Your data matches 17 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001492
(load all 10 compositions to match this statistic)
Mapping
St001492: 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]
 => 3
[1,0,1,1,0,0]
 => 3
[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]
 => 4
[1,0,1,0,1,1,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,0]
 => 4
[1,0,1,1,1,0,0,0]
 => 4
[1,1,0,0,1,0,1,0]
 => 4
[1,1,0,0,1,1,0,0]
 => 4
[1,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => 4
[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]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => 5
[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]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => 5
[1,0,1,1,1,0,0,0,1,0]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => 5
[1,0,1,1,1,1,0,0,0,0]
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => 4
[1,1,0,0,1,1,0,1,0,0]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => 5
[1,1,0,1,0,0,1,0,1,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => 5
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
Matching statistic: St001211
(load all 4 compositions to match this statistic)
Mapping
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,0]
 => 2
[1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 3
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 3
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 3
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 3
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 4
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 4
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 4
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 4
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 4
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 4
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 4
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 5
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 5
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 5
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 4
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 5
Description
The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.
Matching statistic: St000105
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ 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]
 => [2,1] => [2,1] => {{1,2}}
 => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => [1,2] => {{1},{2}}
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [2,3,1] => [3,2,1] => {{1,3},{2}}
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => {{1,2},{3}}
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => {{1},{2,3}}
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [3,1,2] => [3,1,2] => {{1,3},{2}}
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => {{1},{2},{3}}
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [4,1,3,2] => {{1,4},{2},{3}}
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,2,4] => {{1,3},{2},{4}}
 => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,2,3] => {{1},{2,4},{3}}
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4,1,2,3] => {{1,4},{2},{3}}
 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
 => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
 => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
 => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
 => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
 => 4 = 5 - 1
Description
The number of blocks in the set partition. The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
Matching statistic: St000507
Mapping
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ 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] => [[1],[2]]
 => 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]
 => [3,1,2] => [[1,2],[3]]
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => [[1,2],[3]]
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [[1,3],[2]]
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => [[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]
 => [4,1,2,3] => [[1,2,3],[4]]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [[1,2,3],[4]]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [[1,3],[2,4]]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [[1,2,4],[3]]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [[1,2],[3,4]]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [[1,2,4],[3]]
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [[1,2],[3,4]]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [[1,3,4],[2]]
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [[1,3,4],[2]]
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 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]
 => [5,1,2,3,4] => [[1,2,3,4],[5]]
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [[1,2,3,4],[5]]
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [[1,3,4],[2,5]]
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [[1,2,3,5],[4]]
 => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [[1,2,3,4],[5]]
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [[1,2,4],[3,5]]
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [[1,2,4],[3,5]]
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [[1,2,3],[4,5]]
 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [[1,3,4],[2,5]]
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [[1,2,3,5],[4]]
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [[1,3,4],[2,5]]
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [[1,3,5],[2,4]]
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [[1,2,4,5],[3]]
 => 4 = 5 - 1
[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] => [[1,2,3,4],[5]]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [[1,2,3],[4,5]]
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [[1,2,3],[4,5]]
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [[1,3,5],[2,4]]
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => [[1,2,5],[3,4]]
 => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [[1,2,3,5],[4]]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [[1,2,3],[4,5]]
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [[1,2,3],[4,5]]
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [[1,2,4],[3,5]]
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [[1,2,4],[3,5]]
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [[1,2,4],[3,5]]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [[1,3,5],[2,4]]
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [[1,2,5],[3,4]]
 => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [[1,3,4],[2,5]]
 => 3 = 4 - 1
[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] => [[1,2,3,5],[4]]
 => 4 = 5 - 1
Description
The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000250
Mapping
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000250: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1,0]
 => {{1}}
 => ? = 2
[1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => {{1},{2}}
 => 2
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => {{1,2}}
 => 3
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 3
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 3
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 3
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 3
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 4
[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}}
 => 4
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 4
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 3
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 4
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 4
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 4
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 4
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 4
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 3
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 4
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 4
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 4
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 4
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 5
[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}}
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 5
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 5
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => 5
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 5
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => 5
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 4
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 5
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 5
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 5
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 4
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 5
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => 5
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 5
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 5
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 5
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => {{1,2,4},{3},{5}}
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => {{1,2,5},{3,4}}
 => 5
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 5
Description
The number of blocks ([[St000105]]) plus the number of antisingletons ([[St000248]]) of a set partition.
