Processing math: 100%

Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000245
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 1
{{1},{2}}
 => [1,2] => [2,1] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => [1,2,3] => 2
{{1,2},{3}}
 => [2,1,3] => [3,2,1] => [2,3,1] => 1
{{1,3},{2}}
 => [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
 => [1,3,2] => [2,1,3] => [2,1,3] => 1
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => [3,1,2] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 3
{{1,2,3},{4}}
 => [2,3,1,4] => [4,2,3,1] => [2,3,4,1] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [3,2,1,4] => [2,3,1,4] => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [3,2,4,1] => [2,4,1,3] => 2
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => [4,3,2,1] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [4,3,2,1] => [3,2,4,1] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,3,2] => [1,3,4,2] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,4,3,1] => [3,4,1,2] => 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,3,4,2] => [1,4,2,3] => 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [5,2,3,4,1] => [2,3,4,5,1] => 3
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [4,2,3,1,5] => [2,3,4,1,5] => 3
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [4,2,3,5,1] => [2,3,5,1,4] => 3
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 3
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [5,2,1,3,4] => [2,5,4,3,1] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [5,2,4,3,1] => [2,4,3,5,1] => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [3,2,1,4,5] => [2,3,1,4,5] => 3
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [3,2,5,4,1] => [2,4,5,1,3] => 3
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,4,5,3] => [1,2,5,3,4] => 3
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [3,2,1,5,4] => [2,3,1,5,4] => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [3,2,4,1,5] => [2,4,1,3,5] => 3
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [3,2,4,5,1] => [2,5,1,3,4] => 3
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 3
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,1,2,4,3] => [4,5,3,2,1] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [5,3,2,4,1] => [3,2,4,5,1] => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,5,2,3,4] => [1,5,4,3,2] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [4,1,2,3,5] => [4,3,2,1,5] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,5,2,3,1] => [5,1,4,3,2] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,1,2,5,3] => [5,3,2,1,4] => 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [4,3,2,1,5] => [3,2,4,1,5] => 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [4,3,2,5,1] => [3,2,5,1,4] => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,4,3,2,5] => [1,3,4,2,5] => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,1,3,2,4] => [3,5,4,2,1] => 1
Description
The number of ascents of a permutation.
Matching statistic: St000250
(load all 14 compositions to match this statistic)
Mapping
Mp00176: Set partitions —rotate decreasing⟶ Set partitions
St000250: Set partitions ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 80%
Values
{{1}}
 => {{1}}
 => ? = 0 + 2
{{1,2}}
 => {{1,2}}
 => 3 = 1 + 2
{{1},{2}}
 => {{1},{2}}
 => 2 = 0 + 2
{{1,2,3}}
 => {{1,2,3}}
 => 4 = 2 + 2
{{1,2},{3}}
 => {{1,3},{2}}
 => 3 = 1 + 2
{{1,3},{2}}
 => {{1},{2,3}}
 => 3 = 1 + 2
{{1},{2,3}}
 => {{1,2},{3}}
 => 3 = 1 + 2
{{1},{2},{3}}
 => {{1},{2},{3}}
 => 3 = 1 + 2
{{1,2,3,4}}
 => {{1,2,3,4}}
 => 5 = 3 + 2
{{1,2,3},{4}}
