Your data matches 75 different statistics following compositions of up to 3 maps.
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Matching statistic: St001036
(load all 237 compositions to match this statistic)
Mapping
St001036: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 0
[1,0,1,0]
 => 0
[1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => 0
[1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,0,1,1,0,0]
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => 3
[1,1,0,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => 1
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000249
Mapping
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000249: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [1] => {{1}}
 => ? = 0 + 2
[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => {{1,2}}
 => 2 = 0 + 2
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => {{1},{2}}
 => 2 = 0 + 2
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => {{1,3},{2}}
 => 2 = 0 + 2
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => {{1,2,3}}
 => 3 = 1 + 2
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => {{1},{2},{3}}
 => 3 = 1 + 2
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => {{1},{2,3}}
 => 2 = 0 + 2
[1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => {{1,2},{3}}
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => {{1,4},{2,3}}
 => 2 = 0 + 2
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => {{1,4},{2},{3}}
 => 3 = 1 + 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => 4 = 2 + 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => {{1,2,4},{3}}
 => 3 = 1 + 2
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => {{1,3,4},{2}}
 => 3 = 1 + 2
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => {{1},{2,3},{4}}
 => 3 = 1 + 2
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => 4 = 2 + 2
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => {{1},{2,3,4}}
 => 3 = 1 + 2
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => {{1},{2,4},{3}}
 => 2 = 0 + 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => {{1},{2},{3,4}}
 => 3 = 1 + 2
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => {{1,2,3},{4}}
 => 3 = 1 + 2
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => {{1,2},{3},{4}}
 => 3 = 1 + 2
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => {{1,2},{3,4}}
 => 2 = 0 + 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => {{1,3},{2},{4}}
 => 2 = 0 + 2
[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] => {{1,5},{2,4},{3}}
 => 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => {{1,5},{2,3,4}}
 => 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
 => 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => {{1,5},{2},{3,4}}
 => 3 = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => {{1,5},{2,3},{4}}
 => 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => {{1,2,4,5},{3}}
 => 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => {{1,2,5},{3},{4}}
 => 4 = 2 + 2
[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] => {{1,2,5},{3,4}}
 => 3 = 1 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => {{1,2,3,5},{4}}
 => 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => {{1,4,5},{2},{3}}
 => 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => {{1,3,4,5},{2}}
 => 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => {{1,3,5},{2},{4}}
 => 3 = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => {{1,4,5},{2,3}}
 => 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => 3 = 1 + 2
[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] => {{1},{2,3,4},{5}}
 => 4 = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 4 = 2 + 2
[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] => {{1},{2,3},{4},{5}}
 => 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => {{1},{2,4,5},{3}}
 => 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => {{1},{2,3,4,5}}
 => 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
 => 3 = 1 + 2
[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},{2,5},{3,4}}
 => 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => {{1},{2,3,5},{4}}
 => 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 3 = 1 + 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => {{1,2,4},{3},{5}}
 => 3 = 1 + 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
 => ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [8,7,6,4,5,3,2,1] => {{1,8},{2,7},{3,6},{4},{5}}
 => ? = 1 + 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [8,7,4,5,6,3,2,1] => {{1,8},{2,7},{3,4,5,6}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [8,7,4,6,5,3,2,1] => {{1,8},{2,7},{3,4,6},{5}}
 => ? = 1 + 2
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [8,7,5,4,6,3,2,1] => {{1,8},{2,7},{3,5,6},{4}}
 => ? = 1 + 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [8,7,3,5,4,6,2,1] => {{1,8},{2,7},{3},{4,5},{6}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [8,7,3,4,5,6,2,1] => {{1,8},{2,7},{3},{4},{5},{6}}
 => ? = 3 + 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [8,7,3,5,6,4,2,1] => {{1,8},{2,7},{3},{4,5,6}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [8,7,3,6,5,4,2,1] => {{1,8},{2,7},{3},{4,6},{5}}
 => ? = 1 + 2
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [8,7,3,4,6,5,2,1] => {{1,8},{2,7},{3},{4},{5,6}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [8,7,4,5,3,6,2,1] => {{1,8},{2,7},{3,4,5},{6}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [8,7,4,3,5,6,2,1] => {{1,8},{2,7},{3,4},{5},{6}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [8,7,4,3,6,5,2,1] => {{1,8},{2,7},{3,4},{5,6}}
 => ? = 1 + 2
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [8,7,5,4,3,6,2,1] => {{1,8},{2,7},{3,5},{4},{6}}
 => ? = 1 + 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
 => [8,3,6,5,4,7,2,1] => {{1,8},{2,3,6,7},{4,5}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
 => [8,3,6,4,5,7,2,1] => ?
 => ? = 3 + 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
 => [8,3,4,6,5,7,2,1] => ?
 => ? = 3 + 2
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
 => [8,3,5,4,6,7,2,1] => {{1,8},{2,3,5,6,7},{4}}
 => ? = 3 + 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
 => [8,3,7,5,4,6,2,1] => ?
