Processing math: 82%

Your data matches 18 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000237
(load all 3 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => 0
[2,1,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => 2
[3,2,1,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => 2
[3,4,1,2] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => 0
[4,2,1,3] => [1,4,3,2] => 0
[4,2,3,1] => [1,4,2,3] => 0
[4,3,1,2] => [1,4,2,3] => 0
[4,3,2,1] => [1,4,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => 1
Description
The number of small exceedances. This is the number of indices i such that πi=i+1.
Matching statistic: St000247
(load all 3 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => {{1}}
 => ? = 0
[1,2] => [1,2] => [2,1] => {{1,2}}
 => 0
[2,1] => [1,2] => [2,1] => {{1,2}}
 => 0
[1,2,3] => [1,2,3] => [2,3,1] => {{1,2,3}}
 => 0
[1,3,2] => [1,2,3] => [2,3,1] => {{1,2,3}}
 => 0
[2,1,3] => [1,2,3] => [2,3,1] => {{1,2,3}}
 => 0
[2,3,1] => [1,2,3] => [2,3,1] => {{1,2,3}}
 => 0
[3,1,2] => [1,3,2] => [3,2,1] => {{1,3},{2}}
 => 1
[3,2,1] => [1,3,2] => [3,2,1] => {{1,3},{2}}
 => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => {{1,2,4},{3}}
 => 1
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => {{1,2,4},{3}}
 => 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
 => 0
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => {{1,2,4},{3}}
 => 1
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => {{1,2,4},{3}}
 => 1
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => {{1,3,4},{2}}
 => 1
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => {{1,4},{2},{3}}
 => 2
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => {{1,3,4},{2}}
 => 1
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => {{1,4},{2},{3}}
 => 2
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => {{1,3,4},{2}}
 => 1
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => {{1,3,4},{2}}
 => 1
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => {{1,4},{2,3}}
 => 0
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => {{1,2,3,4}}
 => 0
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => {{1,4},{2,3}}
 => 0
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => {{1,2,3,4}}
 => 0
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => {{1,2,3,4}}
 => 0
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => {{1,2,3,4}}
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,5,4,1] => {{1,2,3,5},{4}}
 => 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,5,4,1] => {{1,2,3,5},{4}}
 => 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,5,4,1] => {{1,2,3,5},{4}}
 => 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,5,4,1] => {{1,2,3,5},{4}}
 => 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
 => 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
 => 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
 => 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
 => 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
 => 1
[1,4,5,3,2] => [1,2,4,3,5] => [2,4,3,5,1] => {{1,2,4,5},{3}}
 => 1
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => [3,5,7,8,6,2,4,1] => {{1,3,4,7,8},{2,5,6}}
 => ? = 0
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => [3,5,7,6,4,2,8,1] => {{1,3,7,8},{2,4,5,6}}
 => ? = 0
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => [3,5,7,8,6,2,4,1] => {{1,3,4,7,8},{2,5,6}}
 => ? = 0
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => [3,5,6,7,4,8,2,1] => {{1,3,6,8},{2,4,5,7}}
 => ? = 0
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => [3,5,7,8,6,2,4,1] => {{1,3,4,7,8},{2,5,6}}
 => ? = 0
[8,5,4,6,3,7,2,1] => [1,8,2,5,3,4,6,7] => [3,5,6,4,7,8,2,1] => {{1,3,6,8},{2,5,7},{4}}
 => ? = 1
[7,3,4,5,6,8,2,1] => [1,7,2,3,4,5,6,8] => [3,4,5,6,7,2,8,1] => {{1,3,5,7,8},{2,4,6}}
 => ? = 0
[6,7,5,8,4,2,3,1] => [1,6,2,7,3,5,4,8] => [3,5,7,6,2,4,8,1] => {{1,3,7,8},{2,5},{4,6}}
 => ? = 0
[8,6,4,5,7,2,3,1] => [1,8,2,6,3,4,5,7] => [3,5,6,7,4,8,2,1] => {{1,3,6,8},{2,4,5,7}}
 => ? = 0
[5,6,7,8,2,3,4,1] => [1,5,2,6,3,7,4,8] => [3,5,7,2,4,6,8,1] => {{1,3,7,8},{2,4,5},{6}}
 => ? = 1
[8,6,4,5,3,2,7,1] => [1,8,2,6,3,4,5,7] => [3,5,6,7,4,8,2,1] => {{1,3,6,8},{2,4,5,7}}
 => ? = 0
[8,5,4,6,3,2,7,1] => [1,8,2,5,3,4,6,7] => [3,5,6,4,7,8,2,1] => {{1,3,6,8},{2,5,7},{4}}
 => ? = 1
[8,6,4,3,5,2,7,1] => [1,8,2,6,3,4,5,7] => [3,5,6,7,4,8,2,1] => {{1,3,6,8},{2,4,5,7}}
 => ? = 0
[8,6,3,4,5,2,7,1] => [1,8,2,6,3,4,5,7] => [3,5,6,7,4,8,2,1] => {{1,3,6,8},{2,4,5,7}}
 => ? = 0
[8,4,5,3,6,2,7,1] => [1,8,2,4,3,5,6,7] => [3,5,4,6,7,8,2,1] => {{1,3,4,6,8},{2,5,7}}
