Английская Википедия:Enumerations of specific permutation classes

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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, ... Шаблон:OEIS link algebraic (nonrational) g.f.
Catalan numbers		 Шаблон:Harvtxt
Шаблон:Harvtxt

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, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt

1234
1243
1432
2143

1, 2, 6, 23, 103, 513, 2761, 15767, ... Шаблон:OEIS link holonomic (nonalgebraic) g.f. Шаблон:Harvtxt
1324 1, 2, 6, 23, 103, 513, 2762, 15793, ... Шаблон:OEIS link

No non-recursive formula counting 1324-avoiding permutations is known. A recursive formula was given by Шаблон:Harvtxt. A more efficient algorithm using functional equations was given by Шаблон:Harvtxt, which was enhanced by Шаблон:Harvtxt, and then further enhanced by Шаблон:Harvtxt who give the first 50 terms of the enumeration. Шаблон:Harvtxt 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 Шаблон:Harvtxt.

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, ... Шаблон:OEIS link polynomial, <math>{n\choose 2}+1</math>

231, 321
132, 312
231, 312

1, 2, 4, 8, 16, 32, 64, 128, ... Шаблон:OEIS link rational g.f., <math>2^{n-1}</math>

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 Шаблон:Harvtxt or Шаблон:Harvtxt.

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, ... Шаблон:OEIS link polynomial
132, 4321 1, 2, 5, 13, 31, 66, 127, 225, ... Шаблон:OEIS link polynomial
321, 1324 1, 2, 5, 13, 32, 72, 148, 281, ... Шаблон:OEIS link polynomial
321, 1342 1, 2, 5, 13, 32, 74, 163, 347, ... Шаблон:OEIS link rational g.f.
321, 2143 1, 2, 5, 13, 33, 80, 185, 411, ... Шаблон:OEIS link rational g.f.

132, 4312
132, 4231

1, 2, 5, 13, 33, 81, 193, 449, ... Шаблон:OEIS link rational g.f.
132, 3214 1, 2, 5, 13, 33, 82, 202, 497, ... Шаблон:OEIS link 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, ... Шаблон:OEIS link rational g.f.,
alternate Fibonacci numbers

Classes avoiding two patterns of length 4

Файл:Heatmaps of all non-finite permutation classes avoiding two length four patterns 11 by 5.png
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 Шаблон:Harvtxt; 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 <math>(i,j)</math> represents how many permutations have value <math>j</math> at index <math>i</math>. 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, ... Шаблон:OEIS link finite Erdős–Szekeres theorem
4312, 1234 1, 2, 6, 22, 86, 321, 1085, 3266, ... Шаблон:OEIS link polynomial Шаблон:Harvtxt
4321, 3124 1, 2, 6, 22, 86, 330, 1198, 4087, ... Шаблон:OEIS link rational g.f. Шаблон:Harvtxt
4312, 2134 1, 2, 6, 22, 86, 330, 1206, 4174, ... Шаблон:OEIS link rational g.f. Шаблон:Harvtxt
4321, 1324 1, 2, 6, 22, 86, 332, 1217, 4140, ... Шаблон:OEIS link polynomial Шаблон:Harvtxt
4321, 2143 1, 2, 6, 22, 86, 333, 1235, 4339, ... Шаблон:OEIS link rational g.f. Шаблон:Harvtxt
4312, 1324 1, 2, 6, 22, 86, 335, 1266, 4598, ... Шаблон:OEIS link rational g.f. Шаблон:Harvtxt
4231, 2143 1, 2, 6, 22, 86, 335, 1271, 4680, ... Шаблон:OEIS link rational g.f. Шаблон:Harvtxt
4231, 1324 1, 2, 6, 22, 86, 336, 1282, 4758, ... Шаблон:OEIS link rational g.f. Шаблон:Harvtxt
4213, 2341 1, 2, 6, 22, 86, 336, 1290, 4870, ... Шаблон:OEIS link rational g.f. Шаблон:Harvtxt
4312, 2143 1, 2, 6, 22, 86, 337, 1295, 4854, ... Шаблон:OEIS link rational g.f. Шаблон:Harvtxt
4213, 1243 1, 2, 6, 22, 86, 337, 1299, 4910, ... Шаблон:OEIS link rational g.f. Шаблон:Harvtxt
4321, 3142 1, 2, 6, 22, 86, 338, 1314, 5046, ... Шаблон:OEIS link rational g.f. Шаблон:Harvtxt
4213, 1342 1, 2, 6, 22, 86, 338, 1318, 5106, ... Шаблон:OEIS link rational g.f. Шаблон:Harvtxt
4312, 2341 1, 2, 6, 22, 86, 338, 1318, 5110, ... Шаблон:OEIS link rational g.f. Шаблон:Harvtxt
3412, 2143 1, 2, 6, 22, 86, 340, 1340, 5254, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt

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

1, 2, 6, 22, 86, 342, 1366, 5462, ... Шаблон:OEIS link rational g.f. Шаблон:Harvtxt
4123, 2341 1, 2, 6, 22, 87, 348, 1374, 5335, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt
4231, 3214 1, 2, 6, 22, 87, 352, 1428, 5768, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt
4213, 1432 1, 2, 6, 22, 87, 352, 1434, 5861, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt

4312, 3421
4213, 2431

1, 2, 6, 22, 87, 354, 1459, 6056, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt established the Wilf-equivalence;
Шаблон:Harvtxt determined the enumeration.
4312, 3124 1, 2, 6, 22, 88, 363, 1507, 6241, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt
4231, 3124 1, 2, 6, 22, 88, 363, 1508, 6255, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt
4312, 3214 1, 2, 6, 22, 88, 365, 1540, 6568, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt

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

1, 2, 6, 22, 88, 366, 1552, 6652, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt
4213, 2143 1, 2, 6, 22, 88, 366, 1556, 6720, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt
4312, 3142 1, 2, 6, 22, 88, 367, 1568, 6810, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt
4213, 3421 1, 2, 6, 22, 88, 367, 1571, 6861, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt

4213, 3412
4123, 3142

1, 2, 6, 22, 88, 368, 1584, 6968, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt
4321, 3214 1, 2, 6, 22, 89, 376, 1611, 6901, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt
4213, 3142 1, 2, 6, 22, 89, 379, 1664, 7460, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt
4231, 4123 1, 2, 6, 22, 89, 380, 1677, 7566, ... Шаблон:OEIS link conjectured to not satisfy any ADE, see Шаблон:Harvtxt
4321, 4213 1, 2, 6, 22, 89, 380, 1678, 7584, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt; see also Шаблон:Harvtxt
4123, 3412 1, 2, 6, 22, 89, 381, 1696, 7781, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt
4312, 4123 1, 2, 6, 22, 89, 382, 1711, 7922, ... Шаблон:OEIS link conjectured to not satisfy any ADE, see Шаблон:Harvtxt

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, ... Шаблон:OEIS link Schröder numbers
algebraic (nonrational) g.f.		 Шаблон:Harvtxt, Шаблон:Harvtxt
3412, 2413 1, 2, 6, 22, 90, 395, 1823, 8741, ... Шаблон:OEIS link algebraic (nonrational) g.f. Шаблон:Harvtxt
4321, 4231 1, 2, 6, 22, 90, 396, 1837, 8864, ... Шаблон:OEIS link conjectured to not satisfy any ADE, see Шаблон:Harvtxt

See also

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.

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