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
 => [8,3,7,4,5,6,2,1] => {{1,8},{2,3,7},{4},{5},{6}}
 => ? = 3 + 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => [8,3,7,5,6,4,2,1] => {{1,8},{2,3,7},{4,5,6}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [8,3,7,6,5,4,2,1] => {{1,8},{2,3,7},{4,6},{5}}
 => ? = 1 + 2
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => [8,3,7,4,6,5,2,1] => {{1,8},{2,3,7},{4},{5,6}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [8,3,4,5,7,6,2,1] => {{1,8},{2,3,4,5,7},{6}}
 => ? = 3 + 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
 => [8,3,4,7,5,6,2,1] => ?
 => ? = 3 + 2
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
 => [8,3,4,7,6,5,2,1] => {{1,8},{2,3,4,7},{5,6}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
 => [8,3,5,4,7,6,2,1] => {{1,8},{2,3,5,7},{4},{6}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => [8,6,3,5,4,7,2,1] => ?
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => [8,6,3,4,5,7,2,1] => ?
 => ? = 3 + 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [8,4,3,5,6,7,2,1] => {{1,8},{2,4,5,6,7},{3}}
 => ? = 3 + 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
 => [8,4,3,6,5,7,2,1] => ?
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => [8,5,3,4,6,7,2,1] => {{1,8},{2,5,6,7},{3},{4}}
 => ? = 3 + 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
 => [8,4,3,5,7,6,2,1] => {{1,8},{2,4,5,7},{3},{6}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
 => [8,4,3,7,5,6,2,1] => {{1,8},{2,4,7},{3},{5},{6}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => [8,4,3,7,6,5,2,1] => ?
 => ? = 1 + 2
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
 => [8,5,3,4,7,6,2,1] => {{1,8},{2,5,7},{3},{4},{6}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [8,6,4,5,3,7,2,1] => ?
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
 => [8,6,4,3,5,7,2,1] => {{1,8},{2,6,7},{3,4},{5}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => [8,5,4,3,6,7,2,1] => {{1,8},{2,5,6,7},{3,4}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
 => [8,5,4,3,7,6,2,1] => {{1,8},{2,5,7},{3,4},{6}}
 => ? = 1 + 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [8,6,5,4,3,7,2,1] => {{1,8},{2,6,7},{3,5},{4}}
 => ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
 => [8,2,6,5,4,3,7,1] => {{1,8},{2},{3,6},{4,5},{7}}
 => ? = 2 + 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
 => [8,2,6,4,5,3,7,1] => {{1,8},{2},{3,6},{4},{5},{7}}
 => ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
 => [8,2,4,5,6,3,7,1] => ?
 => ? = 4 + 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
 => [8,2,4,6,5,3,7,1] => ?
 => ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
 => [8,2,5,4,6,3,7,1] => ?
 => ? = 3 + 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
 => [8,2,3,5,4,6,7,1] => {{1,8},{2},{3},{4,5},{6},{7}}
 => ? = 4 + 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [8,2,3,4,5,6,7,1] => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 5 + 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
 => [8,2,3,5,6,4,7,1] => ?
 => ? = 4 + 2
Description
The number of singletons ([[St000247]]) plus the number of antisingletons ([[St000248]]) of a set partition.
Matching statistic: St001499
(load all 66 compositions to match this statistic)
Mapping
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 86%
Values
[1,0]
 => [1,0]
 => ? = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 5 + 1
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St000292
(load all 9 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 71%
Values
[1,0]
 => []
 => => => ? = 0
[1,0,1,0]
 => [1]
 => 10 => 00 => 0
[1,1,0,0]
 => []
 => => => ? = 0
[1,0,1,0,1,0]
 => [2,1]
 => 1010 => 0000 => 0
[1,0,1,1,0,0]
 => [1,1]
 => 110 => 001 => 1
[1,1,0,0,1,0]
 => [2]
 => 100 => 010 => 1
[1,1,0,1,0,0]
 => [1]
 => 10 => 00 => 0
[1,1,1,0,0,0]
 => []
 => => => ? = 0
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 101010 => 000000 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 11010 => 00001 => 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 100110 => 001010 => 2
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 10110 => 00100 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1110 => 0011 => 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => 10100 => 01000 => 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => 1100 => 0101 => 2
[1,1,0,1,0,0,1,0]
 => [3,1]
 => 10010 => 00010 => 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => 1010 => 0000 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => 110 => 001 => 1
[1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => 0110 => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 010 => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => 10 => 00 => 0
[1,1,1,1,0,0,0,0]
 => []
 => => => ? = 0
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 10101010 => 00000000 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 1101010 => 0000001 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 10011010 => 00001010 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 1011010 => 0000100 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 111010 => 000011 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 10100110 => 00101000 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 1100110 => 0010101 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => 10010110 => 00100010 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 1010110 => 0010000 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 110110 => 001001 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 10001110 => 00110110 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 1001110 => 0011010 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 101110 => 001100 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 11110 => 00111 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 1010100 => 0100000 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 110100 => 010001 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1001100 => 0101010 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 101100 => 010100 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 11100 => 01011 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 1010010 => 0001000 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 110010 => 000101 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 1001010 => 0000010 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => 101010 => 000000 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => 11010 => 00001 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 1000110 => 0010110 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => 100110 => 001010 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 10110 => 00100 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 1110 => 0011 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 101000 => 011000 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 11000 => 01101 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 100100 => 010010 => 2
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 10100 => 01000 => 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => => => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => 1010001110 => 0011011000 => ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => 1001001110 => 0011010010 => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,1,1]
 => 1000101110 => 0011000110 => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => 1000011110 => 0011101110 => ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => => => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1]