Matching statistic: St000925
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => {{1}}
 => ? = 2 - 1
[1,0,1,0]
 => [2,1] => [2,1] => {{1,2}}
 => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => [1,2] => {{1},{2}}
 => 2 = 3 - 1
[1,0,1,0,1,0]
 => [2,3,1] => [3,2,1] => {{1,3},{2}}
 => 2 = 3 - 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => {{1,2},{3}}
 => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => {{1},{2,3}}
 => 2 = 3 - 1
[1,1,0,1,0,0]
 => [3,1,2] => [3,1,2] => {{1,3},{2}}
 => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => {{1},{2},{3}}
 => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
 => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
 => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
 => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
 => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
 => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
 => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [4,1,3,2] => {{1,4},{2},{3}}
 => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,2,4] => {{1,3},{2},{4}}
 => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,2,3] => {{1},{2,4},{3}}
 => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4,1,2,3] => {{1,4},{2},{3}}
 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
 => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
 => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
 => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
 => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
 => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
 => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
 => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
 => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
 => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
 => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
 => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
 => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
 => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
 => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
 => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
 => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
 => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
 => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
 => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
 => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
 => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
 => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
 => 4 = 5 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => 4 = 5 - 1
Description
The number of topologically connected components of a set partition. For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$. The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St000245
(load all 4 compositions to match this statistic)
Mapping
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1] => [1] => 0 = 2 - 2
[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => [2,1] => 0 = 2 - 2
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => [1,2] => 1 = 3 - 2
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,1,2] => [2,3,1] => 1 = 3 - 2
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 1 = 3 - 2
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => 1 = 3 - 2
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 1 = 3 - 2
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,3,4,1] => 2 = 4 - 2
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,3,4,2] => 2 = 4 - 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [3,1,4,2] => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [2,3,1,4] => 2 = 4 - 2
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => 2 = 4 - 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [3,4,1,2] => 2 = 4 - 2
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [2,4,1,3] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 1 = 3 - 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => 2 = 4 - 2
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1,2,4] => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => [2,3,4,5,1] => 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,3,4,5,2] => 3 = 5 - 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [3,1,4,5,2] => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => [2,3,4,1,5] => 3 = 5 - 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,2,4,5,3] => 3 = 5 - 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [3,4,1,5,2] => 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,4,2,5,3] => 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [2,3,5,1,4] => 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,1,4,5,3] => 2 = 4 - 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,3,4,2,5] => 3 = 5 - 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,1,2,5,3] => 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [3,1,4,2,5] => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => [2,3,1,4,5] => 3 = 5 - 2