 => ? = 0
[8,6,4,5,2,3,7,1] => [1,8,2,6,3,4,5,7] => [3,5,6,7,4,8,2,1] => {{1,3,6,8},{2,4,5,7}}
 => ? = 0
[8,5,4,3,2,6,7,1] => [1,8,2,5,3,4,6,7] => [3,5,6,4,7,8,2,1] => {{1,3,6,8},{2,5,7},{4}}
 => ? = 1
[8,4,5,3,2,6,7,1] => [1,8,2,4,3,5,6,7] => [3,5,4,6,7,8,2,1] => {{1,3,4,6,8},{2,5,7}}
 => ? = 0
[8,5,3,4,2,6,7,1] => [1,8,2,5,3,4,6,7] => [3,5,6,4,7,8,2,1] => {{1,3,6,8},{2,5,7},{4}}
 => ? = 1
[8,5,4,2,3,6,7,1] => [1,8,2,5,3,4,6,7] => [3,5,6,4,7,8,2,1] => {{1,3,6,8},{2,5,7},{4}}
 => ? = 1
[8,4,5,2,3,6,7,1] => [1,8,2,4,3,5,6,7] => [3,5,4,6,7,8,2,1] => {{1,3,4,6,8},{2,5,7}}
 => ? = 0
[8,4,3,2,5,6,7,1] => [1,8,2,4,3,5,6,7] => [3,5,4,6,7,8,2,1] => {{1,3,4,6,8},{2,5,7}}
 => ? = 0
[8,4,2,3,5,6,7,1] => [1,8,2,4,3,5,6,7] => [3,5,4,6,7,8,2,1] => {{1,3,4,6,8},{2,5,7}}
 => ? = 0
[6,5,4,7,3,2,8,1] => [1,6,2,5,3,4,7,8] => [3,5,6,4,2,7,8,1] => {{1,3,6,7,8},{2,5},{4}}
 => ? = 1
[5,6,4,7,2,3,8,1] => [1,5,2,6,3,4,7,8] => [3,5,6,2,4,7,8,1] => {{1,3,6,7,8},{2,4,5}}
 => ? = 0
[7,8,6,5,4,3,1,2] => [1,7,2,8,3,6,4,5] => [3,5,7,8,6,2,4,1] => {{1,3,4,7,8},{2,5,6}}
 => ? = 0
[7,6,5,8,4,3,1,2] => [1,7,2,6,3,5,4,8] => [3,5,7,6,4,2,8,1] => {{1,3,7,8},{2,4,5,6}}
 => ? = 0
[7,8,6,4,5,3,1,2] => [1,7,2,8,3,6,4,5] => [3,5,7,8,6,2,4,1] => {{1,3,4,7,8},{2,5,6}}
 => ? = 0
[8,6,4,5,7,3,1,2] => [1,8,2,6,3,4,5,7] => [3,5,6,7,4,8,2,1] => {{1,3,6,8},{2,4,5,7}}
 => ? = 0
[7,8,6,5,3,4,1,2] => [1,7,2,8,3,6,4,5] => [3,5,7,8,6,2,4,1] => {{1,3,4,7,8},{2,5,6}}
 => ? = 0
[7,5,6,8,3,4,1,2] => [1,7,2,5,3,6,4,8] => [3,5,7,4,6,2,8,1] => {{1,3,7,8},{2,5,6},{4}}
 => ? = 1
[8,5,4,6,3,7,1,2] => [1,8,2,5,3,4,6,7] => [3,5,6,4,7,8,2,1] => {{1,3,6,8},{2,5,7},{4}}
 => ? = 1
[7,3,4,5,6,8,1,2] => [1,7,2,3,4,5,6,8] => [3,4,5,6,7,2,8,1] => {{1,3,5,7,8},{2,4,6}}
 => ? = 0
[7,8,6,5,4,2,1,3] => [1,7,2,8,3,6,4,5] => [3,5,7,8,6,2,4,1] => {{1,3,4,7,8},{2,5,6}}
 => ? = 0
[7,6,5,8,4,2,1,3] => [1,7,2,6,3,5,4,8] => [3,5,7,6,4,2,8,1] => {{1,3,7,8},{2,4,5,6}}
 => ? = 0
[6,7,5,8,4,2,1,3] => [1,6,2,7,3,5,4,8] => [3,5,7,6,2,4,8,1] => {{1,3,7,8},{2,5},{4,6}}
 => ? = 0
[7,8,6,5,4,1,2,3] => [1,7,2,8,3,6,4,5] => [3,5,7,8,6,2,4,1] => {{1,3,4,7,8},{2,5,6}}
 => ? = 0
[7,6,5,8,4,1,2,3] => [1,7,2,6,3,5,4,8] => [3,5,7,6,4,2,8,1] => {{1,3,7,8},{2,4,5,6}}
 => ? = 0
[6,7,5,8,4,1,2,3] => [1,6,2,7,3,5,4,8] => [3,5,7,6,2,4,8,1] => {{1,3,7,8},{2,5},{4,6}}
 => ? = 0
[7,8,6,5,3,2,1,4] => [1,7,2,8,4,5,3,6] => [3,7,5,6,8,2,4,1] => {{1,3,5,8},{2,4,6,7}}
 => ? = 0
[7,6,5,8,3,2,1,4] => [1,7,2,6,3,5,4,8] => [3,5,7,6,4,2,8,1] => {{1,3,7,8},{2,4,5,6}}
 => ? = 0
[6,7,5,8,3,2,1,4] => [1,6,2,7,3,5,4,8] => [3,5,7,6,2,4,8,1] => {{1,3,7,8},{2,5},{4,6}}
 => ? = 0
[7,5,6,8,3,2,1,4] => [1,7,2,5,3,6,4,8] => [3,5,7,4,6,2,8,1] => {{1,3,7,8},{2,5,6},{4}}
 => ? = 1
[6,5,7,8,3,2,1,4] => [1,6,2,5,3,7,4,8] => [3,5,7,4,2,6,8,1] => {{1,3,7,8},{2,5},{4},{6}}
 => ? = 2
[7,8,6,5,2,3,1,4] => [1,7,2,8,4,5,3,6] => [3,7,5,6,8,2,4,1] => {{1,3,5,8},{2,4,6,7}}
 => ? = 0
[7,6,5,8,2,3,1,4] => [1,7,2,6,3,5,4,8] => [3,5,7,6,4,2,8,1] => {{1,3,7,8},{2,4,5,6}}
 => ? = 0
[7,5,6,8,2,3,1,4] => [1,7,2,5,3,6,4,8] => [3,5,7,4,6,2,8,1] => {{1,3,7,8},{2,5,6},{4}}
 => ? = 1
[5,6,7,8,2,3,1,4] => [1,5,2,6,3,7,4,8] => [3,5,7,2,4,6,8,1] => {{1,3,7,8},{2,4,5},{6}}
 => ? = 1
[7,8,6,5,3,1,2,4] => [1,7,2,8,4,5,3,6] => [3,7,5,6,8,2,4,1] => {{1,3,5,8},{2,4,6,7}}
 => ? = 0
Description
The number of singleton blocks of a set partition.
Matching statistic: St000214
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 2
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 2
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 0
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 0
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 0
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,6,7,3,4,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[1,2,6,7,4,3,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[1,2,6,7,5,3,4] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[1,2,6,7,5,4,3] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[1,3,6,7,2,4,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[1,3,6,7,4,2,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[1,3,6,7,5,2,4] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[1,3,6,7,5,4,2] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[1,6,2,7,4,3,5] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[1,6,2,7,5,3,4] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[1,6,3,7,4,2,5] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[1,6,3,7,5,2,4] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[1,6,4,7,2