 => 101010101010 => 000000000000 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,2,1]
 => 100110101010 => ? => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2,1]
 => 10110101010 => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => 1110101010 => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,2,1]
 => 101001101010 => ? => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,2,1]
 => 11001101010 => ? => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,2,1]
 => 100101101010 => ? => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3,2,1]
 => 10101101010 => ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => 1101101010 => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,3,3,2,1]
 => 100011101010 => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,3,3,2,1]
 => 10011101010 => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,3,3,2,1]
 => 1011101010 => ? => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2,1]
 => 101010011010 => ? => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,2,1]
 => 11010011010 => ? => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2,1]
 => 100110011010 => ? => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2,1]
 => 10110011010 => ? => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,4,4,2,2,1]
 => 1110011010 => ? => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2,1]
 => 101001011010 => ? => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2,1]
 => 11001011010 => ? => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,2,1]
 => 100101011010 => ? => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,2,1]
 => 10101011010 => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [4,4,3,2,2,1]
 => 1101011010 => ? => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,3,3,2,2,1]
 => 100011011010 => ? => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [5,3,3,2,2,1]
 => 10011011010 => ? => ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [4,3,3,2,2,1]
 => 1011011010 => ? => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,2,2,1]
 => 101000111010 => ? => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,5,2,2,2,1]
 => 11000111010 => ? => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [6,4,2,2,2,1]
 => 100100111010 => ? => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [5,4,2,2,2,1]
 => 10100111010 => ? => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [4,4,2,2,2,1]
 => 1100111010 => ? => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [6,3,2,2,2,1]
 => 100010111010 => ? => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [5,3,2,2,2,1]
 => 10010111010 => ? => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,2,2,1]
 => 1010111010 => ? => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2,1]
 => 100001111010 => ? => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,2,2,2,1]
 => 10001111010 => ? => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,2,2,2,1]
 => 1001111010 => ? => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,1]
 => 101010100110 => ? => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,1,1]
 => 11010100110 => ? => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,1,1]
 => 100110100110 => ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,1,1]
 => 10110100110 => ? => ? = 3
Description
The number of ascents of a binary word.
Matching statistic: St000390
(load all 13 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 71%
Values
[1,0]
 => []
 => => => ? = 0
[1,0,1,0]
 => [1]
 => 10 => 00 => 0
[1,1,0,0]
 => []
 => => => ? = 0
[1,0,1,0,1,0]
 => [2,1]
 => 1010 => 0000 => 0
[1,0,1,1,0,0]
 => [1,1]
 => 110 => 001 => 1
[1,1,0,0,1,0]
 => [2]
 => 100 => 010 => 1
[1,1,0,1,0,0]
 => [1]
 => 10 => 00 => 0
[1,1,1,0,0,0]
 => []
 => => => ? = 0
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 101010 => 000000 => 0
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 11010 => 00001 => 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 100110 => 001010 => 2
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 10110 => 00100 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1110 => 0011 => 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => 10100 => 01000 => 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => 1100 => 0101 => 2
[1,1,0,1,0,0,1,0]
 => [3,1]
 => 10010 => 00010 => 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => 1010 => 0000 => 0
[1,1,0,1,1,0,0,0]
 => [1,1]
 => 110 => 001 => 1
[1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => 0110 => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 010 => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => 10 => 00 => 0
[1,1,1,1,0,0,0,0]
 => []
 => => => ? = 0
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 10101010 => 00000000 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 1101010 => 0000001 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 10011010 => 00001010 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 1011010 => 0000100 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 111010 => 000011 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 10100110 => 00101000 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 1100110 => 0010101 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => 10010110 => 00100010 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 1010110 => 0010000 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 110110 => 001001 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 10001110 => 00110110 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 1001110 => 0011010 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 101110 => 001100 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 11110 => 00111 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 1010100 => 0100000 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 110100 => 010001 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1001100 => 0101010 => 3
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 101100 => 010100 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 11100 => 01011 => 2
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 1010010 => 0001000 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 110010 => 000101 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 1001010 => 0000010 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => 101010 => 000000 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => 11010 => 00001 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 1000110 => 0010110 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => 100110 => 001010 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 10110 => 00100 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 1110 => 0011 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 101000 => 011000 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 11000 => 01101 => 2
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 100100 => 010010 => 2
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 10100 => 01000 => 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => => => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,1,1]
 => 1010001110 => 0011011000 => ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,1,1]
 => 1001001110 => 0011010010 => ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,1,1]
 => 1000101110 => 0011000110 => ? = 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [5,1,1,1,1]
 => 1000011110 => 0011101110 => ? = 2
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => => => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1]
 => 101010101010 => 000000000000 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,2,1]