[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] => [1,2,3,5,4] => 3 = 5 - 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => [3,4,5,1,2] => 3 = 5 - 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,4,5,2,3] => 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [4,1,5,2,3] => 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => [3,4,1,2,5] => 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => [2,4,5,1,3] => 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,3,5,2,4] => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => [2,4,1,5,3] => 2 = 4 - 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => [2,3,1,5,4] => 2 = 4 - 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 4 - 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [3,1,5,2,4] => 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => [2,4,1,3,5] => 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => 2 = 4 - 2
[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] => [1,2,4,3,5] => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 6 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,7,1,2,4,5,6] => [3,4,1,5,6,7,2] => ? = 6 - 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,7,3,4,5,6] => [2,1,4,5,6,7,3] => ? = 6 - 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,7,1,4,5,6] => [4,1,2,5,6,7,3] => ? = 6 - 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [2,6,1,3,4,5,7] => [3,1,4,5,6,2,7] => ? = 6 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [4,7,1,2,3,5,6] => [3,4,5,1,6,7,2] => ? = 6 - 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [2,4,7,1,3,5,6] => [4,1,5,2,6,7,3] => ? = 5 - 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [3,6,1,2,4,5,7] => [3,4,1,5,6,2,7] => ? = 6 - 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [3,1,7,2,4,5,6] => [2,4,1,5,6,7,3] => ? = 6 - 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [2,6,1,3,4,7,5] => [3,1,4,5,7,2,6] => ? = 6 - 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [5,1,2,3,6,4,7] => [2,3,4,6,1,5,7] => ? = 7 - 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,7,4,5,6] => [2,1,3,5,6,7,4] => ? = 6 - 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [3,4,7,1,2,5,6] => [4,5,1,2,6,7,3] => ? = 6 - 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 6 - 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [3,6,1,2,4,7,5] => [3,4,1,5,7,2,6] => ? = 6 - 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [2,1,4,7,3,5,6] => [2,1,5,3,6,7,4] => ? = 5 - 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [2,3,1,7,4,5,6] => [3,1,2,5,6,7,4] => ? = 6 - 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,6,3,4,5,7] => [2,1,4,5,6,3,7] => ? = 6 - 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [2,3,6,1,4,5,7] => [4,1,2,5,6,3,7] => ? = 6 - 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [2,5,1,3,4,6,7] => [3,1,4,5,2,6,7] => ? = 6 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [5,7,1,2,3,4,6] => [3,4,5,6,1,7,2] => ? = 6 - 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [2,5,7,1,3,4,6] => [4,1,5,6,2,7,3] => ? = 5 - 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [4,6,1,2,3,5,7] => [3,4,5,1,6,2,7] => ? = 6 - 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [3,5,7,1,2,4,6] => [4,5,1,6,2,7,3] => ? = 5 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 5 - 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => [4,6,1,2,3,7,5] => [3,4,5,1,7,2,6] => ? = 6 - 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [2,1,5,7,3,4,6] => [2,1,5,6,3,7,4] => ? = 5 - 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [2,3,5,7,1,4,6] => [5,1,2,6,3,7,4] => ? = 5 - 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [2,4,6,1,3,5,7] => [4,1,5,2,6,3,7] => ? = 5 - 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => [3,5,1,2,4,6,7] => [3,4,1,5,2,6,7] => ? = 6 - 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [2,6,1,3,7,4,5] => [3,1,4,6,7,2,5] => ? = 6 - 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [5,1,2,6,3,4,7] => [2,3,5,6,1,4,7] => ? = 7 - 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [4,1,7,2,3,5,6] => [2,4,5,1,6,7,3] => ? = 6 - 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,2,7,4,5,6] => [2,3,1,5,6,7,4] => ? = 6 - 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [2,4,1,7,3,5,6] => [3,1,5,2,6,7,4] => ? = 5 - 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [3,1,6,2,4,5,7] => [2,4,1,5,6,3,7] => ? = 6 - 2
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [3,6,1,2,7,4,5] => [3,4,1,6,7,2,5] => ? = 6 - 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [5,1,2,6,3,7,4] => [2,3,5,7,1,4,6] => ? = 7 - 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => [2,1,6,3,4,7,5] => [2,1,4,5,7,3,6] => ? = 6 - 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [3,1,4,7,2,5,6] => [2,5,1,3,6,7,4] => ? = 6 - 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => [2,5,1,3,4,7,6] => [3,1,4,5,2,7,6] => ? = 5 - 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => [2,3,4,1,6,5,7] => ? = 6 - 2
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,6,1,4,7,5] => [4,1,2,5,7,3,6] => ? = 6 - 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
 => [2,5,1,3,6,4,7] => [3,1,4,6,2,5,7] => ? = 6 - 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [4,1,2,5,3,6,7] => [2,3,5,1,4,6,7] => ? = 7 - 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [2,1,3,4,7,5,6] => [2,1,3,4,6,7,5] => ? = 6 - 2
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [4,5,7,1,2,3,6] => [4,5,6,1,2,7,3] => ? = 6 - 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 6 - 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => [3,4,6,1,2,5,7] => [4,5,1,2,6,3,7] => ? = 6 - 2
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [4,6,1,2,7,3,5] => [3,4,6,1,7,2,5] => ? = 6 - 2
Description
The number of ascents of a permutation.
