Your data matches 310 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001181
(load all 9 compositions to match this statistic)
Mapping
St001181: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 0 = 1 - 1
[1,0,1,0]
 => 0 = 1 - 1
[1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => 0 = 1 - 1
Description
Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001085
(load all 47 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 0 = 1 - 1
Description
The number of occurrences of the vincular pattern |21-3 in a permutation. This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St000007
(load all 12 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 1
[1,0,1,0]
 => [1,2] => [1,2] => 1
[1,1,0,0]
 => [2,1] => [1,2] => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,2,3] => 1
[1,1,0,0,1,0]
 => [2,1,3] => [1,2,3] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [1,2,3] => 1
[1,1,1,0,0,0]
 => [3,2,1] => [1,3,2] => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,3,4] => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,2,3,4] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,2,3,4] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,2,4,3] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,2,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,2,3,4] => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [1,2,3,4] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,2,3,4] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [1,2,4,3] => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [1,3,2,4] => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,3,4,2] => 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [1,4,2,3] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,4,2,3] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,4,5] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,3,4,5] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,3,5,4] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,2,3,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,2,3,4,5] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,2,3,4,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,2,3,4,5] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,2,3,5,4] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,2,4,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,2,4,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,2,5,3,4] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,2,5,3,4] => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,2,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,2,3,4,5] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,2,3,4,5] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,2,3,4,5] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [1,2,3,5,4] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [1,2,3,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [1,2,3,4,5] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [1,2,3,4,5] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [1,2,3,5,4] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [1,2,4,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [1,2,4,5,3] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [1,2,5,3,4] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [1,2,5,3,4] => 2
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000092
(load all 7 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 1
[1,0,1,0]
 => [1,2] => [1,2] => 1
[1,1,0,0]
 => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,3,2] => 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,4,3,2] => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,4,3,2] => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,4,3,2] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,4,3,1] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,5,4,3,2] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,5,4,3,2] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,5,4,3,2] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,5,4,3,2] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,5,4,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,5,4,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,5,4,3] => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,5,4,3] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,5,4,3] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,5,1,4,3] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,5,1,4,3] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,5,4,1,3] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,5,4,3,1] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [2,5,4,3,1] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [2,5,4,1,3] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [2,5,4,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [2,5,4,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [2,5,4,3,1] => 1
Description
The number of outer peaks of a permutation. An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$. In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
Matching statistic: St000390
(load all 3 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => 1 => 1
[1,0,1,0]
 => [1,1] => 11 => 1
[1,1,0,0]
 => [2] => 10 => 1
[1,0,1,0,1,0]
 => [1,1,1] => 111 => 1
[1,0,1,1,0,0]
 => [1,2] => 110 => 1
[1,1,0,0,1,0]
 => [2,1] => 101 => 2
[1,1,0,1,0,0]
 => [3] => 100 => 1
[1,1,1,0,0,0]
 => [3] => 100 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1111 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => 1110 => 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => 1101 => 2
[1,0,1,1,0,1,0,0]
 => [1,3] => 1100 => 1
[1,0,1,1,1,0,0,0]
 => [1,3] => 1100 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => 1011 => 2
[1,1,0,0,1,1,0,0]
 => [2,2] => 1010 => 2
[1,1,0,1,0,0,1,0]
 => [3,1] => 1001 => 2
[1,1,0,1,0,1,0,0]
 => [4] => 1000 => 1
[1,1,0,1,1,0,0,0]
 => [4] => 1000 => 1
[1,1,1,0,0,0,1,0]
 => [3,1] => 1001 => 2
[1,1,1,0,0,1,0,0]
 => [4] => 1000 => 1
[1,1,1,0,1,0,0,0]
 => [4] => 1000 => 1
[1,1,1,1,0,0,0,0]
 => [4] => 1000 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 11111 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 11110 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 11101 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => 11100 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 11100 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 11011 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 11010 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => 11001 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => 11000 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => 11000 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 11001 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => 11000 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 11000 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 11000 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 10111 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 10110 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 10101 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => 10100 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 10100 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => 10011 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => 10010 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => 10001 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5] => 10000 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => 10000 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => 10001 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [5] => 10000 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => 10000 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => 10000 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St001011