,3,5] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[1,6,4,7,3,2,5] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[1,6,4,7,5,2,3] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[1,6,4,7,5,3,2] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[2,1,6,7,3,4,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[2,1,6,7,4,3,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[2,1,6,7,5,3,4] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[2,1,6,7,5,4,3] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[2,3,6,7,1,4,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[2,3,6,7,4,1,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[2,3,6,7,5,1,4] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[2,3,6,7,5,4,1] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,6,4,5] => ? = 1
[2,6,1,7,4,3,5] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[2,6,1,7,5,3,4] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[2,6,3,7,4,1,5] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[2,6,3,7,5,1,4] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[2,6,4,7,1,3,5] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[2,6,4,7,3,1,5] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[2,6,4,7,5,1,3] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[2,6,4,7,5,3,1] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,6,3,4,5] => ? = 1
[3,1,7,4,5,6,2] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,1,7,4,6,5,2] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,1,7,5,4,6,2] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,1,7,5,6,4,2] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,2,7,4,5,6,1] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,2,7,4,6,5,1] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,2,7,5,4,6,1] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,2,7,5,6,4,1] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,1,5,6,2] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,1,6,5,2] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,2,5,6,1] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,2,6,5,1] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,5,1,6,2] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,5,2,6,1] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,5,6,1,2] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,5,6,2,1] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[4,1,2,6,3,7,5] => [1,4,6,7,5,3,2] => [1,7,6,2,5,3,4] => [1,5,6,3,7,4,2] => ? = 0
[4,1,2,7,3,5,6] => [1,4,7,6,5,3,2] => [1,7,6,2,5,4,3] => [1,5,7,3,6,4,2] => ? = 0
Description
The number of adjacencies of a permutation. An adjacency of a permutation π is an index i such that π(i)−1=π(i+1). Adjacencies are also known as ''small descents''. This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000648
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [3,2,1] => 1
[3,2,1] => [1,3,2] => [1,3,2] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => [3,4,2,1] => 2
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => [3,4,2,1] => 2
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 0
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [4,2,3,1] => 0
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 0
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => [4,2,3,1] => 0
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => [4,2,3,1] => 0
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => [4,2,3,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => [2,4,5,3,1] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => [2,4,5,3,1] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,6,7,3,4,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => ? = 1
[1,2,6,7,4,3,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => ? = 1
[1,2,6,7,5,3,4] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => ? = 1
[1,2,6,7,5,4,3] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => ? = 1
[1,3,2,4,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,2,4,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,2,5,7,4,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,2,5,7,6,4] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,2,6,4,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,3,2,6,5,4,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,3,2,6,7,4,5] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,3,2,6,7,5,4] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,3,4,2,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,4,2,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,4,5,7,2,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,4,5,7,6,2] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,4,6,2,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,3,4,6,5,2,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,3,4,6,7,2,5] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,3,4,6,7,5,2] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,3,6,7,2,4,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => ? = 1
[1,3,6,7,4,2,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => ? = 1
[1,3,6,7,5,2,4] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => ? = 1
[1,3,6,7,5,4,2] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => ? = 1
[1,6,2,7,4,3,5] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => ? = 1
[1,6,2,7,5,3,4] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => ? = 1
[1,6,3,7,4,2,5] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => ? = 1
[1,6,3,7,5,2,4] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => ? = 1
[1,6,4,7,2,3,5] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => ? = 1
[1,6,4,7,3,2,5] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => ? = 1
[1,6,4,7,5,2,3] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => ? = 1
[1,6,4,7,5,3,2] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [2,6,3,4,7,5,1] => ? = 1
[2,1,3,4,7,5,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[2,1,3,4,7,6,5] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
Description
The number of 2-excedences of a permutation. This is the number of positions 1≤i≤n such that σ(i)=i+2.
Matching statistic: St000248
(load all 4 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
 => ? = 0
[1,2] => [1,2] => {{1},{2}}
 => 0
[2,1] => [1,2] => {{1},{2}}
 => 0
[1,2,3] => [1,2,3] => {{1},{2},{3}}
 => 0
[1,3,2] => [1,2,3] => {{1},{2},{3}}
 => 0
[2,1,3] => [1,2,3] => {{1},{2},{3}}
 => 0
[2,3,1] => [1,2,3] => {{1},{2},{3}}
 => 0
[3,1,2] => [1,3,2] => {{1},{2,3}}
 => 1
[3,2,1] => [1,3,2] => {{1},{2,3}}
 => 1
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,2,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
 => 2
[3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
[3,2,4,1] => [1,3,4,2] => {{1},{2,3,4}}
 => 2
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => 0
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
 => 0
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => 0
[4,2,3,1] => [1,4,2,3] => {{1},{2,4},{3}}
 => 0
[4,3,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
 => 0
[4,3,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,3,2,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 2
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 2
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[1,4,5,3,2] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,6,7,5,4,3,2,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => ? = 1
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => {{1},{2,7},{3},{4,6,8},{5}}
 => ? = 0
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ? = 0
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => {{1},{2,7},{3},{4,6},{5},{8}}
 => ? = 0
[8,7,6,4,5,3,2,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,6,7,4,5,3,2,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => ? = 1
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => {{1},{2,7},{3},{4,6,8},{5}}
 => ? = 0
[8,7,4,5,6,3,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ? = 0
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => {{1},{2,7},{3},{4,6},{5},{8}}
 => ? = 0
[8,7,6,5,3,4,2,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,5,6,7,3,4,2,1] => [1,8,2,5,3,6,4,7] => {{1},{2,8},{3},{4,5},{6},{7}}
 => ? = 1
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => {{1},{2,5},{3},{4,7},{6},{8}}
 => ? = 0
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,7,4,5,3,6,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,7,4,3,5,6,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[8,7,3,4,5,6,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,3,4,5,6,2,1] => [1,7,2,8,3,4,5,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,5,4,6,3,7,2,1] => [1,8,2,5,3,4,6,7] => {{1},{2,8},{3},{4,5},{6},{7}}
 => ? = 1
[8,3,4,5,6,7,2,1] => [1,8,2,3,4,5,6,7] => {{1},{2,8},{3},{4},{5},{6},{7}}
 => ? = 0
[7,5,4,6,3,8,2,1] => [1,7,2,5,3,4,6,8] => {{1},{2,7},{3},{4,5},{6},{8}}
 => ? = 1
[7,3,4,5,6,8,2,1] => [1,7,2,3,4,5,6,8] => {{1},{2,7},{3},{4},{5},{6},{8}}
 => ? = 0
[6,3,4,5,7,8,2,1] => [1,6,8,2,3,4,5,7] => {{1},{2,6},{3,8},{4},{5},{7}}
 => ? = 0