 => {{1,2,4},{3}}
 => 4 = 2 + 2
{{1,2,4},{3}}
 => {{1,3,4},{2}}
 => 4 = 2 + 2
{{1,2},{3,4}}
 => {{1,4},{2,3}}
 => 4 = 2 + 2
{{1,2},{3},{4}}
 => {{1,4},{2},{3}}
 => 4 = 2 + 2
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => 4 = 2 + 2
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => 2 = 0 + 2
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => 3 = 1 + 2
{{1,4},{2,3}}
 => {{1,2},{3,4}}
 => 4 = 2 + 2
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => 4 = 2 + 2
{{1},{2,3},{4}}
 => {{1,2},{3},{4}}
 => 4 = 2 + 2
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => 4 = 2 + 2
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 3 = 1 + 2
{{1},{2},{3,4}}
 => {{1},{2,3},{4}}
 => 4 = 2 + 2
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 4 = 2 + 2
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 6 = 4 + 2
{{1,2,3,4},{5}}
 => {{1,2,3,5},{4}}
 => 5 = 3 + 2
{{1,2,3,5},{4}}
 => {{1,2,4,5},{3}}
 => 5 = 3 + 2
{{1,2,3},{4,5}}
 => {{1,2,5},{3,4}}
 => 5 = 3 + 2
{{1,2,3},{4},{5}}
 => {{1,2,5},{3},{4}}
 => 5 = 3 + 2
{{1,2,4,5},{3}}
 => {{1,3,4,5},{2}}
 => 5 = 3 + 2
{{1,2,4},{3,5}}
 => {{1,3,5},{2,4}}
 => 3 = 1 + 2
{{1,2,4},{3},{5}}
 => {{1,3,5},{2},{4}}
 => 4 = 2 + 2
{{1,2,5},{3,4}}
 => {{1,4,5},{2,3}}
 => 5 = 3 + 2
{{1,2},{3,4,5}}
 => {{1,5},{2,3,4}}
 => 5 = 3 + 2
{{1,2},{3,4},{5}}
 => {{1,5},{2,3},{4}}
 => 5 = 3 + 2
{{1,2,5},{3},{4}}
 => {{1,4,5},{2},{3}}
 => 5 = 3 + 2
{{1,2},{3,5},{4}}
 => {{1,5},{2,4},{3}}
 => 4 = 2 + 2
{{1,2},{3},{4,5}}
 => {{1,5},{2},{3,4}}
 => 5 = 3 + 2
{{1,2},{3},{4},{5}}
 => {{1,5},{2},{3},{4}}
 => 5 = 3 + 2
{{1,3,4,5},{2}}
 => {{1},{2,3,4,5}}
 => 5 = 3 + 2
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => 3 = 1 + 2
{{1,3,4},{2},{5}}
 => {{1},{2,3,5},{4}}
 => 4 = 2 + 2
{{1,3,5},{2,4}}
 => {{1,3},{2,4,5}}
 => 3 = 1 + 2
{{1,3},{2,4,5}}
 => {{1,3,4},{2,5}}
 => 3 = 1 + 2
{{1,3},{2,4},{5}}
 => {{1,3},{2,5},{4}}
 => 3 = 1 + 2
{{1,3,5},{2},{4}}
 => {{1},{2,4,5},{3}}
 => 4 = 2 + 2
{{1,3},{2,5},{4}}
 => {{1,4},{2,5},{3}}
 => 3 = 1 + 2
{{1,3},{2},{4,5}}
 => {{1},{2,5},{3,4}}
 => 4 = 2 + 2
{{1,3},{2},{4},{5}}
 => {{1},{2,5},{3},{4}}
 => 4 = 2 + 2
{{1,4,5},{2,3}}
 => {{1,2},{3,4,5}}
 => 5 = 3 + 2
{{1,4},{2,3,5}}
 => {{1,2,4},{3,5}}
 => 3 = 1 + 2
{{1,4},{2,3},{5}}
 => {{1,2},{3,5},{4}}
 => 4 = 2 + 2
{{1,5},{2,8},{3,6},{4,7}}
 => {{1,7},{2,5},{3,6},{4,8}}
 => ? = 2 + 2
{{1,5},{2,6},{3,7},{4,8}}
 => {{1,5},{2,6},{3,7},{4,8}}
 => ? = 2 + 2
{{1,6},{2,4},{3,7},{5,8}}
 => {{1,3},{2,6},{4,7},{5,8}}
 => ? = 2 + 2
{{1,7},{2,4},{3,6},{5,8}}
 => {{1,3},{2,5},{4,7},{6,8}}
 => ? = 2 + 2
{{1,3},{2,4},{5,7},{6,8}}
 => {{1,3},{2,8},{4,6},{5,7}}
 => ? = 2 + 2
{{1,7},{2,3},{4,8},{5,6}}
 => {{1,2},{3,7},{4,5},{6,8}}
 => ? = 4 + 2
{{1,8},{2,3},{4,7},{5,6}}
 => {{1,2},{3,6},{4,5},{7,8}}
 => ? = 5 + 2
{{1,6},{2,4},{3,8},{5,7}}
 => {{1,3},{2,7},{4,6},{5,8}}
 => ? = 2 + 2
{{1,7},{2,5},{3,8},{4,6}}
 => {{1,4},{2,7},{3,5},{6,8}}
 => ? = 2 + 2
{{1,8},{2,5},{3,7},{4,6}}
 => {{1,4},{2,6},{3,5},{7,8}}
 => ? = 3 + 2
{{1},{2},{3},{4},{5},{6},{7},{8}}
 => {{1},{2},{3},{4},{5},{6},{7},{8}}
 => ? = 6 + 2
{{1},{2},{3},{4},{5},{6},{7,8}}
 => {{1},{2},{3},{4},{5},{6,7},{8}}
 => ? = 6 + 2
{{1},{2},{3},{4},{5},{6,8},{7}}
 => {{1},{2},{3},{4},{5,7},{6},{8}}
 => ? = 5 + 2
{{1},{2},{3},{4},{5},{6,7,8}}
 => {{1},{2},{3},{4},{5,6,7},{8}}
 => ? = 6 + 2
{{1},{2},{3},{4},{5,6,8},{7}}
 => {{1},{2},{3},{4,5,7},{6},{8}}
 => ? = 5 + 2
{{1},{2},{3},{4},{5,6,7,8}}
 => {{1},{2},{3},{4,5,6,7},{8}}
 => ? = 6 + 2
{{1},{2},{3},{4,8},{5},{6},{7}}
 => {{1},{2},{3,7},{4},{5},{6},{8}}
 => ? = 5 + 2
{{1},{2},{3},{4,6,8},{5},{7}}
 => ?