(load all 6 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000292
(load all 7 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => 1 => 0 = 1 - 1
[1,0,1,0]
 => [1,1] => 11 => 0 = 1 - 1
[1,1,0,0]
 => [2] => 10 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,1,1] => 111 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,2] => 110 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1] => 101 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [3] => 100 => 0 = 1 - 1
[1,1,1,0,0,0]
 => [3] => 100 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1111 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => 1110 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => 1101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [1,3] => 1100 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,3] => 1100 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => 1011 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,2] => 1010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
 => [3,1] => 1001 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [4] => 1000 => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
 => [4] => 1000 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3,1] => 1001 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
 => [4] => 1000 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [4] => 1000 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => [4] => 1000 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 11111 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 11110 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 11101 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => 11100 => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 11100 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 11011 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 11010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => 11001 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => 11000 => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => 11000 => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 11001 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => 11000 => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => 11000 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 11000 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 10111 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 10110 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 10101 => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => 10100 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 10100 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => 10011 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => 10010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => 10001 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5] => 10000 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => 10000 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => 10001 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [5] => 10000 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => 10000 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => 10000 => 0 = 1 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000356
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
 => [2,1] => [1,2] => 0 = 1 - 1
[1,1,0,0]
 => [1,2] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [3,2,1] => [2,1,3] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => 0 = 1 - 1
[1,1,0,0,1,0]
 => [3,1,2] => [3,1,2] => 0 = 1 - 1
[1,1,0,1,0,0]
 => [2,1,3] => [1,3,2] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [2,3,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [3,2,1,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [3,1,2,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,3,1,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [2,1,3,4] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [4,2,1,3] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,1,2,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [3,4,1,2] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [3,1,4,2] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,3,4,2] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [4,3,2,1,5] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [4,3,1,2,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [4,2,3,1,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [4,2,1,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [4,1,2,3,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [3,4,2,1,5] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,4,1,2,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,2,4,1,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [3,2,1,4,5] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [3,1,2,4,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [2,3,4,1,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [2,3,1,4,5] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [2,1,3,4,5] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [5,3,2,1,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [5,3,1,2,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [5,2,3,1,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [5,2,1,3,4] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [5,1,2,3,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [3,5,2,1,4] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [3,5,1,2,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [3,2,5,1,4] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [3,2,1,5,4] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [3,1,2,5,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,3,5,1,4] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,3,1,5,4] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [2,1,3,5,4] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,2,3,5,4] => 1 = 2 - 1
Description
The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000386
(load all 15 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1,0]
 => 0 = 1 - 1
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => 0 = 1 - 1
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 2 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000647
(load all 4 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,3,2] => 0 = 1 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 0 = 1 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,4,3,2] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,4,3,2] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,4,3,2] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,4,3] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,4,3,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,4,3,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => 1 = 2 - 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,5,4,3] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,5,4,3] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,5,4,3] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,5,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,5,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,5,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,5,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [2,5,4,1,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [2,5,4,3,1] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [2,5,4,3,1] => 1 = 2 - 1
Description
The number of big descents of a permutation. For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$. The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1]. For the number of small descents, see [[St000214]].
