Your data matches 54 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001504
(load all 35 compositions to match this statistic)
Mapping
St001504: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 2
[1,0,1,0]
 => 2
[1,1,0,0]
 => 3
[1,0,1,0,1,0]
 => 2
[1,0,1,1,0,0]
 => 4
[1,1,0,0,1,0]
 => 3
[1,1,0,1,0,0]
 => 3
[1,1,1,0,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => 2
[1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,0]
 => 4
[1,0,1,1,1,0,0,0]
 => 5
[1,1,0,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => 5
[1,1,1,0,0,0,1,0]
 => 4
[1,1,1,0,0,1,0,0]
 => 4
[1,1,1,0,1,0,0,0]
 => 4
[1,1,1,1,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => 2
[1,0,1,0,1,0,1,1,0,0]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => 5
[1,0,1,1,0,0,1,0,1,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => 6
[1,0,1,1,0,1,0,0,1,0]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => 5
[1,0,1,1,1,0,0,1,0,0]
 => 5
[1,0,1,1,1,0,1,0,0,0]
 => 5
[1,0,1,1,1,1,0,0,0,0]
 => 6
[1,1,0,0,1,0,1,0,1,0]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => 6
[1,1,0,1,0,0,1,0,1,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => 5
[1,1,0,1,1,0,0,0,1,0]
 => 5
[1,1,0,1,1,0,0,1,0,0]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => 5
[1,1,0,1,1,1,0,0,0,0]
 => 6
Description
The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001182
(load all 3 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001182: 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]
 => 3 = 2 + 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5 = 4 + 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]
 => 5 = 4 + 1
[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]
 => 5 = 4 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 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]
 => 6 = 5 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 5 = 4 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[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]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 7 = 6 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => 6 = 5 + 1
[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]
 => 6 = 5 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 7 = 6 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 7 = 6 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 6 = 5 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 6 = 5 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 7 = 6 + 1
Description
Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000673
(load all 3 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [2,1] => [2,1] => 2
[1,0,1,0]
 => [3,1,2] => [2,3,1] => [3,2,1] => 2
[1,1,0,0]
 => [2,3,1] => [3,1,2] => [3,1,2] => 3
[1,0,1,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => [4,2,3,1] => 2
[1,0,1,1,0,0]
 => [3,1,4,2] => [4,2,3,1] => [3,4,2,1] => 4
[1,1,0,0,1,0]
 => [2,4,1,3] => [3,4,1,2] => [4,1,3,2] => 3
[1,1,0,1,0,0]
 => [4,3,1,2] => [2,4,3,1] => [3,2,4,1] => 3
[1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 4
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [2,3,4,5,1] => [5,2,3,4,1] => 2
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [2,5,3,4,1] => [4,2,5,3,1] => 4
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [4,5,2,3,1] => [3,5,2,4,1] => 4
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [3,4,2,5,1] => [5,4,3,2,1] => 4
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [5,2,3,1,4] => [3,5,2,1,4] => 5
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [3,4,5,1,2] => [5,1,3,4,2] => 3
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [5,3,4,1,2] => [4,1,5,3,2] => 5
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [2,4,5,3,1] => [3,2,5,4,1] => 3
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [2,3,5,4,1] => [4,2,3,5,1] => 3
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [5,2,4,3,1] => [3,4,5,2,1] => 5
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [4,5,1,2,3] => [5,1,2,4,3] => 4
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [3,5,4,1,2] => [4,1,3,5,2] => 4
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [2,4,1,5,3] => [5,2,4,1,3] => 4
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 5
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 2
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [2,3,6,4,5,1] => [5,2,3,6,4,1] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [2,5,6,3,4,1] => [4,2,6,3,5,1] => 4
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [2,4,5,3,6,1] => [6,2,5,4,3,1] => 4
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [2,6,3,4,1,5] => [4,2,6,3,1,5] => 5
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [4,5,6,2,3,1] => [3,6,2,4,5,1] => 4
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [6,4,5,2,3,1] => [3,5,2,6,4,1] => 6
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [3,4,2,5,6,1] => [6,4,3,2,5,1] => 4
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [3,4,5,2,6,1] => [6,5,3,4,2,1] => 4
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [6,3,4,2,5,1] => [5,4,6,3,2,1] => 6
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [5,6,2,3,1,4] => [3,6,2,1,5,4] => 5
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [4,6,5,2,3,1] => [3,5,2,4,6,1] => 5
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [3,5,2,4,6,1] => [6,5,3,2,4,1] => 5
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [6,2,3,1,4,5] => [3,6,2,1,4,5] => 6
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [3,4,5,6,1,2] => [6,1,3,4,5,2] => 3
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [3,6,4,5,1,2] => [5,1,3,6,4,2] => 5
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [5,6,3,4,1,2] => [4,1,6,3,5,2] => 5
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [4,5,3,6,1,2] => [6,1,5,4,3,2] => 5
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [6,3,4,1,2,5] => [4,1,6,3,2,5] => 6
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [2,4,5,6,3,1] => [3,2,6,4,5,1] => 3
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [2,6,4,5,3,1] => [3,2,5,6,4,1] => 5
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [2,3,5,6,4,1] => [4,2,3,6,5,1] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [2,6,3,5,4,1] => [4,2,5,6,3,1] => 5
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [5,6,2,4,3,1] => [3,4,6,2,5,1] => 5
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [4,5,2,6,3,1] => [3,6,5,4,2,1] => 5
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,5,2,6,4,1] => [4,6,3,5,2,1] => 5
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [6,2,4,3,1,5] => [3,4,6,2,1,5] => 6
Description
The number of non-fixed points of a permutation. In other words, this statistic is $n$ minus the number of fixed points ([[St000022]]) of $\pi$.
Matching statistic: St001279
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [2] => [2]
 => 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1] => [2,1]
 => 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3] => [3]
 => 3
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => [2,1,1]
 => 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => [2,2]
 => 4
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => [3,1]
 => 3
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => [3,1]
 => 3
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => [4]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [2,1,1,1]
 => 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [2,2,1]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [2,2,1]
 => 4
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [2,2,1]
 => 4
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [3,2]
 => 5
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => [3,1,1]
 => 3
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2] => [3,2]
 => 5
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => [3,1,1]
 => 3
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => [3,1,1]
 => 3
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,2] => [3,2]
 => 5
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1] => [4,1]
 => 4
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1] => [4,1]
 => 4
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,1] => [4,1]
 => 4
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => [5]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1,1] => [2,1,1,1,1]
 => 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,1,1,2] => [2,2,1,1]
 => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,1,2,1] => [2,2,1,1]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,1,2,1] => [2,2,1,1]
 => 4
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,1,3] => [3,2,1]
 => 5
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,2,1,1] => [2,2,1,1]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,2,2] => [2,2,2]
 => 6
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,2,1,1] => [2,2,1,1]
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,2,1,1] => [2,2,1,1]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,2,2] => [2,2,2]
 => 6
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,1] => [3,2,1]
 => 5
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,1] => [3,2,1]
 => 5
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,1] => [3,2,1]
 => 5
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4] => [4,2]
 => 6
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,1,1] => [3,1,1,1]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,2] => [3,2,1]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,1] => [3,2,1]
 => 5
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,2,1] => [3,2,1]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,3] => [3,3]
 => 6
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,1,1,1] => [3,1,1,1]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,1,2] => [3,2,1]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,1,1,1] => [3,1,1,1]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,1,1,1] => [3,1,1,1]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,1,2] => [3,2,1]
 => 5
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,2,1] => [3,2,1]
 => 5
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,2,1] => [3,2,1]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,1] => [3,2,1]
 => 5
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,3] => [3,3]
 => 6
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St000240
(load all 4 compositions to match this statistic)
Mapping
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000240: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1] => [1] => 1 = 2 - 1
[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => [2,3,1] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [3,2,1] => 3 = 4 - 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [2,3,4,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [2,4,3,1] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,3,2,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,2,4,1] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,2,3,1] => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,3,1,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,2,1,4] => 4 = 5 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [2,3,4,5,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => [2,3,5,4,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => [2,5,4,3,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => [2,4,3,5,1] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => [2,5,3,4,1] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [5,3,4,2,1] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => [5,4,3,2,1] => 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [4,3,2,5,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [3,2,4,5,1] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [3,2,5,4,1] => 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [5,3,2,4,1] => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [5,2,4,3,1] => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,2,3,5,1] => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [2,3,4,1,5] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => [2,4,3,1,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [4,3,2,1,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,2,4,1,5] => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [2,3,1,5,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,4,5,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,3,4,5,2] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,3,5,4,2] => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,5,4,3,2] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,4,3,5,2] => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,3,4,2] => 5 = 6 - 1
Description
The number of indices that are not small excedances. A small excedance is an index $i$ for which $\pi_i = i+1$.
Matching statistic: St000722
(load all 2 compositions to match this statistic)
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000722: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1 = 2 - 1
[1,0,1,0]
 => [1,1] => [2] => ([],2)
 => 1 = 2 - 1
[1,1,0,0]
 => [2] => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [1,1,1] => [3] => ([],3)
 => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,2] => [1,2] => ([(1,2)],3)
 => 3 = 4 - 1
[1,1,0,0,1,0]
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => ([],4)
 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,3] => ([(2,3)],4)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 5 - 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => ([],5)
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,4] => ([(3,4)],5)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 6 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 6 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 6 - 1
Description
The number of different neighbourhoods in a graph.
Matching statistic: St000831
Mapping
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000831: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,0]
 => [1] => ? = 2 - 2
[1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 0 = 2 - 2
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 1 = 3 - 2
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0 = 2 - 2
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2 = 4 - 2
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1 = 3 - 2
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1 = 3 - 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 4 - 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 2 = 4 - 2
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2 = 4 - 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 3 = 5 - 2
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 3 = 5 - 2
[1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1 = 3 - 2
[1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 3 = 5 - 2
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2 = 4 - 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 2 = 4 - 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 3 = 5 - 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 4 = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 2 = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 2 = 4 - 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 4 = 6 - 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 3 = 5 - 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 3 = 5 - 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => 3 = 5 - 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 4 = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 3 = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => 4 = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1 = 3 - 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1 = 3 - 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 3 = 5 - 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => 3 = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 3 = 5 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 4 = 6 - 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 2 = 4 - 2
Description
The number of indices that are either descents or recoils. This is, for a permutation $\pi$ of length $n$, this statistics counts the set $$\{ 1 \leq i < n : \pi(i) > \pi(i+1) \text{ or } \pi^{-1}(i) > \pi^{-1}(i+1)\}.$$
Matching statistic: St000454
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 0 = 2 - 2
[1,0,1,0]
 => [1,1] => [2] => ([],2)
 => 0 = 2 - 2
[1,1,0,0]
 => [2] => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
[1,0,1,0,1,0]
 => [1,1,1] => [3] => ([],3)
 => 0 = 2 - 2
[1,0,1,1,0,0]
 => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 2
[1,1,0,0,1,0]
 => [2,1] => [1,2] => ([(1,2)],3)
 => 1 = 3 - 2
[1,1,0,1,0,0]
 => [2,1] => [1,2] => ([(1,2)],3)
 => 1 = 3 - 2
[1,1,1,0,0,0]
 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => ([],4)
 => 0 = 2 - 2
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 4 - 2
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4 - 2
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4 - 2
[1,0,1,1,1,0,0,0]
 => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 - 2
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,3] => ([(2,3)],4)
 => 1 = 3 - 2
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 - 2
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,3] => ([(2,3)],4)
 => 1 = 3 - 2
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,3] => ([(2,3)],4)
 => 1 = 3 - 2
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 - 2
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => ([],5)
 => 0 = 2 - 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 4 - 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4 - 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 - 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4 - 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4 - 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 - 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 - 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 1 = 3 - 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 - 2
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 1 = 3 - 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 1 = 3 - 2
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => 1 = 3 - 2
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 - 2
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 - 2
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,0,0,1,1,0,0,0]
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 - 2
[1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,0,1,0,0,1,0,0]
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,0,1,0,1,0,0,0]
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 4 - 2
[1,1,1,0,1,1,0,0,0,0]
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 - 2
[1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,1,1,0,0,1,0,0,0]
 => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,1,1,0,1,0,0,0,0]
 => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 5 - 2
[1,1,1,1,1,0,0,0,0,0]
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 6 - 2
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000235
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 80%
Values
[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [3,1,2] => 3 = 2 + 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 2 + 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 4 = 3 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5 = 4 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => 4 = 3 + 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 4 = 3 + 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 5 = 4 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 5 = 4 + 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]
 => [4,3,1,6,2,5] => 5 = 4 + 1
[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]
 => [2,4,1,6,3,5] => 5 = 4 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 6 = 5 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 6 = 5 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 4 = 3 + 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]
 => [3,1,6,2,4,5] => 6 = 5 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => 5 = 4 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => 5 = 4 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 5 = 4 + 1
[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]
 => [6,1,2,3,4,5] => 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [4,3,1,5,7,2,6] => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => ? = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,3,4,1,7,2,6] => ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => ? = 6 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [2,5,4,1,7,3,6] => ? = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => ? = 5 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => ? = 5 + 1
[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]
 => [2,5,1,3,7,4,6] => ? = 5 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => ? = 5 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => ? = 5 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => ? = 5 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => ? = 6 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [2,7,4,5,1,3,6] => ? = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => ? = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [2,3,7,5,1,4,6] => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => ? = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => ? = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => ? = 5 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,3,1,7,2,5,6] => ? = 5 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 5 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ? = 6 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => ? = 4 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => ? = 6 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => ? = 4 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [7,3,4,1,2,5,6] => ? = 4 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => ? = 6 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => ? = 4 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [2,7,4,1,3,5,6] => ? = 4 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 4 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => ? = 6 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,1,2,5,3,4,6] => ? = 5 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => ? = 5 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 5 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => ? = 5 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
Description
The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index $i < n$ such that $\pi_i = i+1$, or the index $i = n$ if $\pi_n = 1$.
Matching statistic: St000238
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000238: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 80%
Values
[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [3,1,2] => 3 = 2 + 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 3 = 2 + 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 4 = 3 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 5 = 4 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => 4 = 3 + 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 4 = 3 + 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 5 = 4 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 5 = 4 + 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]
 => [4,3,1,6,2,5] => 5 = 4 + 1
[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]
 => [2,4,1,6,3,5] => 5 = 4 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 6 = 5 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 6 = 5 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 4 = 3 + 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]
 => [3,1,6,2,4,5] => 6 = 5 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => 5 = 4 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => 5 = 4 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 5 = 4 + 1
[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]
 => [6,1,2,3,4,5] => 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [4,3,1,5,7,2,6] => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ? = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => ? = 5 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,3,4,1,7,2,6] => ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => ? = 6 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [2,5,4,1,7,3,6] => ? = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => ? = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => ? = 6 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => ? = 5 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => ? = 5 + 1
[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]
 => [2,5,1,3,7,4,6] => ? = 5 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => ? = 5 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => ? = 5 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => ? = 5 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => ? = 6 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [2,7,4,5,1,3,6] => ? = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => ? = 5 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [2,3,7,5,1,4,6] => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => ? = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => ? = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => ? = 5 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,3,1,7,2,5,6] => ? = 5 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ? = 5 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ? = 6 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [7,1,4,5,2,3,6] => ? = 4 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => ? = 6 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => ? = 4 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [7,3,4,1,2,5,6] => ? = 4 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => ? = 6 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => ? = 4 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [2,7,4,1,3,5,6] => ? = 4 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [2,3,7,1,4,5,6] => ? = 4 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => ? = 6 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,1,2,5,3,4,6] => ? = 5 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => ? = 5 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [7,3,1,2,4,5,6] => ? = 5 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => ? = 5 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 6 + 1
Description
The number of indices that are not small weak excedances. A small weak excedance is an index $i$ such that $\pi_i \in \{i,i+1\}$.
The following 44 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000242The number of indices that are not cyclical small weak excedances. St000670The reversal length of a permutation. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001877Number of indecomposable injective modules with projective dimension 2. 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. St001645The pebbling number of a connected graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001733The number of weak left to right maxima of a Dyck path. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000742The number of big ascents of a permutation after prepending zero. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000491The number of inversions of a set partition. St000565The major index of a set partition. St001330The hat guessing number of a graph. St000474Dyson's crank of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000091The descent variation of a composition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001778The largest greatest common divisor of an element and its image in a permutation. St001806The upper middle entry of a permutation. St000653The last descent of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001497The position of the largest weak excedence of a permutation. St001557The number of inversions of the second entry of a permutation. St001822The number of alignments of a signed permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000173The segment statistic of a semistandard tableau. St000233The number of nestings of a set partition. St000365The number of double ascents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000497The lcb statistic of a set partition. St000562The number of internal points of a set partition. St000624The normalized sum of the minimal distances to a greater element. St000646The number of big ascents of a permutation. St000836The number of descents of distance 2 of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.