 => ? = 4 + 2
{{1},{2},{3,4,5},{6,7,8}}
 => {{1},{2,3,4},{5,6,7},{8}}
 => ? = 6 + 2
{{1},{2},{3,4,5,8},{6},{7}}
 => {{1},{2,3,4,7},{5},{6},{8}}
 => ? = 5 + 2
{{1},{2,3},{4,5},{6,7},{8}}
 => {{1,2},{3,4},{5,6},{7},{8}}
 => ? = 6 + 2
{{1},{2,3},{4,6,8},{5},{7}}
 => ?
 => ? = 4 + 2
{{1},{2,3},{4,5,6,7,8}}
 => {{1,2},{3,4,5,6,7},{8}}
 => ? = 6 + 2
{{1},{2,8},{3},{4},{5},{6},{7}}
 => {{1,7},{2},{3},{4},{5},{6},{8}}
 => ? = 5 + 2
{{1},{2,8},{3},{4},{5,6,7}}
 => ?
 => ? = 5 + 2
{{1},{2,6,8},{3},{4,5},{7}}
 => ?
 => ? = 4 + 2
{{1},{2,4,8},{3},{5},{6},{7}}
 => ?
 => ? = 4 + 2
{{1},{2,8},{3},{4,6,7},{5}}
 => ?
 => ? = 4 + 2
{{1},{2,4,8},{3},{5},{6,7}}
 => ?
 => ? = 4 + 2
{{1},{2,4,6,8},{3},{5},{7}}
 => {{1,3,5,7},{2},{4},{6},{8}}
 => ? = 3 + 2
{{1},{2,4,8},{3},{5,7},{6}}
 => {{1,3,7},{2},{4,6},{5},{8}}
 => ? = 3 + 2
{{1},{2,6,8},{3,5},{4},{7}}
 => {{1,5,7},{2,4},{3},{6},{8}}
 => ? = 3 + 2
{{1},{2,3,8},{4},{5},{6},{7}}
 => {{1,2,7},{3},{4},{5},{6},{8}}
 => ? = 5 + 2
{{1},{2,3,7},{4},{5,6},{8}}
 => ?
 => ? = 5 + 2
{{1},{2,3,5,7},{4},{6},{8}}
 => {{1,2,4,6},{3},{5},{7},{8}}
 => ? = 4 + 2
{{1},{2,8},{3,5,7},{4},{6}}
 => {{1,7},{2,4,6},{3},{5},{8}}
 => ? = 3 + 2
{{1},{2,8},{3,7},{4,6},{5}}
 => {{1,7},{2,6},{3,5},{4},{8}}
 => ? = 3 + 2
{{1},{2,3,4,5,6,8},{7}}
 => {{1,2,3,4,5,7},{6},{8}}
 => ? = 5 + 2
{{1,2},{3},{4},{5},{6},{7},{8}}
 => {{1,8},{2},{3},{4},{5},{6},{7}}
 => ? = 6 + 2
{{1,2},{3},{4},{5},{6},{7,8}}
 => {{1,8},{2},{3},{4},{5},{6,7}}
 => ? = 6 + 2
{{1,2},{3,4},{5,6},{7},{8}}
 => {{1,8},{2,3},{4,5},{6},{7}}
 => ? = 6 + 2
{{1,2},{3,4,5,8},{6,7}}
 => {{1,8},{2,3,4,7},{5,6}}
 => ? = 5 + 2
{{1,3},{2},{4,6,8},{5},{7}}
 => {{1},{2,8},{3,5,7},{4},{6}}
 => ? = 3 + 2
{{1,3},{2},{4,8},{5,7},{6}}
 => {{1},{2,8},{3,7},{4,6},{5}}
 => ? = 3 + 2
{{1,8},{2},{3},{4},{5},{6},{7}}
 => {{1},{2},{3},{4},{5},{6},{7,8}}
 => ? = 6 + 2
{{1,7,8},{2},{3},{4},{5},{6}}
 => {{1},{2},{3},{4},{5},{6,7,8}}
 => ? = 6 + 2
{{1,8},{2},{3},{4},{5},{6,7}}
 => {{1},{2},{3},{4},{5,6},{7,8}}
 => ? = 6 + 2
{{1,6,7,8},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5,6,7,8}}
 => ? = 6 + 2
{{1,8},{2},{3},{4},{5,6,7}}
 => {{1},{2},{3},{4,5,6},{7,8}}
 => ? = 6 + 2
Description
The number of blocks ([[St000105]]) plus the number of antisingletons ([[St000248]]) of a set partition.
