Enumerations of specific permutation classes
Introduction 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. β  sequence enumerating ''Avn''(β)  OEIS  type of sequence  exact enumeration reference  123 231 1, 2, 5, 14, 42, 132, 429, 1430, ...    algebraic (nonrational) g.f.  Catalan numbers
Classes avoiding one pattern of length 4 There are seven symmetry classes and three Wilf classes for single permutations of length four. β  sequence enumerating ''Avn''(β)  OEIS  type of sequence  exact enumeration reference  1342  2413 1, 2, 6, 23, 103, 512, 2740, 15485, ...    algebraic (nonrational) g.f.    1234  1243  1432 2143 1, 2, 6, 23, 103, 513, 2761, 15767, ...    holonomic (nonalgebraic) g.f.   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.
Classes avoiding two patterns of length 3 There are five symmetry classes and three Wilf classes, all of which were enumerated in . 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, ...    polynomial,   231, 321 132, 312 231, 312 1, 2, 4, 8, 16,  32, 64, 128, ...    rational g.f.,
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  or . B  sequence enumerating ''Avn''(B)  OEIS  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
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 ; 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  represents how many permutations have value  at index . Higher resolution versions can be obtained at  PermPal B  sequence enumerating ''Avn''(B)  OEIS  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, ...    rational g.f.    4312, 2134  1, 2, 6, 22, 86, 330, 1206, 4174, ...    rational g.f.    4321, 1324  1, 2, 6, 22, 86, 332, 1217, 4140, ...    polynomial    4321, 2143  1, 2, 6, 22, 86, 333, 1235, 4339, ...    rational g.f.    4312, 1324  1, 2, 6, 22, 86, 335, 1266, 4598, ...    rational g.f.    4231, 2143  1, 2, 6, 22, 86, 335, 1271, 4680, ...    rational g.f.    4231, 1324  1, 2, 6, 22, 86, 336, 1282, 4758, ...    rational g.f.    4213, 2341  1, 2, 6, 22, 86, 336, 1290, 4870, ...    rational g.f.    4312, 2143  1, 2, 6, 22, 86, 337, 1295, 4854, ...    rational g.f.    4213, 1243  1, 2, 6, 22, 86, 337, 1299, 4910, ...    rational g.f.    4321, 3142  1, 2, 6, 22, 86, 338, 1314, 5046, ...    rational g.f.    4213, 1342  1, 2, 6, 22, 86, 338, 1318, 5106, ...    rational g.f.    4312, 2341  1, 2, 6, 22, 86, 338, 1318, 5110, ...    rational g.f.   3412, 2143  1, 2, 6, 22, 86, 340, 1340, 5254, ...    algebraic (nonrational) g.f.    4321, 4123 4321, 3412 4123, 3214 4123, 2143 1, 2, 6, 22, 86, 342, 1366, 5462, ...    rational g.f.    4123, 2341  1, 2, 6, 22, 87, 348, 1374, 5335, ...    algebraic (nonrational) g.f.    4231, 3214  1, 2, 6, 22, 87, 352, 1428, 5768, ...    algebraic (nonrational) g.f.     4213, 1432  1, 2, 6, 22, 87, 352, 1434, 5861, ...    algebraic (nonrational) g.f.    4312, 3421 4213, 2431 1, 2, 6, 22, 87, 354, 1459, 6056, ...    algebraic (nonrational) g.f.   established the Wilf-equivalence; determined the enumeration. 4312, 3124  1, 2, 6, 22, 88, 363, 1507, 6241, ...    algebraic (nonrational) g.f.    4231, 3124  1, 2, 6, 22, 88, 363, 1508, 6255, ...    algebraic (nonrational) g.f.    4312, 3214  1, 2, 6, 22, 88, 365, 1540, 6568, ...    algebraic (nonrational) g.f.    4231, 3412 4231, 3142 4213, 3241 4213, 3124 4213, 2314 1, 2, 6, 22, 88, 366, 1552, 6652, ...    algebraic (nonrational) g.f.    4213, 2143  1, 2, 6, 22, 88, 366, 1556, 6720, ...    algebraic (nonrational) g.f.    4312, 3142  1, 2, 6, 22, 88, 367, 1568, 6810, ...    algebraic (nonrational) g.f.    4213, 3421  1, 2, 6, 22, 88, 367, 1571, 6861, ...    algebraic (nonrational) g.f.    4213, 3412 4123, 3142 1, 2, 6, 22, 88, 368, 1584, 6968, ...    algebraic (nonrational) g.f.    4321, 3214  1, 2, 6, 22, 89, 376, 1611, 6901, ...    algebraic (nonrational) g.f.    4213, 3142  1, 2, 6, 22, 89, 379, 1664, 7460, ...    algebraic (nonrational) g.f.    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, ...    algebraic (nonrational) g.f.  ; see also   4123, 3412  1, 2, 6, 22, 89, 381, 1696, 7781, ...    algebraic (nonrational) g.f.    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, 32143142, 2413 1, 2, 6, 22, 90, 394, 1806, 8558, ...    Schröder numbersalgebraic (nonrational) g.f.  ,   3412, 2413  1, 2, 6, 22, 90, 395, 1823, 8741, ...    algebraic (nonrational) g.f.    4321, 4231  1, 2, 6, 22, 90, 396, 1837, 8864, ...    conjectured to not satisfy any ADE, see
See also * Baxter permutation* Riffle shuffle permutation
References *.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*.*. * . *.*.*.*. * . *.*.*.*.*.*. *.
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.