 | 1, 2, 5, 13, 33, 81, 193, 449, ... | rational g.f. | ||
| 132, 3214 | 1, 2, 5, 13, 33, 82, 202, 497, ... | rational g.f. | ||
| 321, 2341 321, 3412 321, 3142 132, 1234 132, 4213 132, 4123 132, 3124 132, 2134 132, 3412 | 1, 2, 5, 13, 34, 89, 233, 610, ... | rational g.f., alternate Fibonacci numbers |
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 ; 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)
j
i
| B | sequence enumerating Avn(B) | type of sequence | exact enumeration reference | ||
|---|---|---|---|---|---|
| 4321, 1234 | 1, 2, 6, 22, 86, 306, 882, 1764, ... | finite | Erdős–Szekeres theorem | ||
| 4312, 1234 | 1, 2, 6, 22, 86, 321, 1085, 3266, ... | polynomial | |||
| 4321, 3124 | 1, 2, 6, 22, 86, 330, 1198, 4087, ... | ||||
| 4312, 2134 | 1, 2, 6, 22, 86, 330, 1206, 4174, ... | ||||
| 4321, 1324 | 1, 2, 6, 22, 86, 332, 1217, 4140, ... | polynomial | |||
| 4321, 2143 | 1, 2, 6, 22, 86, 333, 1235, 4339, ... | ||||
| 4312, 1324 | 1, 2, 6, 22, 86, 335, 1266, 4598, ... | ||||
| 4231, 2143 | 1, 2, 6, 22, 86, 335, 1271, 4680, ... | ||||
| 4231, 1324 | 1, 2, 6, 22, 86, 336, 1282, 4758, ... | ||||
| 4213, 2341 | 1, 2, 6, 22, 86, 336, 1290, 4870, ... | ||||
| 4312, 2143 | 1, 2, 6, 22, 86, 337, 1295, 4854, ... | ||||
| 4213, 1243 | 1, 2, 6, 22, 86, 337, 1299, 4910, ... | ||||
| 4321, 3142 | 1, 2, 6, 22, 86, 338, 1314, 5046, ... | ||||
| 4213, 1342 | 1, 2, 6, 22, 86, 338, 1318, 5106, ... | ||||
| 4312, 2341 | 1, 2, 6, 22, 86, 338, 1318, 5110, ... | ||||
| 1, 2, 6, 22, 86, 340, 1340, 5254, ... | |||||
| 4321, 4123 4321, 3412 4123, 3214 4123, 2143 | 1, 2, 6, 22, 86, 342, 1366, 5462, ... | ||||
| 4123, 2341 | 1, 2, 6, 22, 87, 348, 1374, 5335, ... | ||||
| 4231, 3214 | 1, 2, 6, 22, 87, 352, 1428, 5768, ... | ||||
| 4213, 1432 | 1, 2, 6, 22, 87, 352, 1434, 5861, ... | ||||
| 4312, 3421 4213, 2431 | 1, 2, 6, 22, 87, 354, 1459, 6056, ... | established the Wilf-equivalence; determined the enumeration. | |||
| 4312, 3124 | 1, 2, 6, 22, 88, 363, 1507, 6241, ... | ||||
| 4231, 3124 | 1, 2, 6, 22, 88, 363, 1508, 6255, ... | ||||
| 4312, 3214 | 1, 2, 6, 22, 88, 365, 1540, 6568, ... | ||||
| 4231, 3412 4231, 3142 4213, 3241 4213, 3124 4213, 2314 | 1, 2, 6, 22, 88, 366, 1552, 6652, ... | ||||
| 4213, 2143 | 1, 2, 6, 22, 88, 366, 1556, 6720, ... | ||||
| 4312, 3142 | 1, 2, 6, 22, 88, 367, 1568, 6810, ... | ||||
| 4213, 3421 | 1, 2, 6, 22, 88, 367, 1571, 6861, ... | ||||
| 4213, 3412 4123, 3142 | 1, 2, 6, 22, 88, 368, 1584, 6968, ... | ||||
| 4321, 3214 | 1, 2, 6, 22, 89, 376, 1611, 6901, ... | ||||
| 4213, 3142 | 1, 2, 6, 22, 89, 379, 1664, 7460, ... | ||||
| 4231, 4123 | 1, 2, 6, 22, 89, 380, 1677, 7566, ... | conjectured to not satisfy any ADE, see | |||
| 4321, 4213 | 1, 2, 6, 22, 89, 380, 1678, 7584, ... |
| |||
| 4123, 3412 | 1, 2, 6, 22, 89, 381, 1696, 7781, ... | ||||
| 4312, 4123 | 1, 2, 6, 22, 89, 382, 1711, 7922, ... | conjectured to not satisfy any ADE, see | |||
| 4321, 4312 4312, 4231 4312, 4213 4312, 3412 4231, 4213 4213, 4132 4213, 4123 4213, 2413 4213, 3214 3142, 2413 | 1, 2, 6, 22, 90, 394, 1806, 8558, ... | , | |||
| 3412, 2413 | 1, 2, 6, 22, 90, 395, 1823, 8741, ... | ||||
| 4321, 4231 | 1, 2, 6, 22, 90, 396, 1837, 8864, ... | conjectured to not satisfy any ADE, see |
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.