[8,7,6,5,4,2,3,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[8,6,7,5,4,2,3,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => ? = 1
[8,7,5,6,4,2,3,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[8,6,5,7,4,2,3,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ? = 0
[6,7,5,8,4,2,3,1] => [1,6,2,7,3,5,4,8] => {{1},{2,6},{3},{4,7},{5},{8}}
 => ? = 0
[8,7,6,4,5,2,3,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[8,6,7,4,5,2,3,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => ? = 1
[6,7,8,4,5,2,3,1] => [1,6,2,7,3,8,4,5] => {{1},{2,6,8},{3},{4,7},{5}}
 => ? = 0
[8,7,4,5,6,2,3,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[8,6,4,5,7,2,3,1] => [1,8,2,6,3,4,5,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ? = 0
[6,7,4,5,8,2,3,1] => [1,6,2,7,3,4,5,8] => {{1},{2,6},{3},{4,7},{5},{8}}
 => ? = 0
[8,6,7,5,3,2,4,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => ? = 1
[8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ? = 0
[8,5,6,7,3,2,4,1] => [1,8,2,5,3,6,4,7] => {{1},{2,8},{3},{4,5},{6},{7}}
 => ? = 1
[8,6,7,5,2,3,4,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => ? = 1
Description
The number of anti-singletons of a set partition. An anti-singleton of a set partition S is an index i such that i and i+1 (considered cyclically) are both in the same block of S. For noncrossing set partitions, this is also the number of singletons of the image of S under the Kreweras complement.
Matching statistic: St000502
(load all 3 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
 => ? = 0
[1,2] => [1,2] => {{1},{2}}
 => 0
[2,1] => [1,2] => {{1},{2}}
 => 0
[1,2,3] => [1,2,3] => {{1},{2},{3}}
 => 0
[1,3,2] => [1,2,3] => {{1},{2},{3}}
 => 0
[2,1,3] => [1,2,3] => {{1},{2},{3}}
 => 0
[2,3,1] => [1,2,3] => {{1},{2},{3}}
 => 0
[3,1,2] => [1,3,2] => {{1},{2,3}}
 => 1
[3,2,1] => [1,3,2] => {{1},{2,3}}
 => 1
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,2,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 0
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
 => 1
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
 => 2
[3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
[3,2,4,1] => [1,3,4,2] => {{1},{2,3,4}}
 => 2
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
 => 1
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => 0
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
 => 0
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => 0
[4,2,3,1] => [1,4,2,3] => {{1},{2,4},{3}}
 => 0
[4,3,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
 => 0
[4,3,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,3,2,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 1
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 2
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 2
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[1,4,5,3,2] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 1
[8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,6,7,5,4,3,2,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => ? = 1
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => {{1},{2,7},{3},{4,6,8},{5}}
 => ? = 0
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ? = 0
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => {{1},{2,7},{3},{4,6},{5},{8}}
 => ? = 0
[8,7,6,4,5,3,2,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,6,7,4,5,3,2,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => ? = 1
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => {{1},{2,7},{3},{4,6,8},{5}}
 => ? = 0
[8,7,4,5,6,3,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ? = 0
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => {{1},{2,7},{3},{4,6},{5},{8}}
 => ? = 0
[8,7,6,5,3,4,2,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,5,6,7,3,4,2,1] => [1,8,2,5,3,6,4,7] => {{1},{2,8},{3},{4,5},{6},{7}}
 => ? = 1
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => {{1},{2,5},{3},{4,7},{6},{8}}
 => ? = 0
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,7,4,5,3,6,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,7,4,3,5,6,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[8,7,3,4,5,6,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[7,8,3,4,5,6,2,1] => [1,7,2,8,3,4,5,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ? = 0
[8,5,4,6,3,7,2,1] => [1,8,2,5,3,4,6,7] => {{1},{2,8},{3},{4,5},{6},{7}}
 => ? = 1
[8,3,4,5,6,7,2,1] => [1,8,2,3,4,5,6,7] => {{1},{2,8},{3},{4},{5},{6},{7}}
 => ? = 0
[7,5,4,6,3,8,2,1] => [1,7,2,5,3,4,6,8] => {{1},{2,7},{3},{4,5},{6},{8}}
 => ? = 1
[7,3,4,5,6,8,2,1] => [1,7,2,3,4,5,6,8] => {{1},{2,7},{3},{4},{5},{6},{8}}
