A permutation pattern records the relative order of a subsequence of a permutation. Let 
denote the number of permutations
in the symmetric group 
which avoid 
as a subpattern, for positive
integers 
and 
, where "
contains 
as a subpattern" is interpreted to mean that there
exist 
such that for 
,
 |
iff 
.
For example, a permutation contains the pattern (123) iff it has an ascending subsequence
of length three. Here, note that members need not actually be consecutive,
merely ascending (Wilf 1997). Therefore, of the 
permutations of 
, all but 
(i.e., 
, 
, 
, 
, and 
) contain the pattern (12)
(i.e., an increasing subsequence of length two).
The following table gives the numbers of pattern-matching permutations of 
, 
, ..., 
numbers for various patterns 
of length 
.
| pattern | OEIS | number of pattern-matching permutations |
| 1 | A000142 | 1, 2, 6, 24, 120, 720, 5040, ... |
| 12 | A033312 | 1, 5, 23, 119, 719, 5039, 40319, ... |
 | A056986 | 1, 10, 78, 588, 4611, 38890, ... |
| 1234 | A158005 | 1, 17, 207, 2279, 24553, ... |
| 1324 | A158009 | 1, 17, 207, 2278, 24527, ... |
| 1342 | A158006 | 1, 17, 208, 2300, 24835, ... |
The following table gives the numbers of pattern-avoiding permutations of 
for various sets of patterns.
| Wilf class | OEIS | number of pattern-avoiding permutations |
 | A000108 | 1, 2, 5, 14, 42, 132, ... |
| 123, 132, 213 | A000045 | 1, 2, 3, 5, 8, 13, 21, 34, ... |
| 123, 132, 231 | A000027 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... |
| 132, 231, 321 | A000027 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... |
| 123, 132, 3214 | A000073 | 1, 2, 4, 7, 13, 24, 44, 81, 149, ... |
| 123, 132, 3241 | A000071 | 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, ... |
| 123, 132, 3412 | A000124 | 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, ... |
123,
231,  | A004275 | 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, ... |
123, 231,
 | A000124 | 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, ... |
| 123, 231, 4321 | 1, 2, 4, 6, 3, 1, 0, ... | |
| 132, 213, 1234 | A000073 | 1, 2, 4, 7, 13, 24, 44, 81, 149, ... |
213,
231,  | A000124 | 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, ... |
Abbreviations used in the tables are summarized below. Each abbreviation denotes a choice of one pattern from its class, with the other patterns in that row also avoided. Every such choice gives the stated counting sequence.
| abbreviation | patterns in class |
 | 123, 132, 213, 231, 312, 321 |
 | 1432, 2143, 3214, 4132, 4213, 4312 |
 | 1234, 1243, 1324, 1342, 1423, 2134, 2314, 2341, 2413, 2431, 3124, 3142, 3241, 3412, 3421, 4123, 4231 |
 | 1234, 1243, 1423, 1432 |
Brignall (2026) proved the upper bound
 |
improving the earlier bound 13.5 for the exponential growth rate of 1324-avoiding permutations. Brignall (2026) reports substantial assistance from ChatGPT in developing statistics, reasoning about proofs, and producing code and Lean formalizations.
and at Least Two Patterns from 
." 31 Jul 2000.