Matching statistic: St000702
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 70%
Values
{{1}}
 => [1] => [1] => [1] => ? = 0 + 1
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 2 = 1 + 1
{{1},{2}}
 => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
{{1,2,3}}
 => [2,3,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
{{1,2},{3}}
 => [2,1,3] => [3,2,1] => [3,2,1] => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
{{1},{2,3}}
 => [1,3,2] => [2,1,3] => [2,1,3] => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => [3,1,2] => 2 = 1 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [4,2,3,1] => [4,2,3,1] => 3 = 2 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [3,2,4,1] => [4,2,1,3] => 3 = 2 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => [2,3,4,1] => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,4,3,1] => [4,1,3,2] => 3 = 2 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,3,4,2] => [1,4,2,3] => 3 = 2 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 3 = 2 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3 = 2 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [5,2,3,4,1] => [5,2,3,4,1] => 4 = 3 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [4,2,3,1,5] => [4,2,3,1,5] => 4 = 3 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [4,2,3,5,1] => [5,2,3,1,4] => 4 = 3 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 3 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [5,2,1,3,4] => [3,2,4,5,1] => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [5,2,4,3,1] => [5,2,4,3,1] => 3 = 2 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,4,3] => [1,2,5,4,3] => 4 = 3 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [3,2,1,4,5] => [3,2,1,4,5] => 4 = 3 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [3,2,5,4,1] => [5,2,1,4,3] => 4 = 3 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,4,5,3] => [1,2,5,3,4] => 4 = 3 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [3,2,1,5,4] => [3,2,1,5,4] => 3 = 2 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [3,2,4,1,5] => [4,2,1,3,5] => 4 = 3 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [3,2,4,5,1] => [5,2,1,3,4] => 4 = 3 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 3 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,1,2,4,3] => [2,3,5,4,1] => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [5,3,2,4,1] => [5,3,2,4,1] => 3 = 2 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [4,1,2,3,5] => [2,3,4,1,5] => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,5,2,3,1] => [5,3,4,1,2] => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,1,2,5,3] => [2,3,5,1,4] => 2 = 1 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [4,3,2,1,5] => [4,3,2,1,5] => 3 = 2 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [4,3,2,5,1] => [5,3,2,1,4] => 3 = 2 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,4,3,2,5] => [1,4,3,2,5] => 4 = 3 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,1,3,2,4] => [2,4,3,5,1] => 2 = 1 + 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [5,4,3,2,1] => [5,4,3,2,1] => 3 = 2 + 1
{{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => [7,2,3,4,5,6,1] => [7,2,3,4,5,6,1] => ? = 5 + 1
{{1,2,3,4,5},{6,7}}
 => [2,3,4,5,1,7,6] => [6,2,3,4,5,1,7] => [6,2,3,4,5,1,7] => ? = 5 + 1
{{1,2,3,4,5},{6},{7}}
 => [2,3,4,5,1,6,7] => [6,2,3,4,5,7,1] => [7,2,3,4,5,1,6] => ? = 5 + 1
{{1,2,3,4},{5,6,7}}
 => [2,3,4,1,6,7,5] => [5,2,3,4,1,6,7] => [5,2,3,4,1,6,7] => ? = 5 + 1
{{1,2,3,4},{5,6},{7}}
 => [2,3,4,1,6,5,7] => [5,2,3,4,7,6,1] => [7,2,3,4,1,6,5] => ? = 5 + 1
{{1,2,3,4},{5,7},{6}}
 => [2,3,4,1,7,6,5] => [5,2,3,4,1,7,6] => [5,2,3,4,1,7,6] => ? = 4 + 1
{{1,2,3,4},{5},{6},{7}}
 => [2,3,4,1,5,6,7] => [5,2,3,4,6,7,1] => [7,2,3,4,1,5,6] => ? = 5 + 1
{{1,2,3},{4,5,6,7}}
 => [2,3,1,5,6,7,4] => [4,2,3,1,5,6,7] => [4,2,3,1,5,6,7] => ? = 5 + 1
{{1,2,3},{4,5,6},{7}}
 => [2,3,1,5,6,4,7] => [4,2,3,7,5,6,1] => [7,2,3,1,5,6,4] => ? = 5 + 1
{{1,2,3},{4,5,7},{6}}
