Your data matches 242 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000217
(load all 10 compositions to match this statistic)
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000217: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => 0
[1,1,0,0]
 => [2,1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => 0
[1,1,0,0,1,0]
 => [2,1,3] => 0
[1,1,0,1,0,0]
 => [2,3,1] => 0
[1,1,1,0,0,0]
 => [3,1,2] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 3
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 1
Description
The number of occurrences of the pattern 312 in a permutation.
Matching statistic: St000436
(load all 9 compositions to match this statistic)
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000436: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => 0
[1,1,0,0]
 => [2,1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => 0
[1,1,0,0,1,0]
 => [2,1,3] => 0
[1,1,0,1,0,0]
 => [2,3,1] => 1
[1,1,1,0,0,0]
 => [3,1,2] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 0
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 6
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 0
Description
The number of occurrences of the pattern 231 or of the pattern 321 in a permutation.
Matching statistic: St000437
(load all 10 compositions to match this statistic)
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000437: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => 0
[1,1,0,0]
 => [2,1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => 0
[1,1,0,0,1,0]
 => [2,1,3] => 0
[1,1,0,1,0,0]
 => [2,3,1] => 0
[1,1,1,0,0,0]
 => [3,1,2] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 3
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 1
Description
The number of occurrences of the pattern 312 or of the pattern 321 in a permutation.
Matching statistic: St000218
(load all 4 compositions to match this statistic)
Mapping
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => [1,2] => 0
[1,1,0,0]
 => [1,2] => [2,1] => 0
[1,0,1,0,1,0]
 => [2,1,3] => [3,1,2] => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,3,2] => 0
[1,1,0,0,1,0]
 => [3,1,2] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [1,3,2] => [2,3,1] => 0
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [3,4,1,2] => 0
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => 0
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [4,1,3,2] => 0
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,4,3,2] => 0
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,4,3,1] => 0
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [3,4,2,1] => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [5,3,4,1,2] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [5,3,1,4,2] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [3,5,4,1,2] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [3,5,1,4,2] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [3,1,5,4,2] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [4,3,5,1,2] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [4,3,1,5,2] => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [4,5,1,3,2] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,1,5,3,2] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [5,4,1,3,2] => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [5,1,4,3,2] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,5,4,3,2] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [5,2,4,1,3] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [5,2,1,4,3] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,5,4,1,3] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,5,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,5,4,3] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [4,2,5,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [4,2,1,5,3] => 4
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [4,5,2,1,3] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [4,2,5,3,1] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [5,4,2,1,3] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [5,2,4,3,1] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [2,5,4,3,1] => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [3,2,5,1,4] => 4
Description
The number of occurrences of the pattern 213 in a permutation.
Matching statistic: St000220
(load all 3 compositions to match this statistic)
Mapping
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => [1,2] => 0
[1,1,0,0]
 => [1,2] => [2,1] => 0
[1,0,1,0,1,0]
 => [2,1,3] => [3,1,2] => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,3,2] => 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,1,3] => 0
[1,1,0,1,0,0]
 => [1,3,2] => [2,3,1] => 0
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [3,4,1,2] => 0
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => 0
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [4,1,3,2] => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [4,2,1,3] => 0
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [3,2,1,4] => 0
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [3,2,4,1] => 0
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [3,4,2,1] => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [5,3,4,1,2] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [5,3,1,4,2] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [3,5,4,1,2] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [3,5,1,4,2] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [3,1,5,4,2] => 4
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [4,3,5,1,2] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [4,3,1,5,2] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [4,5,1,3,2] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,1,5,3,2] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [5,4,1,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [5,1,4,3,2] => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,5,4,3,2] => 6
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [5,2,4,1,3] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [5,2,1,4,3] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,5,4,1,3] => 3
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,5,1,4,3] => 4
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,5,4,3] => 6
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [4,2,5,1,3] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [4,2,1,5,3] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [4,5,2,1,3] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [4,2,5,3,1] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [5,4,2,1,3] => 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [5,2,4,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [2,5,4,3,1] => 3
[1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [3,2,5,1,4] => 2
Description
The number of occurrences of the pattern 132 in a permutation.
Matching statistic: St000423
(load all 4 compositions to match this statistic)
Mapping
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000423: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => [1,2] => 0
[1,1,0,0]
 => [1,2] => [2,1] => 0
[1,0,1,0,1,0]
 => [2,1,3] => [3,1,2] => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,3,2] => 1
[1,1,0,0,1,0]
 => [3,1,2] => [2,1,3] => 0
[1,1,0,1,0,0]
 => [1,3,2] => [2,3,1] => 0
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [3,4,1,2] => 0
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => 0
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [4,1,3,2] => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [4,2,1,3] => 0
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [3,2,1,4] => 0
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [3,2,4,1] => 0
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [3,4,2,1] => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [5,3,4,1,2] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [5,3,1,4,2] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [3,5,4,1,2] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [3,5,1,4,2] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [3,1,5,4,2] => 4
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [4,3,5,1,2] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [4,3,1,5,2] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [4,5,1,3,2] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,1,5,3,2] => 3
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [5,4,1,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [5,1,4,3,2] => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,5,4,3,2] => 6
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [5,2,4,1,3] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [5,2,1,4,3] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,5,4,1,3] => 3
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,5,1,4,3] => 4
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,5,4,3] => 6
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [4,2,5,1,3] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [4,2,1,5,3] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [4,5,2,1,3] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [4,2,5,3,1] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [5,4,2,1,3] => 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [5,2,4,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [2,5,4,3,1] => 3
[1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [3,2,5,1,4] => 2
Description
The number of occurrences of the pattern 123 or of the pattern 132 in a permutation.
Matching statistic: St000428
(load all 5 compositions to match this statistic)
Mapping
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000428: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [2,1] => [1,2] => 0
[1,1,0,0]
 => [1,2] => [2,1] => 0
[1,0,1,0,1,0]
 => [2,1,3] => [3,1,2] => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,3,2] => 0
[1,1,0,0,1,0]
 => [3,1,2] => [2,1,3] => 1
[1,1,0,1,0,0]
 => [1,3,2] => [2,3,1] => 0
[1,1,1,0,0,0]
 => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [3,4,1,2] => 0
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => 0
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [4,1,3,2] => 0
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,4,3,2] => 0
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [2,4,3,1] => 0
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [3,4,2,1] => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [5,3,4,1,2] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [5,3,1,4,2] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [3,5,4,1,2] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [3,5,1,4,2] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [3,1,5,4,2] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [4,3,5,1,2] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [4,3,1,5,2] => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [4,5,1,3,2] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,1,5,3,2] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [5,4,1,3,2] => 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [5,1,4,3,2] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,5,4,3,2] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [5,2,4,1,3] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [5,2,1,4,3] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,5,4,1,3] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [2,5,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [2,1,5,4,3] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [4,2,5,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [4,2,1,5,3] => 4
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [4,5,2,1,3] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [4,2,5,3,1] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [5,4,2,1,3] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [5,2,4,3,1] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [2,5,4,3,1] => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [3,2,5,1,4] => 4
Description
The number of occurrences of the pattern 123 or of the pattern 213 in a permutation.
Matching statistic: St001398
(load all 2 compositions to match this statistic)
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001398: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => [2,1] => ([],2)
 => 0
[1,1,0,0]
 => [2,1] => [1,2] => ([(0,1)],2)
 => 0
[1,0,1,0,1,0]
 => [1,2,3] => [3,2,1] => ([],3)
 => 0
[1,0,1,1,0,0]
 => [1,3,2] => [2,3,1] => ([(1,2)],3)
 => 0
[1,1,0,0,1,0]
 => [2,1,3] => [3,1,2] => ([(1,2)],3)
 => 0
[1,1,0,1,0,0]
 => [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => 1
[1,1,1,0,0,0]
 => [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4,3,2,1] => ([],4)
 => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
 => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
 => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
 => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 0
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 6
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => 0
Description
Number of subsets of size 3 of elements in a poset that form a "v". For a finite poset $(P,\leq)$, this is the number of sets $\{x,y,z\} \in \binom{P}{3}$ that form a "v"-subposet (i.e., a subposet consisting of a bottom element covered by two incomparable elements).
Matching statistic: St000219
(load all 7 compositions to match this statistic)
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000219: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1,2] => ? ∊ {0,0}
[1,1,0,0]
 => [2,1] => ? ∊ {0,0}
[1,0,1,0,1,0]
 => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => 0
[1,1,0,0,1,0]
 => [2,1,3] => 0
[1,1,0,1,0,0]
 => [2,3,1] => 1
[1,1,1,0,0,0]
 => [3,1,2] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 0
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 6
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 0
[1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 0
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => 1
Description
The number of occurrences of the pattern 231 in a permutation.
