Let 
denote the number of permutations on the symmetric
group 
which avoid 
as a subpattern, 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
partitions 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 | 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, 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 above table are summarized below.
| abbreviation | patterns in class |
 | 123, 132, 213, 232, 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 |
and at Least Two Patterns from 
." 31 Jul 2000.