 => [2,3,1,5,7,6,4] => [4,2,3,1,5,7,6] => [4,2,3,1,5,7,6] => ? = 4 + 1
{{1,2},{3,4,5,6,7}}
 => [2,1,4,5,6,7,3] => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 5 + 1
{{1,2},{3},{4,5,6,7}}
 => [2,1,3,5,6,7,4] => [3,2,4,1,5,6,7] => [4,2,1,3,5,6,7] => ? = 5 + 1
{{1,2},{3},{4,5},{6},{7}}
 => [2,1,3,5,4,6,7] => [3,2,4,6,5,7,1] => [7,2,1,3,5,4,6] => ? = 5 + 1
{{1,2},{3},{4},{5,6,7}}
 => [2,1,3,4,6,7,5] => [3,2,4,5,1,6,7] => [5,2,1,3,4,6,7] => ? = 5 + 1
{{1,2},{3},{4},{5,6},{7}}
 => [2,1,3,4,6,5,7] => [3,2,4,5,7,6,1] => [7,2,1,3,4,6,5] => ? = 5 + 1
{{1,2},{3},{4},{5},{6,7}}
 => [2,1,3,4,5,7,6] => [3,2,4,5,6,1,7] => [6,2,1,3,4,5,7] => ? = 5 + 1
{{1,2},{3},{4},{5},{6},{7}}
 => [2,1,3,4,5,6,7] => [3,2,4,5,6,7,1] => [7,2,1,3,4,5,6] => ? = 5 + 1
{{1,3,4,5,6},{2},{7}}
 => [3,2,4,5,6,1,7] => [7,3,2,4,5,6,1] => [7,3,2,4,5,6,1] => ? = 4 + 1
{{1,3,4,5},{2,6,7}}
 => [3,6,4,5,1,7,2] => [6,1,2,4,5,3,7] => [2,3,6,4,5,1,7] => ? = 3 + 1
{{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 1 + 1
{{1,3,5},{2,4},{6,7}}
 => [3,4,5,2,1,7,6] => [6,5,2,3,4,1,7] => [6,3,4,5,2,1,7] => ? = 2 + 1
{{1,3},{2,4,5,6,7}}
 => [3,4,1,5,6,7,2] => [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => ? = 3 + 1
{{1,3,5},{2,6,7},{4}}
 => [3,6,5,4,1,7,2] => [6,1,2,5,4,3,7] => [2,3,6,5,4,1,7] => ? = 2 + 1
{{1,3},{2,7},{4},{5},{6}}
 => [3,7,1,4,5,6,2] => [4,1,2,5,6,7,3] => [2,3,7,1,4,5,6] => ? = 3 + 1
{{1,4,5,6},{2,3},{7}}
 => [4,3,2,5,6,1,7] => [7,4,3,2,5,6,1] => [7,4,3,2,5,6,1] => ? = 4 + 1
{{1,4,5},{2,3,6,7}}
 => [4,3,6,5,1,7,2] => [6,1,3,2,5,4,7] => [2,4,3,6,5,1,7] => ? = 3 + 1
{{1,5,6,7},{2,3,4}}
 => [5,3,4,2,6,7,1] => [1,5,3,4,2,6,7] => [1,5,3,4,2,6,7] => ? = 5 + 1
{{1,5,6},{2,3,4},{7}}
 => [5,3,4,2,6,1,7] => [7,5,3,4,2,6,1] => [7,5,3,4,2,6,1] => ? = 4 + 1
{{1,5,7},{2,3,4,6}}
 => [5,3,4,6,7,2,1] => [1,7,3,4,2,5,6] => [1,5,3,4,6,7,2] => ? = 3 + 1
{{1,5,7},{2,3,4},{6}}
 => [5,3,4,2,7,6,1] => [1,5,3,4,2,7,6] => [1,5,3,4,2,7,6] => ? = 4 + 1
{{1,6,7},{2,3,4,5}}
 => [6,3,4,5,2,7,1] => [1,6,3,4,5,2,7] => [1,6,3,4,5,2,7] => ? = 5 + 1
{{1,6},{2,3,4,5},{7}}
 => [6,3,4,5,2,1,7] => [7,6,3,4,5,2,1] => [7,6,3,4,5,2,1] => ? = 4 + 1
{{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => [1,7,3,4,5,6,2] => [1,7,3,4,5,6,2] => ? = 5 + 1
{{1},{2,3,4,5,6,7}}
 => [1,3,4,5,6,7,2] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 5 + 1
{{1},{2,3,4,5,6},{7}}
 => [1,3,4,5,6,2,7] => [2,7,3,4,5,6,1] => [7,1,3,4,5,6,2] => ? = 5 + 1
{{1,7},{2,3,4,5},{6}}
 => [7,3,4,5,2,6,1] => [1,6,3,4,5,7,2] => [1,7,3,4,5,2,6] => ? = 5 + 1
{{1,6,7},{2,3,4},{5}}
 => [6,3,4,2,5,7,1] => [1,5,3,4,6,2,7] => [1,6,3,4,2,5,7] => ? = 5 + 1
{{1,7},{2,3,4,6},{5}}
 => [7,3,4,6,5,2,1] => [1,7,3,4,6,5,2] => [1,7,3,4,6,5,2] => ? = 4 + 1
{{1,7},{2,3,4},{5,6}}
 => [7,3,4,2,6,5,1] => [1,5,3,4,7,6,2] => [1,7,3,4,2,6,5] => ? = 5 + 1
{{1,7},{2,3,4},{5},{6}}
 => [7,3,4,2,5,6,1] => [1,5,3,4,6,7,2] => [1,7,3,4,2,5,6] => ? = 5 + 1
{{1},{2,3,4,7},{5},{6}}
 => [1,3,4,7,5,6,2] => [2,1,3,4,6,7,5] => [2,1,3,4,7,5,6] => ? = 4 + 1
{{1,5,6,7},{2,3},{4}}
 => [5,3,2,4,6,7,1] => [1,4,3,5,2,6,7] => [1,5,3,2,4,6,7] => ? = 5 + 1
{{1,5,7},{2,3,6},{4}}
 => [5,3,6,4,7,2,1] => [1,7,3,5,2,4,6] => [1,5,3,6,4,7,2] => ? = 3 + 1
{{1,5,7},{2,3},{4,6}}
 => [5,3,2,6,7,4,1] => [1,4,3,7,2,5,6] => [1,5,3,2,6,7,4] => ? = 3 + 1
{{1,5,7},{2,3},{4},{6}}
 => [5,3,2,4,7,6,1] => [1,4,3,5,2,7,6] => [1,5,3,2,4,7,6] => ? = 4 + 1
{{1,6,7},{2,3,5},{4}}
 => [6,3,5,4,2,7,1] => [1,6,3,5,4,2,7] => [1,6,3,5,4,2,7] => ? = 4 + 1
{{1,7},{2,3,5,6},{4}}
 => [7,3,5,4,6,2,1] => [1,7,3,5,4,6,2] => [1,7,3,5,4,6,2] => ? = 4 + 1
{{1,7},{2,3,5},{4,6}}
 => [7,3,5,6,2,4,1] => [1,6,3,7,4,5,2] => [1,7,3,5,6,2,4] => ? = 3 + 1
{{1,7},{2,3,5},{4},{6}}
 => [7,3,5,4,2,6,1] => [1,6,3,5,4,7,2] => [1,7,3,5,4,2,6] => ? = 4 + 1
Description
The number of weak deficiencies of a permutation. This is defined as wdec(σ)=#{i:σ(i)≤i}. The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
Matching statistic: St000213
(load all 8 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 60%
Values
{{1}}
 => [1] => [1] => 1 = 0 + 1
{{1,2}}