Matching statistic: St001266
(load all 3 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001266: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 64%●distinct values known / distinct values provided: 45%
Values
[1,0,1,0]
 => [1]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,0,0]
 => []
 => ?
 => ?
 => ? = 0
[1,0,1,0,1,0]
 => [2,1]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => [2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,0,1,0,0]
 => [1]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,1,0,0,0]
 => []
 => ?
 => ?
 => ? = 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [3]
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [1]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,1,1,0,0,0,0]
 => []
 => ?
 => ?
 => ? = 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? ∊ {0,0,1,1,1,3,4,6,6}
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ? ∊ {0,0,1,1,1,3,4,6,6}
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,1,1,1,3,4,6,6}
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => ? ∊ {0,0,1,1,1,3,4,6,6}
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 4
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => ? ∊ {0,0,1,1,1,3,4,6,6}
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 3
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? ∊ {0,0,1,1,1,3,4,6,6}
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? ∊ {0,0,1,1,1,3,4,6,6}
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,1,1,1,3,4,6,6}
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,1,3,4,6,6}
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [5,5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [7,7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [3,3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [5,5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [8,3,3]
 => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [9,4]
 => [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [7,5]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [7,3,2]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [9,2]
 => [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [7,4,1,1]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [3,3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [3,3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [6,3,3,1]
 => [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [8,3,1]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [5,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,2,2,1,1]
 => [7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => [7,2,1,1]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => [3,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,3,1,1,1]
 => [3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1,1]
 => [6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => [2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [7,4,3]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [8,5]
 => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,1,0,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2]
 => [7,4,1]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => [8,4,1]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => [8,4]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,1,0,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2]
 => [7,2,2,1]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2]
 => [8,3]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,1,0,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2]
 => [7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => [8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2]
 => [8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2]
 => [8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,9,9,9,10,12}
Description
The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra.
The following 232 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000454The largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001964The interval resolution global dimension of a poset. St000137The Grundy value of an integer partition. St000944The 3-degree of an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000693The modular (standard) major index of a standard tableau. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000984The number of boxes below precisely one peak. St001480The number of simple summands of the module J^2/J^3. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000815The number of semistandard Young tableaux of partition weight of given shape. St000941The number of characters of the symmetric group whose value on the partition is even. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000940The number of characters of the symmetric group whose value on the partition is zero. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000909The number of maximal chains of maximal size in a poset. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001470The cyclic holeyness of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001877Number of indecomposable injective modules with projective dimension 2. St001624The breadth of a lattice. St000100The number of linear extensions of a poset. St000307The number of rowmotion orbits of a poset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St001330The hat guessing number of a graph. St000441The number of successions of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000650The number of 3-rises of a permutation. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001731The factorization defect of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St000527The width of the poset. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001525The number of symmetric hooks on the diagonal of a partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001651The Frankl number of a lattice. St001902The number of potential covers of a poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000455The second largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001095The number of non-isomorphic posets with precisely one further covering relation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000929The constant term of the character polynomial of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001248Sum of the even parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001571The Cartan determinant of the integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000928The sum of the coefficients of the character polynomial of an integer partition. St000181The number of connected components of the Hasse diagram for the poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001413Half the length of the longest even length palindromic prefix of a binary word. St001557The number of inversions of the second entry of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001846The number of elements which do not have a complement in the lattice. St000741The Colin de Verdière graph invariant. St000260The radius of a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000933The number of multipartitions of sizes given by an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000058The order of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001820The size of the image of the pop stack sorting operator. St000355The number of occurrences of the pattern 21-3. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001271The competition number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001060The distinguishing index of a graph. St001487The number of inner corners of a skew partition. St000264The girth of a graph, which is not a tree. St000284The Plancherel distribution on integer partitions. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000934The 2-degree of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001568The smallest positive integer that does not appear twice in the partition. St001644The dimension of a graph. St000570The Edelman-Greene number of a permutation. St000920The logarithmic height of a Dyck path. St000091The descent variation of a composition. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001866The nesting alignments of a signed permutation. St000237The number of small exceedances. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000884The number of isolated descents of a permutation. St001394The genus of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001821The sorting index of a signed permutation. St001979The size of the permutation set corresponding to the alternating sign matrix variety. St000764The number of strong records in an integer composition. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000534The number of 2-rises of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000883The number of longest increasing subsequences of a permutation.