Matching statistic: St000702
(load all 3 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ? = 2 - 1
[1,0,1,0]
 => [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
 => [1,2] => [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
 => [2,3,1] => [3,2,1] => 2 = 3 - 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => 2 = 3 - 1
[1,1,0,1,0,0]
 => [3,1,2] => [3,1,2] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [4,2,3,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,2,1,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4,2,1,3] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,4,3,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [4,1,3,2] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [4,3,2,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,2,4] => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,2,3] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4,1,2,3] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [5,2,3,1,4] => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [5,2,1,4,3] => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [5,2,4,3,1] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [4,2,1,3,5] => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,1,5,3,4] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,5,3,4,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,5,3,2,4] => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [5,1,3,4,2] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [4,1,3,2,5] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [5,3,2,4,1] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [5,4,3,2,1] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [4,3,2,1,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,1,2,5,4] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [5,1,3,2,4] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [5,3,2,1,4] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [3,1,2,4,5] => 4 = 5 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,6,1,7] => [6,2,3,4,5,1,7] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,3,4,5,1,7,6] => [5,2,3,4,1,7,6] => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,5,1,6,7] => [5,2,3,4,1,6,7] => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,3,4,1,6,7,5] => [4,2,3,1,7,6,5] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,4,1,6,5,7] => [4,2,3,1,6,5,7] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [2,3,4,6,1,7,5] => [7,2,3,4,1,6,5] => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,4,6,7,1,5] => [7,2,3,4,6,5,1] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [2,3,4,6,1,5,7] => [6,2,3,4,1,5,7] => ? = 7 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [2,3,4,1,5,7,6] => [4,2,3,1,5,7,6] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,3,4,1,7,5,6] => [4,2,3,1,7,5,6] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,7,1,5,6] => [7,2,3,4,1,5,6] => ? = 7 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,1,5,6,7] => [4,2,3,1,5,6,7] => ? = 7 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,3,1,5,6,7,4] => [3,2,1,7,5,6,4] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,3,1,5,6,4,7] => [3,2,1,6,5,4,7] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [2,3,1,5,4,7,6] => [3,2,1,5,4,7,6] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [2,3,1,5,7,4,6] => [3,2,1,7,5,4,6] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,3,1,5,4,6,7] => [3,2,1,5,4,6,7] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [2,3,5,1,6,7,4] => [7,2,3,1,5,6,4] => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [2,3,5,1,6,4,7] => [6,2,3,1,5,4,7] => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [2,3,5,6,1,7,4] => [7,2,3,5,4,6,1] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,3,5,6,7,1,4] => [7,2,3,6,5,4,1] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,6,1,4,7] => [6,2,3,5,4,1,7] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,7,6] => [5,2,3,1,4,7,6] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,3,5,1,7,4,6] => [7,2,3,1,5,4,6] => ? = 7 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,3,5,7,1,4,6] => [7,2,3,5,4,1,6] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,3,5,1,4,6,7] => [5,2,3,1,4,6,7] => ? = 7 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [2,3,1,4,6,7,5] => [3,2,1,4,7,6,5] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,3,1,4,6,5,7] => [3,2,1,4,6,5,7] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [2,3,1,6,4,7,5] => [3,2,1,7,4,6,5] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,3,1,6,7,4,5] => [3,2,1,7,6,5,4] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [2,3,1,6,4,5,7] => [3,2,1,6,4,5,7] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [2,3,6,1,4,7,5] => [7,2,3,1,4,6,5] => ? = 7 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,3,6,1,7,4,5] => [7,2,3,1,6,5,4] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,3,6,7,1,4,5] => [7,2,3,6,4,5,1] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,6,1,4,5,7] => [6,2,3,1,4,5,7] => ? = 7 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,3,1,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,3,1,4,7,5,6] => [3,2,1,4,7,5,6] => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [2,3,1,7,4,5,6] => [3,2,1,7,4,5,6] => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,7,1,4,5,6] => [7,2,3,1,4,5,6] => ? = 7 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 7 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,4,5,6,7,3] => [2,1,7,4,5,6,3] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,4,5,6,3,7] => [2,1,6,4,5,3,7] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,4,5,7,3,6] => [2,1,7,4,5,3,6] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,7,5] => [2,1,4,3,7,6,5] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 5 - 1