 => [2,1] => [1,2] => 2 = 1 + 1
{{1},{2}}
 => [1,2] => [2,1] => 1 = 0 + 1
{{1,2,3}}
 => [2,3,1] => [1,2,3] => 3 = 2 + 1
{{1,2},{3}}
 => [2,1,3] => [1,3,2] => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => [2,1,3] => 2 = 1 + 1
{{1},{2,3}}
 => [1,3,2] => [3,2,1] => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => 2 = 1 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => 4 = 3 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,4,3] => 3 = 2 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,3,2,4] => 3 = 2 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,3,2] => 3 = 2 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,3,4,2] => 3 = 2 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,3,4] => 3 = 2 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,1,4,3] => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,1,4] => 3 = 2 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,3,1] => 3 = 2 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,2,4,1] => 3 = 2 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,1,4] => 3 = 2 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,2,1] => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,4,3,1] => 3 = 2 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => 3 = 2 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => 5 = 4 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,5,4] => 4 = 3 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,4,3,5] => 4 = 3 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,5,4,3] => 4 = 3 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,4,5,3] => 4 = 3 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,3,2,4,5] => 4 = 3 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,5,2,3,4] => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,3,2,5,4] => 3 = 2 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,4,3,2,5] => 4 = 3 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,5,3,4,2] => 4 = 3 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,4,3,5,2] => 4 = 3 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,3,4,2,5] => 4 = 3 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,5,4,3,2] => 3 = 2 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,3,5,4,2] => 4 = 3 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,3,4,5,2] => 4 = 3 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,1,3,4,5] => 4 = 3 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,1,3,2,4] => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,1,3,5,4] => 3 = 2 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,1,2,3,5] => 2 = 1 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [5,1,2,4,3] => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,1,2,5,3] => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,1,4,3,5] => 3 = 2 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,1,4,2,3] => 2 = 1 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,1,5,4,3] => 3 = 2 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,1,4,5,3] => 3 = 2 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,1,4,5] => 4 = 3 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,2,1,3,4] => 2 = 1 + 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 6 + 1
{{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => [1,2,3,4,5,7,6] => ? = 5 + 1
{{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => [1,2,3,4,6,5,7] => ? = 5 + 1
{{1,2,3,4,5},{6,7}}
 => [2,3,4,5,1,7,6] => [1,2,3,4,7,6,5] => ? = 5 + 1
{{1,2,3,4,5},{6},{7}}
 => [2,3,4,5,1,6,7] => [1,2,3,4,6,7,5] => ? = 5 + 1
{{1,2,3,4,6,7},{5}}
 => [2,3,4,6,5,7,1] => [1,2,3,5,4,6,7] => ? = 5 + 1
{{1,2,3,4,7},{5,6}}
 => [2,3,4,7,6,5,1] => [1,2,3,6,5,4,7] => ? = 5 + 1
{{1,2,3,4},{5,6,7}}
 => [2,3,4,1,6,7,5] => [1,2,3,7,5,6,4] => ? = 5 + 1
{{1,2,3,4},{5,6},{7}}
 => [2,3,4,1,6,5,7] => [1,2,3,6,5,7,4] => ? = 5 + 1
{{1,2,3,4,7},{5},{6}}
 => [2,3,4,7,5,6,1] => [1,2,3,5,6,4,7] => ? = 5 + 1
{{1,2,3,4},{5,7},{6}}
 => [2,3,4,1,7,6,5] => [1,2,3,7,6,5,4] => ? = 4 + 1
{{1,2,3,4},{5},{6},{7}}