The following 300 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000099The number of valleys of a permutation, including the boundary. St000159The number of distinct parts of the integer partition. St000201The number of leaf nodes in a binary tree. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000340The number of non-final maximal constant sub-paths of length greater than one. St000522The number of 1-protected nodes of a rooted tree. St000659The number of rises of length at least 2 of a Dyck path. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000991The number of right-to-left minima of a permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001732The number of peaks visible from the left. St000023The number of inner peaks of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000291The number of descents of a binary word. St000358The number of occurrences of the pattern 31-2. St000504The cardinality of the first block of a set partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001840The number of descents of a set partition. St000353The number of inner valleys of a permutation. St000779The tier of a permutation. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000837The number of ascents of distance 2 of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001354The number of series nodes in the modular decomposition of a graph. St001829The common independence number of a graph. St000256The number of parts from which one can substract 2 and still get an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000764The number of strong records in an integer composition. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000664The number of right ropes of a permutation. St000451The length of the longest pattern of the form k 1 2. St001568The smallest positive integer that does not appear twice in the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000834The number of right outer peaks of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St001432The order dimension of the partition. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000668The least common multiple of the parts of the partition. St001128The exponens consonantiae of a partition. St000759The smallest missing part in an integer partition. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St001955The number of natural descents for set-valued two row standard Young tableaux. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000260The radius of a connected graph. St001389The number of partitions of the same length below the given integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001061The number of indices that are both descents and recoils of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000298The order dimension or Dushnik-Miller dimension of a poset. St000632The jump number of the poset. St000640The rank of the largest boolean interval in a poset. St000989The number of final rises of a permutation. St001875The number of simple modules with projective dimension at most 1. St000455The second largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000937The number of positive values of the symmetric group character corresponding to the partition. St000374The number of exclusive right-to-left minima of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000245The number of ascents of a permutation. St000035The number of left outer peaks of a permutation. St000542The number of left-to-right-minima of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001665The number of pure excedances of a permutation. St001728The number of invisible descents of a permutation. St000307The number of rowmotion orbits of a poset. St000906The length of the shortest maximal chain in a poset. St000028The number of stack-sorts needed to sort a permutation. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000649The number of 3-excedences of a permutation. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000744The length of the path to the largest entry in a standard Young tableau. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001964The interval resolution global dimension of a poset. St000654The first descent of a permutation. St001597The Frobenius rank of a skew 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. St001737The number of descents of type 2 in a permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000732The number of double deficiencies of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000877The depth of the binary word interpreted as a path. St000031The number of cycles in the cycle decomposition of a permutation. St001330The hat guessing number of a graph. St000308The height of the tree associated to a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000731The number of double exceedences of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000633The size of the automorphism group of a poset. St000022The number of fixed points of a permutation. St000534The number of 2-rises of a permutation. St001624The breadth of a lattice. St000100The number of linear extensions of a poset. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001399The distinguishing number of a poset. St000761The number of ascents in an integer composition. St001877Number of indecomposable injective modules with projective dimension 2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001487The number of inner corners of a skew partition. St000015The number of peaks of a Dyck path. St000456The monochromatic index of a connected graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000864The number of circled entries of the shifted recording tableau of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001530The depth of a Dyck path. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000062The length of the longest increasing subsequence of the permutation. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000166The depth minus 1 of an ordered tree. St000252The number of nodes of degree 3 of a binary tree. St000317The cycle descent number of a permutation. St000338The number of pixed points of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000094The depth of an ordered tree. St000527The width of the poset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001960The number of descents of a permutation minus one if its first entry is not one. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001571The Cartan determinant of the integer partition. 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. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001118The acyclic chromatic index of a graph. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001820The size of the image of the pop stack sorting operator. St001890The maximum magnitude of the Möbius function of a poset. St000914The sum of the values of the Möbius function of a poset. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000993The multiplicity of the largest part of an integer partition. St001280The number of parts of an integer partition that are at least two. St001498The normalised height of a Nakayama algebra with magnitude 1. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000145The Dyson rank of a partition. St000181The number of connected components of the Hasse diagram for the poset. St000284The Plancherel distribution on integer partitions. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000681The Grundy value of Chomp on Ferrers diagrams. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000366The number of double descents of a permutation. St001846The number of elements which do not have a complement in the lattice. St001720The minimal length of a chain of small intervals in a lattice. St000741The Colin de Verdière graph invariant. 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. St001268The size of the largest ordinal summand in the 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. 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. St001397Number of pairs of incomparable elements in a finite poset. St001862The number of crossings of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001715The number of non-records in a permutation. St000884The number of isolated descents of a permutation. St001435The number of missing boxes in the first row. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000663The number of right floats of a permutation. St001535The number of cyclic alignments of a permutation. St001096The size of the overlap set of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000920The logarithmic height of a Dyck path. St000768The number of peaks in an integer composition. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St000669The number of permutations obtained by switching ascents or descents of size 2. St000711The number of big exceedences of a permutation. St000765The number of weak records in an integer composition. St000805The number of peaks of the associated bargraph. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000658The number of rises of length 2 of a Dyck path. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001394The genus of a permutation. St001438The number of missing boxes of a skew partition. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000883The number of longest increasing subsequences of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000237The number of small exceedances. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation.