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.

βsequence enumerating Avn(β)OEIStype of sequenceexact enumeration reference

123
231

1, 2, 5, 14, 42, 132, 429, 1430, ...A000108algebraic (nonrational) g.f.
Catalan numbers
MacMahon (1916)
Knuth (1968)

Classes avoiding one pattern of length 4

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

βsequence enumerating Avn(β)OEIStype of sequenceexact enumeration reference

1342
2413

1, 2, 6, 23, 103, 512, 2740, 15485, ...A022558algebraic (nonrational) g.f.Bóna (1997)

1234
1243
1432
2143

1, 2, 6, 23, 103, 513, 2761, 15767, ...A005802holonomic (nonalgebraic) g.f.Gessel (1990)
13241, 2, 6, 23, 103, 513, 2762, 15793, ...A061552

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.

Classes avoiding two patterns of length 3

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

Bsequence enumerating Avn(B)OEIStype of sequence
123, 3211, 2, 4, 4, 0, 0, 0, 0, ...n/afinite
213, 3211, 2, 4, 7, 11, 16, 22, 29, ...A000124polynomial,

231, 321
132, 312
231, 312

1, 2, 4, 8, 16, 32, 64, 128, ...A000079rational 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 Atkinson (1999) or West (1996).

Bsequence enumerating Avn(B)OEIStype of sequence
321, 12341, 2, 5, 13, 25, 25, 0, 0, ...n/afinite
321, 21341, 2, 5, 13, 30, 61, 112, 190, ...A116699polynomial
132, 43211, 2, 5, 13, 31, 66, 127, 225, ...A116701polynomial
321, 13241, 2, 5, 13, 32, 72, 148, 281, ...A179257polynomial
321, 13421, 2, 5, 13, 32, 74, 163, 347, ...A116702rational g.f.
321, 21431, 2, 5, 13, 33, 80, 185, 411, ...A088921rational g.f.

132, 4312
132, 4231

1, 2, 5, 13, 33, 81, 193, 449, ...A005183rational g.f.
132, 32141, 2, 5, 13, 33, 82, 202, 497, ...A116703rational 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, ...A001519rational 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 Albert et al. (2018); 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

Bsequence enumerating Avn(B)OEIStype of sequenceexact enumeration reference
4321, 12341, 2, 6, 22, 86, 306, 882, 1764, ...A206736finiteErdős–Szekeres theorem
4312, 12341, 2, 6, 22, 86, 321, 1085, 3266, ...A116705polynomialKremer & Shiu (2003)
4321, 31241, 2, 6, 22, 86, 330, 1198, 4087, ...A116708rational g.f.Kremer & Shiu (2003)
4312, 21341, 2, 6, 22, 86, 330, 1206, 4174, ...A116706rational g.f.Kremer & Shiu (2003)
4321, 13241, 2, 6, 22, 86, 332, 1217, 4140, ...A165524polynomialVatter (2012)
4321, 21431, 2, 6, 22, 86, 333, 1235, 4339, ...A165525rational g.f.Albert, Atkinson & Brignall (2012)
4312, 13241, 2, 6, 22, 86, 335, 1266, 4598, ...A165526rational g.f.Albert, Atkinson & Brignall (2012)
4231, 21431, 2, 6, 22, 86, 335, 1271, 4680, ...A165527rational g.f.Albert, Atkinson & Brignall (2011)
4231, 13241, 2, 6, 22, 86, 336, 1282, 4758, ...A165528rational g.f.Albert, Atkinson & Vatter (2009)
4213, 23411, 2, 6, 22, 86, 336, 1290, 4870, ...A116709rational g.f.Kremer & Shiu (2003)
4312, 21431, 2, 6, 22, 86, 337, 1295, 4854, ...A165529rational g.f.Albert, Atkinson & Brignall (2012)
4213, 12431, 2, 6, 22, 86, 337, 1299, 4910, ...A116710rational g.f.Kremer & Shiu (2003)
4321, 31421, 2, 6, 22, 86, 338, 1314, 5046, ...A165530rational g.f.Vatter (2012)
4213, 13421, 2, 6, 22, 86, 338, 1318, 5106, ...A116707rational g.f.Kremer & Shiu (2003)
4312, 23411, 2, 6, 22, 86, 338, 1318, 5110, ...A116704rational g.f.Kremer & Shiu (2003)
3412, 21431, 2, 6, 22, 86, 340, 1340, 5254, ...A029759algebraic (nonrational) g.f.Atkinson (1998)