 => 100110101010 => ? => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2,1]
 => 10110101010 => ? => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => 1110101010 => ? => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,2,1]
 => 101001101010 => ? => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,2,1]
 => 11001101010 => ? => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,2,1]
 => 100101101010 => ? => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3,2,1]
 => 10101101010 => ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => 1101101010 => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,3,3,2,1]
 => 100011101010 => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,3,3,2,1]
 => 10011101010 => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,3,3,2,1]
 => 1011101010 => ? => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2,1]
 => 101010011010 => ? => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,2,1]
 => 11010011010 => ? => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2,1]
 => 100110011010 => ? => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2,1]
 => 10110011010 => ? => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,4,4,2,2,1]
 => 1110011010 => ? => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2,1]
 => 101001011010 => ? => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2,1]
 => 11001011010 => ? => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,2,1]
 => 100101011010 => ? => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,2,1]
 => 10101011010 => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [4,4,3,2,2,1]
 => 1101011010 => ? => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,3,3,2,2,1]
 => 100011011010 => ? => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [5,3,3,2,2,1]
 => 10011011010 => ? => ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [4,3,3,2,2,1]
 => 1011011010 => ? => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,2,2,1]
 => 101000111010 => ? => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,5,2,2,2,1]
 => 11000111010 => ? => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [6,4,2,2,2,1]
 => 100100111010 => ? => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [5,4,2,2,2,1]
 => 10100111010 => ? => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [4,4,2,2,2,1]
 => 1100111010 => ? => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [6,3,2,2,2,1]
 => 100010111010 => ? => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [5,3,2,2,2,1]
 => 10010111010 => ? => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,2,2,1]
 => 1010111010 => ? => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2,1]
 => 100001111010 => ? => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,2,2,2,1]
 => 10001111010 => ? => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,2,2,2,1]
 => 1001111010 => ? => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,1]
 => 101010100110 => ? => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,1,1]
 => 11010100110 => ? => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,1,1]
 => 100110100110 => ? => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,1,1]
 => 10110100110 => ? => ? = 3
Description
The number of runs of ones in a binary word.
Matching statistic: St000340
(load all 59 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 86%
Values
[1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 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]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0,1,0]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,1,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => ? = 4 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => ? = 4 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 4 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 4 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 3 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 3 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => ? = 4 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,0,0,1,0]
 => ? = 4 + 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => ? = 3 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,1,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
 => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
 => ? = 3 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
 => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,1,0,1,0,0]
 => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0,1,1,0,0]
 => ? = 4 + 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? = 3 + 1
Description
The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St000312
Mapping
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000312: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 86%
Values
[1,0]
 => [1,0]
 => [[]]
 => ([(0,1)],2)
 => 2 = 0 + 2
[1,0,1,0]
 => [1,1,0,0]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,1,0,0]
 => [1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 1 + 2
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3 = 1 + 2
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3 = 1 + 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 2 + 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3 = 1 + 2
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3 = 1 + 2
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3 = 1 + 2
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 2 + 2
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3 = 1 + 2
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3 = 1 + 2
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3 = 1 + 2
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3 = 1 + 2
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[[],[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[[[],[],[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[[[],[[]]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[[[[]],[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[[],[[]],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3 = 1 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[[[],[]],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[[[]],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[[[]]],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4 = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3 = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3 = 1 + 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[[[[[[[]]]]]]]]
 => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
 => ? = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [[[[[[],[],[]]]]]]
 => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [[[[[[],[[]]]]]]]
 => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
 => ? = 1 + 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [[[[[[[]],[]]]]]]
 => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
 => ? = 1 + 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [[[[[],[[]],[]]]]]
 => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [[[[[],[[],[]]]]]]
 => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [[[[[],[],[[]]]]]]
 => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [[[[[[],[]],[]]]]]
 => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [[[[[[]],[],[]]]]]
 => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [[[[[[]],[[]]]]]]
 => ([(0,6),(1,5),(2,4),(3,4),(3,7),(5,7),(6,7)],8)
 => ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [[[[],[[[]]],[]]]]
 => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [[[[],[[],[]],[]]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 3 + 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [[[[],[],[],[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 4 + 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [[[[],[],[[]],[]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 3 + 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [[[[],[[]],[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 3 + 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [[[[],[[[]],[]]]]]
 => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [[[[],[[],[],[]]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ? = 3 + 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [[[[],[[[],[]]]]]]
 => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [[[[],[[[[]]]]]]]
 => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
 => ? = 1 + 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [[[[],[[],[[]]]]]]
 => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [[[[],[],[],[[]]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 3 + 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [[[[],[],[[],[]]]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 3 + 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [[[[],[],[[[]]]]]]
 => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [[[[],[[]],[[]]]]]
 => ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [[[[[],[[]]],[]]]]
 => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [[[[[],[],[]],[]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ? = 3 + 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [[[[[]],[],[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 3 + 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [[[[[]],[[]],[]]]]
 => ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [[[[[],[]],[],[]]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 3 + 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [[[[[]],[],[[]]]]]
 => ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [[[[[]],[[],[]]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [[[[[]],[[[]]]]]]
 => ([(0,6),(1,5),(2,4),(3,4),(3,7),(5,7),(6,7)],8)
 => ? = 1 + 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [[[[[],[]],[[]]]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [[[[[[],[]]],[]]]]
 => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [[[[[[]],[]],[]]]]
 => ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [[[[[[]]],[],[]]]]
 => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
 => ? = 2 + 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [[[[[[]]],[[]]]]]
 => ([(0,6),(1,5),(2,4),(3,4),(3,7),(5,7),(6,7)],8)
 => ? = 1 + 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [[[[[[[]]]],[]]]]
 => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
 => ? = 1 + 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [[[],[[[[]]]],[]]]
 => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
 => ? = 2 + 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [[[],[[],[],[]],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ? = 4 + 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [[[],[[],[[]]],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ? = 3 + 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [[[],[[[]],[]],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ? = 3 + 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [[[],[],[[]],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 4 + 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [[[],[],[[],[]],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 4 + 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [[[],[],[],[[]],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 4 + 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [[[],[[],[]],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 4 + 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [[[],[[]],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 4 + 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [[[],[[]],[[]],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 3 + 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [[[],[[[[]]],[]]]]
 => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
 => ? = 2 + 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [[[],[[[],[]],[]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 3 + 2
Description
The number of leaves in a graph. That is, the number of vertices of a graph that have degree 1.
Matching statistic: St000619
(load all 17 compositions to match this statistic)
Mapping
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 71%
Values
[1,0]
 => [1,0]
 => [1] => [1] => ? = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => [1,2] => 1 = 0 + 1
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [1,3,2] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [3,2,1] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => [2,3,1] => 1 = 0 + 1
[1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => [3,1,2] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [1,2,4,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,4,3,2] => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [1,3,4,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,4,2,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [4,2,3,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4,3,2,1] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,3,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,3,4,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,4,2,1] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [4,1,3,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,4,1,2] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [4,1,2,3] => 1 = 0 + 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] => [1,2,3,4,5] => 1 = 0 + 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] => [1,2,3,5,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => [1,2,5,4,3] => 3 = 2 + 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] => [1,2,4,5,3] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => [1,2,5,3,4] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [1,5,3,4,2] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,5,4,3,2] => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [1,3,5,4,2] => 3 = 2 + 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] => [1,3,4,5,2] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [1,4,5,3,2] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => [1,5,2,4,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [1,5,4,2,3] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [1,4,5,2,3] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [1,5,2,3,4] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [5,2,3,4,1] => 2 = 1 + 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] => [5,2,4,3,1] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [5,4,3,2,1] => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [5,3,4,2,1] => 3 = 2 + 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] => [5,4,2,3,1] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [2,5,3,4,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,5,4,3,1] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [2,3,5,4,1] => 2 = 1 + 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] => [2,3,4,5,1] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [2,4,5,3,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [4,5,3,2,1] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,5,4,2,1] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [3,4,5,2,1] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [5,1,3,4,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [2,6,5,4,3,7,1] => [1,7,3,4,5,6,2] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [2,5,6,4,3,7,1] => [1,7,3,4,6,5,2] => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [2,4,5,6,3,7,1] => [1,7,3,6,5,4,2] => ? = 4 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [2,4,6,5,3,7,1] => [1,7,3,5,6,4,2] => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [2,5,4,6,3,7,1] => [1,7,3,6,4,5,2] => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [2,3,5,4,6,7,1] => [1,7,6,4,5,3,2] => ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => ? = 5 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [2,3,5,6,4,7,1] => [1,7,4,6,5,3,2] => ? = 4 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,4,7,1] => [1,7,4,5,6,3,2] => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,4,6,5,7,1] => [1,7,5,6,4,3,2] => ? = 4 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [2,4,5,3,6,7,1] => [1,7,6,3,5,4,2] => ? = 4 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,4,3,5,6,7,1] => [1,7,6,5,3,4,2] => ? = 4 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [2,4,3,6,5,7,1] => [1,7,5,6,3,4,2] => ? = 3 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [2,5,4,3,6,7,1] => [1,7,6,3,4,5,2] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [2,3,5,4,7,6,1] => [1,6,7,4,5,3,2] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,5,7,6,1] => [1,6,7,5,4,3,2] => ? = 4 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [2,3,5,6,7,4,1] => [1,4,7,6,5,3,2] => ? = 4 + 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [2,3,6,5,7,4,1] => [1,4,7,5,6,3,2] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,4,6,7,5,1] => [1,5,7,6,4,3,2] => ? = 4 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [2,3,5,7,6,4,1] => [1,4,6,7,5,3,2] => ? = 3 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,4,7,6,5,1] => [1,5,6,7,4,3,2] => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [2,4,5,3,7,6,1] => [1,6,7,3,5,4,2] => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [2,4,3,5,7,6,1] => [1,6,7,5,3,4,2] => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [2,4,3,6,7,5,1] => [1,5,7,6,3,4,2] => ? = 3 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [2,4,3,7,6,5,1] => [1,5,6,7,3,4,2] => ? = 2 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [2,5,4,3,7,6,1] => [1,6,7,3,4,5,2] => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [3,6,5,4,2,7,1] => [1,7,2,4,5,6,3] => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [3,5,6,4,2,7,1] => [1,7,2,4,6,5,3] => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,6,2,7,1] => [1,7,2,6,5,4,3] => ? = 4 + 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [3,4,6,5,2,7,1] => [1,7,2,5,6,4,3] => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => [3,5,4,6,2,7,1] => [1,7,2,6,4,5,3] => ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [3,2,5,4,6,7,1] => [1,7,6,4,5,2,3] => ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,7,1] => [1,7,6,5,4,2,3] => ? = 4 + 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [3,2,5,6,4,7,1] => [1,7,4,6,5,2,3] => ? = 3 + 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [3,2,6,5,4,7,1] => [1,7,4,5,6,2,3] => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [3,2,4,6,5,7,1] => [1,7,5,6,4,2,3] => ? = 3 + 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [3,4,5,2,6,7,1] => [1,7,6,2,5,4,3] => ? = 4 + 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [3,4,2,5,6,7,1] => [1,7,6,5,2,4,3] => ? = 4 + 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
 => [3,4,2,6,5,7,1] => [1,7,5,6,2,4,3] => ? = 3 + 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [3,5,4,2,6,7,1] => [1,7,6,2,4,5,3] => ? = 3 + 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,7,6,1] => [1,6,7,4,5,2,3] => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [3,2,4,5,7,6,1] => [1,6,7,5,4,2,3] => ? = 3 + 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => [3,2,5,6,7,4,1] => [1,4,7,6,5,2,3] => ? = 3 + 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [3,2,6,5,7,4,1] => [1,4,7,5,6,2,3] => ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [3,2,4,6,7,5,1] => [1,5,7,6,4,2,3] => ? = 3 + 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2 + 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [3,2,4,7,6,5,1] => [1,5,6,7,4,2,3] => ? = 2 + 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [3,4,5,2,7,6,1] => [1,6,7,2,5,4,3] => ? = 3 + 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => [3,4,2,5,7,6,1] => [1,6,7,5,2,4,3] => ? = 3 + 1
Description
The number of cyclic descents of a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St000291
(load all 12 compositions to match this statistic)
Mapping
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 71%
Values
[1,0]
 => [1,0]
 => 10 => 10 => 1 = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => 1100 => 0110 => 1 = 0 + 1
[1,1,0,0]
 => [1,0,1,0]
 => 1010 => 1100 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 110100 => 011100 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 110010 => 101100 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 101100 => 011010 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 111000 => 001110 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 101010 => 111000 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 01111000 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 10111000 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 01011010 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => 00111010 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 11011000 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 01110100 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 10110100 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 01011100 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 00011110 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 10011100 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 01110010 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 00110110 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 00111100 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 11110000 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 0111110000 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 1011110000 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 0101110010 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 0011110010 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 1101110000 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 0110110100 => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 1010110100 => 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 0101110100 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 0001110110 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 1001110100 => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 0110110010 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0010110110 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 0011110100 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 1110110000 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 0111101000 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 1011101000 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0101101010 => 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 0011101010 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 1101101000 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => 0110111000 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 1010111000 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 0100111100 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 0001111100 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 1000111100 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0100111010 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 0010111010 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 0010111100 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 1100111000 => 2 = 1 + 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]
 => 11010101010100 => 01111111000000 => ? = 0 + 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]
 => 11010101001100 => 01011111000010 => ? = 2 + 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]
 => 11010101011000 => 00111111000010 => ? = 1 + 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]
 => 11010101001010 => 11011111000000 => ? = 1 + 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]
 => 11010100110100 => 01101111000100 => ? = 2 + 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]
 => 11010100110010 => 10101111000100 => ? = 3 + 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]
 => 11010101100100 => 01011111000100 => ? = 2 + 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]
 => 11010101110000 => 00011111000110 => ? = 1 + 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]
 => 11010101100010 => 10011111000100 => ? = 2 + 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]
 => 11010100101100 => 01101111000010 => ? = 2 + 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]
 => 11010100111000 => 00101111000110 => ? = 2 + 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]
 => 11010101101000 => 00111111000100 => ? = 1 + 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]
 => 11010100101010 => 11101111000000 => ? = 1 + 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]
 => 11010011010100 => 01110111001000 => ? = 2 + 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]
 => 11010011010010 => 10110111001000 => ? = 3 + 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]
 => 11010011001100 => 01010111001010 => ? = 4 + 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]
 => 11010011011000 => 00110111001010 => ? = 3 + 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]
 => 11010011001010 => 11010111001000 => ? = 3 + 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]
 => 11010110010100 => 01101111001000 => ? = 2 + 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]
 => 11010110010010 => 10101111001000 => ? = 3 + 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]
 => 11010111000100 => 01001111001100 => ? = 2 + 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]
 => 11010111010000 => 00011111001100 => ? = 1 + 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]
 => 11010111000010 => 10001111001100 => ? = 2 + 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]
 => 11010110001100 => 01001111001010 => ? = 3 + 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]
 => 11010110011000 => 00101111001010 => ? = 3 + 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]
 => 11010111001000 => 00101111001100 => ? = 2 + 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]
 => 11010110001010 => 11001111001000 => ? = 2 + 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]
 => 11010010110100 => 01110111000100 => ? = 2 + 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]
 => 11010010110010 => 10110111000100 => ? = 3 + 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]
 => 11010011100100 => 01010111001100 => ? = 3 + 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]
 => 11010011110000 => 00010111001110 => ? = 2 + 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]
 => 11010011100010 => 10010111001100 => ? = 3 + 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]
 => 11010110100100 => 01011111001000 => ? = 2 + 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]
 => 11010110110000 => 00011111001010 => ? = 2 + 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]
 => 11010111100000 => 00001111001110 => ? = 1 + 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]
 => 11010110100010 => 10011111001000 => ? = 2 + 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]
 => 11010010101100 => 01110111000010 => ? = 2 + 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]
 => 11010010111000 => 00110111000110 => ? = 2 + 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]
 => 11010011101000 => 00110111001100 => ? = 2 + 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]
 => 11010110101000 => 00111111001000 => ? = 1 + 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]
 => 11010010101010 => 11110111000000 => ? = 1 + 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]
 => 11001101010100 => 01111011010000 => ? = 2 + 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]
 => 11001101001100 => 01011011010010 => ? = 4 + 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]
 => 11001101011000 => 00111011010010 => ? = 3 + 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]
 => 11001101001010 => 11011011010000 => ? = 3 + 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]
 => 11001100110100 => 01101011010100 => ? = 4 + 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]
 => 11001100110010 => 10101011010100 => ? = 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]
 => 11001101100100 => 01011011010100 => ? = 4 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => 11001101110000 => 00011011010110 => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => 11001101100010 => 10011011010100 => ? = 4 + 1
Description
The number of descents of a binary word.
Matching statistic: St000925
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 71%
Values
[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => {{1},{2}}
 => 2 = 0 + 2
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 2 = 0 + 2
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2 = 0 + 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2 = 0 + 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 3 = 1 + 2
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 3 = 1 + 2
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2 = 0 + 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 2 = 0 + 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 3 = 1 + 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 4 = 2 + 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 3 = 1 + 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 3 = 1 + 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 3 = 1 + 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 4 = 2 + 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 3 = 1 + 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2 = 0 + 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 3 = 1 + 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 3 = 1 + 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 3 = 1 + 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 2 = 0 + 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3,4,5,6}}
 => 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => {{1},{2,3,5,6},{4}}
 => 3 = 1 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => {{1},{2,4,6},{3},{5}}
 => 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => {{1},{2,4,5,6},{3}}
 => 3 = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => {{1},{2,5,6},{3,4}}
 => 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => {{1},{2},{3,6},{4,5}}
 => 4 = 2 + 2
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3},{4,6},{5}}
 => 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => {{1},{2},{3,4,6},{5}}
 => 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => {{1},{2},{3,4,5,6}}
 => 3 = 1 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3},{4,5,6}}
 => 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => {{1},{2,3},{4},{5,6}}
 => 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => {{1},{2,3},{4,6},{5}}
 => 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => {{1},{2,3},{4,5,6}}