Description
The number of weak deficiencies of a permutation. This is defined as $$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$$ The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
Matching statistic: St000672
(load all 2 compositions to match this statistic)
Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1] => 0 = 2 - 2
[1,0,1,0]
 => [2,1] => [2,1] => [2,1] => 0 = 2 - 2
[1,1,0,0]
 => [1,2] => [1,2] => [1,2] => 1 = 3 - 2
[1,0,1,0,1,0]
 => [2,3,1] => [3,2,1] => [2,3,1] => 1 = 3 - 2
[1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1 = 3 - 2
[1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => [1,3,2] => 1 = 3 - 2
[1,1,0,1,0,0]
 => [3,1,2] => [3,1,2] => [3,1,2] => 1 = 3 - 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,2,3] => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 2 = 4 - 2
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 2 = 4 - 2
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1 = 3 - 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 2 = 4 - 2
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 4 - 2
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 2 = 4 - 2
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 4 - 2
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [4,1,3,2] => [3,4,1,2] => 2 = 4 - 2
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [4,3,2,1] => [3,2,4,1] => 1 = 3 - 2
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 2 = 4 - 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 4 - 2
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2 = 4 - 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3 = 5 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 3 = 5 - 2
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2 = 4 - 2
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [5,2,3,1,4] => [2,3,5,1,4] => 3 = 5 - 2
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 3 = 5 - 2
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2 = 4 - 2
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2 = 4 - 2
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [5,2,1,4,3] => [2,4,5,1,3] => 3 = 5 - 2
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [5,2,4,3,1] => [2,4,3,5,1] => 2 = 4 - 2
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [4,2,1,3,5] => [2,4,1,3,5] => 3 = 5 - 2
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2 = 4 - 2
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => 2 = 4 - 2
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => [2,5,1,3,4] => 3 = 5 - 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 3 = 5 - 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 3 = 5 - 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 3 = 5 - 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 4 - 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,2,4] => 3 = 5 - 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 3 = 5 - 2
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [5,1,3,4,2] => [3,4,5,1,2] => 3 = 5 - 2
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [4,1,3,2,5] => [3,4,1,2,5] => 3 = 5 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [5,3,2,4,1] => [3,2,4,5,1] => 2 = 4 - 2
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [5,4,3,2,1] => [3,4,2,5,1] => 2 = 4 - 2
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [4,3,2,1,5] => [3,2,4,1,5] => 2 = 4 - 2
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 2 = 4 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [5,1,3,2,4] => [3,5,1,2,4] => 3 = 5 - 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [5,3,2,1,4] => [3,2,5,1,4] => 2 = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 3 = 5 - 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,3,4,1,6,7,5] => [4,2,3,1,7,6,5] => [2,3,4,1,6,7,5] => ? = 6 - 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,4,1,6,5,7] => [4,2,3,1,6,5,7] => [2,3,4,1,6,5,7] => ? = 6 - 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,4,6,7,1,5] => [7,2,3,4,6,5,1] => [2,3,4,6,5,7,1] => ? = 6 - 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [2,3,4,6,1,5,7] => [6,2,3,4,1,5,7] => [2,3,4,6,1,5,7] => ? = 7 - 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,3,4,1,7,5,6] => [4,2,3,1,7,5,6] => [2,3,4,1,7,5,6] => ? = 6 - 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,3,1,5,6,7,4] => [3,2,1,7,5,6,4] => [2,3,1,5,6,7,4] => ? = 6 - 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,3,1,5,6,4,7] => [3,2,1,6,5,4,7] => [2,3,1,5,6,4,7] => ? = 6 - 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [2,3,1,5,4,7,6] => [3,2,1,5,4,7,6] => [2,3,1,5,4,7,6] => ? = 5 - 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [2,3,1,5,7,4,6] => [3,2,1,7,5,4,6] => [2,3,1,5,7,4,6] => ? = 6 - 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,3,1,5,4,6,7] => [3,2,1,5,4,6,7] => [2,3,1,5,4,6,7] => ? = 6 - 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [2,3,5,1,6,4,7] => [6,2,3,1,5,4,7] => [2,3,5,6,1,4,7] => ? = 7 - 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [2,3,5,6,1,7,4] => [7,2,3,5,4,6,1] => [2,3,5,4,6,7,1] => ? = 6 - 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,3,5,6,7,1,4] => [7,2,3,6,5,4,1] => [2,3,5,6,4,7,1] => ? = 6 - 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,6,1,4,7] => [6,2,3,5,4,1,7] => [2,3,5,4,6,1,7] => ? = 6 - 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,3,5,1,4,7,6] => [5,2,3,1,4,7,6] => [2,3,5,1,4,7,6] => ? = 6 - 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,3,5,1,7,4,6] => [7,2,3,1,5,4,6] => [2,3,5,7,1,4,6] => ? = 7 - 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,3,5,7,1,4,6] => [7,2,3,5,4,1,6] => [2,3,5,4,7,1,6] => ? = 6 - 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,3,5,1,4,6,7] => [5,2,3,1,4,6,7] => [2,3,5,1,4,6,7] => ? = 7 - 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [2,3,1,4,6,7,5] => [3,2,1,4,7,6,5] => [2,3,1,4,6,7,5] => ? = 6 - 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,3,1,4,6,5,7] => [3,2,1,4,6,5,7] => [2,3,1,4,6,5,7] => ? = 6 - 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [2,3,1,6,4,7,5] => [3,2,1,7,4,6,5] => [2,3,1,6,7,4,5] => ? = 6 - 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,3,1,6,7,4,5] => [3,2,1,7,6,5,4] => [2,3,1,6,5,7,4] => ? = 5 - 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [2,3,1,6,4,5,7] => [3,2,1,6,4,5,7] => [2,3,1,6,4,5,7] => ? = 6 - 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [2,3,6,1,4,7,5] => [7,2,3,1,4,6,5] => [2,3,6,7,1,4,5] => ? = 7 - 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,3,6,1,7,4,5] => [7,2,3,1,6,5,4] => [2,3,6,5,7,1,4] => ? = 6 - 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,3,6,7,1,4,5] => [7,2,3,6,4,5,1] => [2,3,6,4,5,7,1] => ? = 6 - 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,6,1,4,5,7] => [6,2,3,1,4,5,7] => [2,3,6,1,4,5,7] => ? = 7 - 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,3,1,4,5,7,6] => [3,2,1,4,5,7,6] => [2,3,1,4,5,7,6] => ? = 6 - 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,3,1,4,7,5,6] => [3,2,1,4,7,5,6] => [2,3,1,4,7,5,6] => ? = 6 - 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [2,3,1,7,4,5,6] => [3,2,1,7,4,5,6] => [2,3,1,7,4,5,6] => ? = 6 - 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,7,1,4,5,6] => [7,2,3,1,4,5,6] => [2,3,7,1,4,5,6] => ? = 7 - 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,4,5,6,7,3] => [2,1,7,4,5,6,3] => [2,1,4,5,6,7,3] => ? = 6 - 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,4,5,6,3,7] => [2,1,6,4,5,3,7] => [2,1,4,5,6,3,7] => ? = 6 - 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => [2,1,4,5,3,7,6] => ? = 5 - 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [2,1,4,5,7,3,6] => [2,1,7,4,5,3,6] => [2,1,4,5,7,3,6] => ? = 6 - 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,4,5,3,6,7] => [2,1,5,4,3,6,7] => [2,1,4,5,3,6,7] => ? = 6 - 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,7,5] => [2,1,4,3,7,6,5] => [2,1,4,3,6,7,5] => ? = 5 - 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => [2,1,4,3,6,5,7] => ? = 5 - 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,1,4,6,3,7,5] => [2,1,7,4,3,6,5] => [2,1,4,6,7,3,5] => ? = 6 - 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,1,4,6,7,3,5] => [2,1,7,4,6,5,3] => [2,1,4,6,5,7,3] => ? = 5 - 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,1,4,6,3,5,7] => [2,1,6,4,3,5,7] => [2,1,4,6,3,5,7] => ? = 6 - 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 5 - 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,4,3,7,5,6] => [2,1,4,3,7,5,6] => [2,1,4,3,7,5,6] => ? = 5 - 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,1,4,7,3,5,6] => [2,1,7,4,3,5,6] => [2,1,4,7,3,5,6] => ? = 6 - 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ? = 6 - 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [2,4,1,5,6,7,3] => [7,2,1,4,5,6,3] => [2,4,5,6,7,1,3] => ? = 7 - 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [2,4,1,5,6,3,7] => [6,2,1,4,5,3,7] => [2,4,5,6,1,3,7] => ? = 7 - 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [2,4,1,5,3,7,6] => [5,2,1,4,3,7,6] => [2,4,5,1,3,7,6] => ? = 6 - 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [2,4,1,5,7,3,6] => [7,2,1,4,5,3,6] => [2,4,5,7,1,3,6] => ? = 7 - 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,3,6,7] => [5,2,1,4,3,6,7] => [2,4,5,1,3,6,7] => ? = 7 - 2
Description
The number of minimal elements in Bruhat order not less than the permutation. The minimal elements in question are biGrassmannian, that is $$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$ for some $(r,a,b)$. This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000213
(load all 6 compositions to match this statistic)
Mapping
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%
Values
[1,0]
 => [1,0]
 => [1] => 1 = 2 - 1