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

1, 2, 6, 22, 86, 342, 1366, 5462, ...A047849rational g.f.Kremer & Shiu (2003)
4123, 23411, 2, 6, 22, 87, 348, 1374, 5335, ...A165531algebraic (nonrational) g.f.Atkinson, Sagan & Vatter (2012)
4231, 32141, 2, 6, 22, 87, 352, 1428, 5768, ...A165532algebraic (nonrational) g.f.Miner (2016)
4213, 14321, 2, 6, 22, 87, 352, 1434, 5861, ...A165533algebraic (nonrational) g.f.Miner (2016)

4312, 3421
4213, 2431

1, 2, 6, 22, 87, 354, 1459, 6056, ...A164651algebraic (nonrational) g.f.Le (2005) established the Wilf-equivalence;
Callan (2013a) determined the enumeration.
4312, 31241, 2, 6, 22, 88, 363, 1507, 6241, ...A165534algebraic (nonrational) g.f.Pantone (2017)
4231, 31241, 2, 6, 22, 88, 363, 1508, 6255, ...A165535algebraic (nonrational) g.f.Albert, Atkinson & Vatter (2014)
4312, 32141, 2, 6, 22, 88, 365, 1540, 6568, ...A165536algebraic (nonrational) g.f.Miner (2016)

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

1, 2, 6, 22, 88, 366, 1552, 6652, ...A032351algebraic (nonrational) g.f.Bóna (1998)
4213, 21431, 2, 6, 22, 88, 366, 1556, 6720, ...A165537algebraic (nonrational) g.f.Bevan (2016b)
4312, 31421, 2, 6, 22, 88, 367, 1568, 6810, ...A165538algebraic (nonrational) g.f.Albert, Atkinson & Vatter (2014)
4213, 34211, 2, 6, 22, 88, 367, 1571, 6861, ...A165539algebraic (nonrational) g.f.Bevan (2016a)

4213, 3412
4123, 3142

1, 2, 6, 22, 88, 368, 1584, 6968, ...A109033algebraic (nonrational) g.f.Le (2005)
4321, 32141, 2, 6, 22, 89, 376, 1611, 6901, ...A165540algebraic (nonrational) g.f.Bevan (2016a)
4213, 31421, 2, 6, 22, 89, 379, 1664, 7460, ...A165541algebraic (nonrational) g.f.Albert, Atkinson & Vatter (2014)
4231, 41231, 2, 6, 22, 89, 380, 1677, 7566, ...A165542conjectured to not satisfy any ADE, see Albert et al. (2018)
4321, 42131, 2, 6, 22, 89, 380, 1678, 7584, ...A165543algebraic (nonrational) g.f.Callan (2013b); see also Bloom & Vatter (2016)
4123, 34121, 2, 6, 22, 89, 381, 1696, 7781, ...A165544algebraic (nonrational) g.f.Miner & Pantone (2018)
4312, 41231, 2, 6, 22, 89, 382, 1711, 7922, ...A165545conjectured to not satisfy any ADE, see Albert et al. (2018)

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, ...A006318Schröder numbers
algebraic (nonrational) g.f.
Kremer (2000), Kremer (2003)
3412, 24131, 2, 6, 22, 90, 395, 1823, 8741, ...A165546algebraic (nonrational) g.f.Miner & Pantone (2018)
4321, 42311, 2, 6, 22, 90, 396, 1837, 8864, ...A053617conjectured to not satisfy any ADE, see Albert et al. (2018)

See also

References

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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