 => [2,3,4,1,5,6,7] => [1,2,3,5,6,7,4] => ? = 5 + 1
{{1,2,3,5,6,7},{4}}
 => [2,3,5,4,6,7,1] => [1,2,4,3,5,6,7] => ? = 5 + 1
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => [1,2,6,3,4,5,7] => ? = 3 + 1
{{1,2,3,5,7},{4},{6}}
 => [2,3,5,4,7,6,1] => [1,2,4,3,6,5,7] => ? = 4 + 1
{{1,2,3,6,7},{4,5}}
 => [2,3,6,5,4,7,1] => [1,2,5,4,3,6,7] => ? = 5 + 1
{{1,2,3,7},{4,5,6}}
 => [2,3,7,5,6,4,1] => [1,2,6,4,5,3,7] => ? = 5 + 1
{{1,2,3},{4,5,6,7}}
 => [2,3,1,5,6,7,4] => [1,2,7,4,5,6,3] => ? = 5 + 1
{{1,2,3},{4,5,6},{7}}
 => [2,3,1,5,6,4,7] => [1,2,6,4,5,7,3] => ? = 5 + 1
{{1,2,3,7},{4,5},{6}}
 => [2,3,7,5,4,6,1] => [1,2,5,4,6,3,7] => ? = 5 + 1
{{1,2,3},{4,5,7},{6}}
 => [2,3,1,5,7,6,4] => [1,2,7,4,6,5,3] => ? = 4 + 1
{{1,2,3,6,7},{4},{5}}
 => [2,3,6,4,5,7,1] => [1,2,4,5,3,6,7] => ? = 5 + 1
{{1,2,3,7},{4,6},{5}}
 => [2,3,7,6,5,4,1] => [1,2,6,5,4,3,7] => ? = 4 + 1
{{1,2,3,7},{4},{5,6}}
 => [2,3,7,4,6,5,1] => [1,2,4,6,5,3,7] => ? = 5 + 1
{{1,2,3,7},{4},{5},{6}}
 => [2,3,7,4,5,6,1] => [1,2,4,5,6,3,7] => ? = 5 + 1
{{1,2,4,5,6,7},{3}}
 => [2,4,3,5,6,7,1] => [1,3,2,4,5,6,7] => ? = 5 + 1
{{1,2,4,5,7},{3,6}}
 => [2,4,6,5,7,3,1] => [1,6,2,4,3,5,7] => ? = 3 + 1
{{1,2,4,5,7},{3},{6}}
 => [2,4,3,5,7,6,1] => [1,3,2,4,6,5,7] => ? = 4 + 1
{{1,2,4,6,7},{3,5}}
 => [2,4,5,6,3,7,1] => [1,5,2,3,4,6,7] => ? = 3 + 1
{{1,2,4,7},{3,5,6}}
 => [2,4,5,7,6,3,1] => [1,6,2,3,5,4,7] => ? = 3 + 1
{{1,2,4,7},{3,5},{6}}
 => [2,4,5,7,3,6,1] => [1,5,2,3,6,4,7] => ? = 3 + 1
{{1,2,4,6,7},{3},{5}}
 => [2,4,3,6,5,7,1] => [1,3,2,5,4,6,7] => ? = 4 + 1
{{1,2,4,7},{3,6},{5}}
 => [2,4,6,7,5,3,1] => [1,6,2,5,3,4,7] => ? = 3 + 1
{{1,2,4,7},{3},{5,6}}
 => [2,4,3,7,6,5,1] => [1,3,2,6,5,4,7] => ? = 4 + 1
{{1,2,4,7},{3},{5},{6}}
 => [2,4,3,7,5,6,1] => [1,3,2,5,6,4,7] => ? = 4 + 1
{{1,2,5,6,7},{3,4}}
 => [2,5,4,3,6,7,1] => [1,4,3,2,5,6,7] => ? = 5 + 1
{{1,2,5,7},{3,4,6}}
 => [2,5,4,6,7,3,1] => [1,6,3,2,4,5,7] => ? = 3 + 1
{{1,2,5,7},{3,4},{6}}
 => [2,5,4,3,7,6,1] => [1,4,3,2,6,5,7] => ? = 4 + 1
{{1,2,6,7},{3,4,5}}
 => [2,6,4,5,3,7,1] => [1,5,3,4,2,6,7] => ? = 5 + 1
{{1,2,7},{3,4,5,6}}
 => [2,7,4,5,6,3,1] => [1,6,3,4,5,2,7] => ? = 5 + 1
{{1,2},{3,4,5,6,7}}
 => [2,1,4,5,6,7,3] => [1,7,3,4,5,6,2] => ? = 5 + 1
{{1,2,7},{3,4,5},{6}}
 => [2,7,4,5,3,6,1] => [1,5,3,4,6,2,7] => ? = 5 + 1
{{1,2,6,7},{3,4},{5}}
 => [2,6,4,3,5,7,1] => [1,4,3,5,2,6,7] => ? = 5 + 1
{{1,2,7},{3,4,6},{5}}
 => [2,7,4,6,5,3,1] => [1,6,3,5,4,2,7] => ? = 4 + 1
{{1,2,7},{3,4},{5,6}}
 => [2,7,4,3,6,5,1] => [1,4,3,6,5,2,7] => ? = 5 + 1
{{1,2,7},{3,4},{5},{6}}
 => [2,7,4,3,5,6,1] => [1,4,3,5,6,2,7] => ? = 5 + 1
{{1,2,5,6,7},{3},{4}}
 => [2,5,3,4,6,7,1] => [1,3,4,2,5,6,7] => ? = 5 + 1
{{1,2,5,7},{3,6},{4}}
 => [2,5,6,4,7,3,1] => [1,6,4,2,3,5,7] => ? = 3 + 1
{{1,2,5,7},{3},{4,6}}
 => [2,5,3,6,7,4,1] => [1,3,6,2,4,5,7] => ? = 3 + 1
{{1,2,5,7},{3},{4},{6}}
 => [2,5,3,4,7,6,1] => [1,3,4,2,6,5,7] => ? = 4 + 1
Description
The number of weak exceedances (also weak excedences) of a permutation. This is defined as wex(σ)=#{i:σ(i)≥i}. The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of σ.
Matching statistic: St001863
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
{{1}}
 => [1] => [1] => [1] => 1 = 0 + 1
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 2 = 1 + 1
{{1},{2}}
 => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
{{1,2,3}}
 => [2,3,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
{{1,2},{3}}
 => [2,1,3] => [1,3,2] => [1,3,2] => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => [2,1,3] => [2,1,3] => 2 = 1 + 1
{{1},{2,3}}