[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 1 = 2 - 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 2 = 3 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2 = 3 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2 = 3 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 3 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2 = 3 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3 = 4 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 3 = 4 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 3 = 4 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3 = 4 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3 = 4 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 4 = 5 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 3 = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 4 = 5 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 4 = 5 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 4 = 5 - 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 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 4 = 5 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 3 = 4 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => 4 = 5 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 4 = 5 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 4 = 5 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 3 = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 4 = 5 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 4 = 5 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 4 = 5 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 3 = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 3 = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 3 = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 4 = 5 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [2,3,4,5,6,1,7] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [2,3,4,5,1,7,6] => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,7,6,1] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [2,3,4,5,1,6,7] => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [2,3,4,1,6,7,5] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [2,3,4,1,6,5,7] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,6,5,7,1] => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,4,7,6,5,1] => ? = 6 - 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [2,3,4,6,5,1,7] => ? = 7 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [2,3,4,1,5,7,6] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [2,3,4,1,7,6,5] => ? = 6 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,4,7,5,6,1] => ? = 7 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [2,3,4,1,5,6,7] => ? = 7 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,6,7,4] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [2,3,1,5,6,4,7] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [2,3,1,5,4,7,6] => ? = 5 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [2,3,1,5,7,6,4] => ? = 6 - 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [2,3,1,5,4,6,7] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [2,3,5,4,6,7,1] => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [2,3,5,4,6,1,7] => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,4,7,1] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [2,3,7,5,6,4,1] => ? = 6 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [2,3,6,5,4,1,7] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
 => [2,3,5,4,1,7,6] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [2,3,5,4,7,6,1] => ? = 7 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,7,5,4,6,1] => ? = 6 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [2,3,5,4,1,6,7] => ? = 7 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [2,3,1,4,6,7,5] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [2,3,1,4,6,5,7] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
 => [2,3,1,6,5,7,4] => ? = 6 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [2,3,1,7,6,5,4] => ? = 5 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1,6,5,4,7] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [2,3,6,4,5,7,1] => ? = 7 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [2,3,7,4,6,5,1] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,7,6,5,4,1] => ? = 6 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => [2,3,6,4,5,1,7] => ? = 7 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [2,3,1,4,5,7,6] => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,7,6,5] => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => [2,3,1,7,5,6,4] => ? = 6 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [2,3,7,4,5,6,1] => ? = 7 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [2,3,1,4,5,6,7] => ? = 7 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [2,1,4,5,6,7,3] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [2,1,4,5,6,3,7] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [2,1,4,5,3,7,6] => ? = 5 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [2,1,4,5,7,6,3] => ? = 6 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [2,1,4,5,3,6,7] => ? = 6 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,3,6,7,5] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,6,5,7] => ? = 5 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => [2,1,4,6,5,7,3] => ? = 6 - 1
Description
The number of weak exceedances (also weak excedences) of a permutation. This is defined as $$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$ The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000155The number of exceedances (also excedences) of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001863The number of weak excedances of a signed permutation. St001626The number of maximal proper sublattices of a lattice.