 => ? = 0
[6,3,4,5,7,8,2,1] => [1,6,8,2,3,4,5,7] => {{1},{2,6},{3,8},{4},{5},{7}}
 => ? = 0
[8,7,6,5,4,2,3,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[8,6,7,5,4,2,3,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => ? = 1
[8,7,5,6,4,2,3,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[8,6,5,7,4,2,3,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ? = 0
[6,7,5,8,4,2,3,1] => [1,6,2,7,3,5,4,8] => {{1},{2,6},{3},{4,7},{5},{8}}
 => ? = 0
[8,7,6,4,5,2,3,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[8,6,7,4,5,2,3,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => ? = 1
[6,7,8,4,5,2,3,1] => [1,6,2,7,3,8,4,5] => {{1},{2,6,8},{3},{4,7},{5}}
 => ? = 0
[8,7,4,5,6,2,3,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ? = 0
[8,6,4,5,7,2,3,1] => [1,8,2,6,3,4,5,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ? = 0
[6,7,4,5,8,2,3,1] => [1,6,2,7,3,4,5,8] => {{1},{2,6},{3},{4,7},{5},{8}}
 => ? = 0
[8,6,7,5,3,2,4,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => ? = 1
[8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ? = 0
[8,5,6,7,3,2,4,1] => [1,8,2,5,3,6,4,7] => {{1},{2,8},{3},{4,5},{6},{7}}
 => ? = 1
[8,6,7,5,2,3,4,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => ? = 1
Description
The number of successions of a set partitions. This is the number of indices i such that i and i+1 belonging to the same block.
Matching statistic: St000504
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
 => {{1}}
 => ? = 0 + 1
[1,2] => [1,2] => {{1},{2}}
 => {{1},{2}}
 => 1 = 0 + 1
[2,1] => [1,2] => {{1},{2}}
 => {{1},{2}}
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 1 = 0 + 1
[1,3,2] => [1,2,3] => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 1 = 0 + 1
[2,1,3] => [1,2,3] => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 1 = 0 + 1
[2,3,1] => [1,2,3] => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 1 = 0 + 1
[3,1,2] => [1,3,2] => {{1},{2,3}}
 => {{1,3},{2}}
 => 2 = 1 + 1
[3,2,1] => [1,3,2] => {{1},{2,3}}
 => {{1,3},{2}}
 => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 3 = 2 + 1
[3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => {{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 3 = 2 + 1
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1 = 0 + 1
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1 = 0 + 1
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1 = 0 + 1
[4,2,3,1] => [1,4,2,3] => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1 = 0 + 1
[4,3,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1 = 0 + 1
[4,3,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 3 = 2 + 1
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 3 = 2 + 1
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => 2 = 1 + 1
[8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ?
 => ? = 0 + 1
[8,6,7,5,4,3,2,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => {{1,7},{2,6},{3},{4,8},{5}}
 => ? = 1 + 1
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => {{1},{2,7},{3},{4,6,8},{5}}
 => ?
 => ? = 0 + 1
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ?
 => ? = 0 + 1
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ?
 => ? = 0 + 1
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => {{1},{2,7},{3},{4,6},{5},{8}}
 => ?
 => ? = 0 + 1
[8,7,6,4,5,3,2,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ?
 => ? = 0 + 1
[8,6,7,4,5,3,2,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => {{1,7},{2,6},{3},{4,8},{5}}
 => ? = 1 + 1
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => {{1},{2,7},{3},{4,6,8},{5}}
 => ?
 => ? = 0 + 1
[8,7,4,5,6,3,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ?
 => ? = 0 + 1
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => {{1},{2,7},{3},{4,6},{5},{8}}
 => ?
 => ? = 0 + 1
[8,7,6,5,3,4,2,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ?
 => ? = 0 + 1
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ?
 => ? = 0 + 1
[8,5,6,7,3,4,2,1] => [1,8,2,5,3,6,4,7] => {{1},{2,8},{3},{4,5},{6},{7}}
 => ?
 => ? = 1 + 1
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => {{1},{2,5},{3},{4,7},{6},{8}}
 => {{1},{2},{3,5,7},{4},{6},{8}}
 => ? = 0 + 1
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ?
 => ? = 0 + 1
[8,7,4,5,3,6,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ?
 => ? = 0 + 1
[8,7,4,3,5,6,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[8,7,3,4,5,6,2,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[7,8,3,4,5,6,2,1] => [1,7,2,8,3,4,5,6] => {{1},{2,7},{3},{4,8},{5},{6}}
 => ?