 => 3 = 1 + 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => {{1},{2,3,4},{5,6}}
 => 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => {{1,2},{3,4,5},{6}}
 => 3 = 1 + 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => {{1,2},{3},{4,5},{6}}
 => 4 = 2 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => {{1,3},{2},{4},{5},{6}}
 => 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => {{1,3},{2},{4,5},{6}}
 => 4 = 2 + 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => {{1,3,4},{2},{5},{6}}
 => 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => {{1,2},{3,6},{4,5}}
 => 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => {{1,2},{3},{4,6},{5}}
 => 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => {{1,2},{3,4,6},{5}}
 => 3 = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4,5,6}}
 => 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4,5,6}}
 => 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => {{1,3},{2},{4},{5,6}}
 => 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => {{1,3},{2},{4,6},{5}}
 => 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => {{1,3},{2},{4,5,6}}
 => 3 = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => {{1,3,4},{2},{5,6}}
 => 3 = 1 + 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => {{1},{2,3,4,6,8},{5},{7}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => {{1},{2,3,4,7,8},{5,6}}
 => ? = 1 + 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => {{1},{2,3,5,8},{4},{6,7}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => {{1},{2,3,6,8},{4},{5},{7}}
 => ? = 3 + 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => {{1},{2,3,5,6,8},{4},{7}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => {{1},{2,3,6,7,8},{4},{5}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => {{1},{2,3,7,8},{4,5},{6}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => {{1},{2,3,6,8},{4,5},{7}}
 => ? = 2 + 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => {{1},{2,3,6,7,8},{4,5}}
 => ? = 1 + 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => {{1},{2,3,7,8},{4,5,6}}
 => ? = 1 + 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => {{1},{2,4,8},{3},{5,6,7}}
 => ? = 2 + 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => {{1},{2,4,8},{3},{5},{6,7}}
 => ? = 3 + 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => {{1},{2,5,8},{3},{4},{6},{7}}
 => ? = 4 + 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => {{1},{2,5,8},{3},{4},{6,7}}
 => ? = 3 + 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => {{1},{2,6,8},{3},{4,5},{7}}
 => ? = 3 + 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => {{1},{2,4,5,8},{3},{6,7}}
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => {{1},{2,4,6,8},{3},{5},{7}}
 => ? = 3 + 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => {{1},{2,4,5,6,8},{3},{7}}
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => {{1},{2,4,6,7,8},{3},{5}}
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => {{1},{2,5,7,8},{3},{4},{6}}
 => ? = 3 + 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => {{1},{2,5,6,8},{3},{4},{7}}
 => ? = 3 + 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => {{1},{2,5,6,7,8},{3},{4}}
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => {{1},{2,6,7,8},{3},{4,5}}
 => ? = 2 + 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => {{1},{2,7,8},{3,4},{5,6}}
 => ? = 2 + 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => {{1},{2,7,8},{3,5},{4},{6}}
 => ? = 3 + 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => {{1},{2,5,8},{3,4},{6},{7}}
 => ? = 3 + 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => {{1},{2,5,8},{3,4},{6,7}}
 => ? = 2 + 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => {{1},{2,6,8},{3,5},{4},{7}}
 => ? = 3 + 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => {{1},{2,5,7,8},{3,4},{6}}
 => ? = 2 + 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => {{1},{2,5,6,8},{3,4},{7}}
 => ? = 2 + 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => {{1},{2,5,6,7,8},{3,4}}
 => ? = 1 + 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => {{1},{2,6,7,8},{3,5},{4}}
 => ? = 2 + 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => {{1},{2,7,8},{3,6},{4,5}}
 => ? = 2 + 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => {{1},{2,7,8},{3,4,5},{6}}
 => ? = 2 + 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => {{1},{2,6,8},{3,4,5},{7}}
 => ? = 2 + 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => {{1},{2,6,7,8},{3,4,5}}
 => ? = 1 + 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => {{1},{2,7,8},{3,4,5,6}}
 => ? = 1 + 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => {{1},{2},{3,8},{4,5,6,7}}
 => ? = 2 + 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => {{1},{2},{3,8},{4,6,7},{5}}
 => ? = 3 + 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => {{1},{2},{3,8},{4},{5,7},{6}}
 => ? = 4 + 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => {{1},{2},{3,8},{4},{5,6,7}}
 => ? = 3 + 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => {{1},{2},{3,8},{4,5},{6,7}}
 => ? = 3 + 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => {{1},{2},{3},{4,8},{5,6},{7}}
 => ? = 4 + 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => {{1},{2},{3},{4},{5,8},{6},{7}}
 => ? = 5 + 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => {{1},{2},{3},{4,8},{5,7},{6}}
 => ? = 4 + 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => {{1},{2},{3},{4,8},{5,6,7}}
 => ? = 3 + 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => {{1},{2},{3},{4},{5,8},{6,7}}
 => ? = 4 + 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,4},{5},{6,8},{7}}
 => ? = 4 + 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => {{1},{2},{3,4},{5,8},{6},{7}}
 => ? = 4 + 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => {{1},{2},{3,4},{5,8},{6,7}}
 => ? = 3 + 2
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]].
The following 65 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000871The number of very big ascents of a permutation. St000053The number of valleys of the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000024The number of double up and double down steps of a Dyck path. St000105The number of blocks in the set partition. St000636The hull number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001883The mutual visibility number of a graph. St000996The number of exclusive left-to-right maxima of a permutation. St000245The number of ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St000167The number of leaves of an ordered tree. St000672The number of minimal elements in Bruhat order not less than the permutation. St000159The number of distinct parts of the integer partition. St001488The number of corners of a skew partition. St000824The sum of the number of descents and the number of recoils of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000052The number of valleys of a Dyck path not on the x-axis. St001902The number of potential covers of a poset. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000665The number of rafts of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001960The number of descents of a permutation minus one if its first entry is not one. St001964The interval resolution global dimension of a poset. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000035The number of left outer peaks of a permutation. St000317The cycle descent number of a permutation. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St001637The number of (upper) dissectors of a poset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000075The orbit size of a standard tableau under promotion.