 => [1,3,2] => [3,2,1] => [3,2,1] => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => [2,3,1] => 2 = 1 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 3 = 2 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 3 = 2 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,2,4,1] => [3,2,4,1] => 3 = 2 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,1,4] => [2,3,1,4] => 3 = 2 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,4,3,1] => [2,4,3,1] => 3 = 2 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 3 = 2 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 3 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 4 = 3 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => 4 = 3 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 3 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,5,2,3,4] => [1,5,2,3,4] => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 4 = 3 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 4 = 3 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,4,3,5,2] => [1,4,3,5,2] => 4 = 3 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,3,4,2,5] => [1,3,4,2,5] => 4 = 3 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 3 = 2 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,3,5,4,2] => [1,3,5,4,2] => 4 = 3 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => 4 = 3 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 3 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 2 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,1,4,5,3] => [2,1,4,5,3] => ? = 2 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 3 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 + 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 2 + 1
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 3 + 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 3 + 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [4,2,3,5,1] => [4,2,3,5,1] => ? = 3 + 1
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [3,2,4,1,5] => [3,2,4,1,5] => ? = 3 + 1
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 2 + 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [3,2,5,4,1] => [3,2,5,4,1] => ? = 3 + 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => ? = 3 + 1
{{1,4,5},{2},{3}}
 => [4,2,3,5,1] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 3 + 1
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 + 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1 + 1
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [2,3,1,5,4] => [2,3,1,5,4] => ? = 2 + 1
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2 + 1
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2 + 1
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 1 + 1
{{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => [4,3,2,5,1] => [4,3,2,5,1] => ? = 2 + 1
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [2,4,3,1,5] => [2,4,3,1,5] => ? = 3 + 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2 + 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [2,5,3,4,1] => [2,5,3,4,1] => ? = 3 + 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? = 3 + 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 3 + 1
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 2 + 1
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 2 + 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? = 3 + 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 3 + 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 5 + 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 4 + 1
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 4 + 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? = 4 + 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? = 4 + 1
{{1,2,3,5,6},{4}}
 => [2,3,5,4,6,1] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? = 4 + 1
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? = 2 + 1
{{1,2,3,5},{4},{6}}
 => [2,3,5,4,1,6] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? = 3 + 1
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? = 4 + 1
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 4 + 1
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => [1,2,5,4,6,3] => [1,2,5,4,6,3] => ? = 4 + 1
{{1,2,3,6},{4},{5}}
 => [2,3,6,4,5,1] => [1,2,4,5,3,6] => [1,2,4,5,3,6] => ? = 4 + 1
{{1,2,3},{4,6},{5}}
 => [2,3,1,6,5,4] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => ? = 3 + 1
Description
The number of weak excedances of a signed permutation. For a signed permutation π∈Hn, this is |{i∈[n]∣π(i)≥i}|.