 => ? = 0 + 1
[8,5,4,6,3,7,2,1] => [1,8,2,5,3,4,6,7] => {{1},{2,8},{3},{4,5},{6},{7}}
 => ?
 => ? = 1 + 1
[8,3,4,5,6,7,2,1] => [1,8,2,3,4,5,6,7] => {{1},{2,8},{3},{4},{5},{6},{7}}
 => {{1},{2},{3},{4},{5},{6,8},{7}}
 => ? = 0 + 1
[7,5,4,6,3,8,2,1] => [1,7,2,5,3,4,6,8] => {{1},{2,7},{3},{4,5},{6},{8}}
 => ?
 => ? = 1 + 1
[7,3,4,5,6,8,2,1] => [1,7,2,3,4,5,6,8] => {{1},{2,7},{3},{4},{5},{6},{8}}
 => ?
 => ? = 0 + 1
[6,3,4,5,7,8,2,1] => [1,6,8,2,3,4,5,7] => {{1},{2,6},{3,8},{4},{5},{7}}
 => ?
 => ? = 0 + 1
[8,7,6,5,4,2,3,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[8,6,7,5,4,2,3,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => {{1,7},{2,6},{3},{4,8},{5}}
 => ? = 1 + 1
[8,7,5,6,4,2,3,1] => [1,8,2,7,3,5,4,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[8,6,5,7,4,2,3,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ?
 => ? = 0 + 1
[6,7,5,8,4,2,3,1] => [1,6,2,7,3,5,4,8] => {{1},{2,6},{3},{4,7},{5},{8}}
 => ?
 => ? = 0 + 1
[8,7,6,4,5,2,3,1] => [1,8,2,7,3,6,4,5] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[8,6,7,4,5,2,3,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => {{1,7},{2,6},{3},{4,8},{5}}
 => ? = 1 + 1
[6,7,8,4,5,2,3,1] => [1,6,2,7,3,8,4,5] => {{1},{2,6,8},{3},{4,7},{5}}
 => {{1},{2,8},{3,7},{4,6},{5}}
 => ? = 0 + 1
[8,7,4,5,6,2,3,1] => [1,8,2,7,3,4,5,6] => {{1},{2,8},{3},{4,7},{5},{6}}
 => ?
 => ? = 0 + 1
[8,6,4,5,7,2,3,1] => [1,8,2,6,3,4,5,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ?
 => ? = 0 + 1
[6,7,4,5,8,2,3,1] => [1,6,2,7,3,4,5,8] => {{1},{2,6},{3},{4,7},{5},{8}}
 => ?
 => ? = 0 + 1
[8,6,7,5,3,2,4,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => {{1,7},{2,6},{3},{4,8},{5}}
 => ? = 1 + 1
[8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => {{1},{2,8},{3},{4,6},{5},{7}}
 => ?
 => ? = 0 + 1
[8,5,6,7,3,2,4,1] => [1,8,2,5,3,6,4,7] => {{1},{2,8},{3},{4,5},{6},{7}}
 => ?
 => ? = 1 + 1
[8,6,7,5,2,3,4,1] => [1,8,2,6,3,7,4,5] => {{1},{2,8},{3},{4,6,7},{5}}
 => {{1,7},{2,6},{3},{4,8},{5}}
 => ? = 1 + 1
Description
The cardinality of the first block of a set partition. The number of partitions of {1,…,n} into k blocks in which the first block has cardinality j+1 is given by \binom{n-1}{j}S(n-j-1,k-1), see [1, Theorem 1.1] and the references therein. Here, S(n,k) are the ''Stirling numbers of the second kind'' counting all set partitions of \{1,\ldots,n\} into k blocks [2].
Matching statistic: St001810
(load all 2 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001810: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,2,3] => [2,3,1] => 0
[2,1,3] => [1,2,3] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [1,3,2] => [3,2,1] => 1
[3,2,1] => [1,3,2] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => 0
[1,4,2,3] => [1,2,4,3] => [2,4,3,1] => 1
[1,4,3,2] => [1,2,4,3] => [2,4,3,1] => 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => 0
[2,4,1,3] => [1,2,4,3] => [2,4,3,1] => 1
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => 1
[3,1,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[3,1,4,2] => [1,3,4,2] => [4,2,3,1] => 2
[3,2,1,4] => [1,3,2,4] => [3,2,4,1] => 1
[3,2,4,1] => [1,3,4,2] => [4,2,3,1] => 2
[3,4,1,2] => [1,3,2,4] => [3,2,4,1] => 1
[3,4,2,1] => [1,3,2,4] => [3,2,4,1] => 1
[4,1,2,3] => [1,4,3,2] => [4,3,2,1] => 0
[4,1,3,2] => [1,4,2,3] => [3,4,2,1] => 0
[4,2,1,3] => [1,4,3,2] => [4,3,2,1] => 0
[4,2,3,1] => [1,4,2,3] => [3,4,2,1] => 0
[4,3,1,2] => [1,4,2,3] => [3,4,2,1] => 0
[4,3,2,1] => [1,4,2,3] => [3,4,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,5,3,4,1] => 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,3,4,7,5,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,3,5,7,4,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,3,5,7,6,4] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,4,3,7,5,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,4,3,7,6,5] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,2,4,5,7,3,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,4,5,7,6,3] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,2,6,7,3,4,5] => [1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => ? = 1
[1,2,6,7,4,3,5] => [1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => ? = 1
[1,2,6,7,5,3,4] => [1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => ? = 1
[1,2,6,7,5,4,3] => [1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => ? = 1
[1,3,2,4,5,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,3,2,4,5,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,3,2,4,6,5,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,3,2,4,6,7,5] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,3,2,4,7,5,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,2,4,7,6,5] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,2,5,4,6,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,3,2,5,4,7,6] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,3,2,5,6,4,7] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,3,2,5,6,7,4] => [1,2,3,4,5,6,7] => [2,3,4,5,6,7,1] => ? = 0
[1,3,2,5,7,4,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,2,5,7,6,4] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 1
[1,3,2,6,4,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
[1,3,2,6,5,4,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 1
Description
The number of fixed points of a permutation smaller than its largest moved point.
Matching statistic: St000894
(load all 2 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [[1]]
 => ? = 0
[1,2] => [1,2] => [2,1] => [[0,1],[1,0]]
 => 0
[2,1] => [1,2] => [2,1] => [[0,1],[1,0]]
 => 0
[1,2,3] => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 0
[1,3,2] => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 0
[2,1,3] => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 0
[2,3,1] => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 0
[3,1,2] => [1,3,2] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 1
[3,2,1] => [1,3,2] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => 0
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 1
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 2
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 1
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 2
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 1
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 1
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 0
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => 0
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 0
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => 0
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => 0
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 2
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 2
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 1
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,2,5,3,4,6] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 1
[1,2,5,4,3,6] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 1
[1,2,5,6,3,4] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 1
[1,2,5,6,4,3] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 1
[1,2,6,3,5,4] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => ? = 0
[1,2,6,4,5,3] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => ? = 0
[1,2,6,5,3,4] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => ? = 0
[1,2,6,5,4,3] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => ? = 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
[1,3,5,2,4,6] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 1
[1,3,5,4,2,6] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 1
[1,3,5,6,2,4] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 1
[1,3,5,6,4,2] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 1
[1,3,6,2,5,4] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => ? = 0
[1,3,6,4,5,2] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => ? = 0
[1,3,6,5,2,4] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => ? = 0
[1,3,6,5,4,2] => [1,2,3,6,4,5] => [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
 => ? = 0
[1,4,2,3,5,6] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
[1,4,2,3,6,5] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
[1,4,2,5,3,6] => [1,2,4,5,3,6] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 2
[1,4,2,6,5,3] => [1,2,4,6,3,5] => [2,3,6,4,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
 => ? = 1
[1,4,3,2,5,6] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
[1,4,3,2,6,5] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
[1,4,3,5,2,6] => [1,2,4,5,3,6] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 2
[1,4,3,6,5,2] => [1,2,4,6,3,5] => [2,3,6,4,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
 => ? = 1
[1,4,5,2,3,6] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
[1,4,5,2,6,3] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
[1,4,5,3,2,6] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
[1,4,5,3,6,2] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
[1,4,5,6,2,3] => [1,2,4,6,3,5] => [2,3,6,4,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
 => ? = 1
[1,4,5,6,3,2] => [1,2,4,6,3,5] => [2,3,6,4,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
 => ? = 1
[1,4,6,5,2,3] => [1,2,4,5,3,6] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 2
[1,4,6,5,3,2] => [1,2,4,5,3,6] => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 2
[1,5,2,3,4,6] => [1,2,5,4,3,6] => [2,3,6,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 0
Description
The trace of an alternating sign matrix.
Matching statistic: St000259
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => ([],1)
 => ([],1)
 => 0
[1,2] => [1,2] => ([],2)
 => ([],1)
 => 0
[2,1] => [1,2] => ([],2)
 => ([],1)
 => 0
[1,2,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[1,3,2] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,1,3] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[2,3,1] => [1,2,3] => ([],3)
 => ([],1)
 => 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,2,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,2,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,3,4,2] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[2,1,3,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,1,4,3] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,1,4] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,3,4,1] => [1,2,3,4] => ([],4)
 => ([],1)
 => 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 1
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 1
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? = 0
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => 0
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph.