Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000022
(load all 18 compositions to match this statistic)
Mapping
Mp00091: Set partitions —rotate increasing⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => [1] => [1] => 1
{{1,2}}
 => {{1,2}}
 => [2,1] => [1,2] => 2
{{1},{2}}
 => {{1},{2}}
 => [1,2] => [2,1] => 0
{{1,2,3}}
 => {{1,2,3}}
 => [2,3,1] => [1,2,3] => 3
{{1,2},{3}}
 => {{1},{2,3}}
 => [1,3,2] => [2,1,3] => 1
{{1,3},{2}}
 => {{1,2},{3}}
 => [2,1,3] => [3,2,1] => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [3,2,1] => [1,3,2] => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,2,3] => [2,3,1] => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => 4
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => [1,3,4,2] => [2,1,3,4] => 2
{{1,2,4},{3}}
 => {{1,2,3},{4}}
 => [2,3,1,4] => [4,2,3,1] => 2
{{1,2},{3,4}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => [1,4,3,2] => 2
{{1,2},{3},{4}}
 => {{1},{2,3},{4}}
 => [1,3,2,4] => [2,4,3,1] => 1
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => 2
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => [1,4,3,2] => [2,1,4,3] => 0
{{1,4},{2,3}}
 => {{1,2},{3,4}}
 => [2,1,4,3] => [3,2,1,4] => 2
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => [1,3,2,4] => 2
{{1},{2,3},{4}}
 => {{1},{2},{3,4}}
 => [1,2,4,3] => [2,3,1,4] => 1
{{1,4},{2},{3}}
 => {{1,2},{3},{4}}
 => [2,1,3,4] => [3,2,4,1] => 1
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => [3,2,1,4] => [4,3,2,1] => 0
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => [1,3,4,2] => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => 0
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => 5
{{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => [2,1,3,4,5] => 3
{{1,2,3,5},{4}}
 => {{1,2,3,4},{5}}
 => [2,3,4,1,5] => [5,2,3,4,1] => 3
{{1,2,3},{4,5}}
 => {{1,5},{2,3,4}}
 => [5,3,4,2,1] => [1,5,3,4,2] => 3
{{1,2,3},{4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [2,5,3,4,1] => 2
{{1,2,4,5},{3}}
 => {{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,3,5,4] => 3
{{1,2,4},{3,5}}
 => {{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,1,3,2,4] => 1
{{1,2,4},{3},{5}}
 => {{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [2,1,3,5,4] => 1
{{1,2,5},{3,4}}
 => {{1,2,3},{4,5}}
 => [2,3,1,5,4] => [4,2,3,1,5] => 3
{{1,2},{3,4,5}}
 => {{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,4,3,2,5] => 3
{{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [2,4,3,1,5] => 2
{{1,2,5},{3},{4}}
 => {{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [4,2,3,5,1] => 2
{{1,2},{3,5},{4}}
 => {{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [5,4,3,2,1] => 1
{{1,2},{3},{4,5}}
 => {{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [1,4,3,5,2] => 2
{{1,2},{3},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [2,4,3,5,1] => 1
{{1,3,4,5},{2}}
 => {{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,3,5] => 3
{{1,3,4},{2,5}}
 => {{1,3},{2,4,5}}
 => [3,4,1,5,2] => [4,1,2,3,5] => 1
{{1,3,4},{2},{5}}
 => {{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [2,1,4,3,5] => 1
{{1,3,5},{2,4}}
 => {{1,2,4},{3,5}}
 => [2,4,5,1,3] => [5,2,1,3,4] => 1
{{1,3},{2,4,5}}
 => {{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,5,2,3,4] => 1
{{1,3},{2,4},{5}}
 => {{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [2,5,1,3,4] => 0
{{1,3,5},{2},{4}}
 => {{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [5,2,4,3,1] => 1
{{1,3},{2,5},{4}}
 => {{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,5,2,3,1] => 0
{{1,3},{2},{4,5}}
 => {{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,5,4,3,2] => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => [2,5,4,3,1] => 0
{{1,4,5},{2,3}}
 => {{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,4,3] => 3
{{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,1,2,4,3] => 1
Description
The number of fixed points of a permutation.
Matching statistic: St000475
(load all 8 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1]
 => 1
{{1,2}}
 => [2,1] => [1,2] => [1,1]
 => 2
{{1},{2}}
 => [1,2] => [2,1] => [2]
 => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => [1,1,1]
 => 3
{{1,2},{3}}
 => [2,1,3] => [3,2,1] => [2,1]
 => 1
{{1,3},{2}}
 => [3,2,1] => [1,3,2] => [2,1]
 => 1
{{1},{2,3}}
 => [1,3,2] => [2,1,3] => [2,1]
 => 1
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => [3]
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
 => 4
{{1,2,3},{4}}
 => [2,3,1,4] => [4,2,3,1] => [2,1,1]
 => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => [2,1,1]
 => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [3,2,1,4] => [2,1,1]
 => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [3,2,4,1] => [3,1]
 => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,2,4] => [2,1,1]
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => [4]
 => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [4,3,2,1] => [2,2]
 => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,3,2] => [2,1,1]
 => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [2,1,3,4] => [2,1,1]
 => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,4,3,1] => [3,1]
 => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,3,4,2] => [3,1]
 => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [2,1,4,3] => [2,2]
 => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,3,1,4] => [3,1]
 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => [4]
 => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
 => 5
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [5,2,3,4,1] => [2,1,1,1]
 => 3
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,3,5,4] => [2,1,1,1]
 => 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [4,2,3,1,5] => [2,1,1,1]
 => 3
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [4,2,3,5,1] => [3,1,1]
 => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,3,5] => [2,1,1,1]
 => 3
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [5,2,1,3,4] => [4,1]
 => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [5,2,4,3,1] => [2,2,1]
 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,4,3] => [2,1,1,1]
 => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [3,2,1,4,5] => [2,1,1,1]
 => 3
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [3,2,5,4,1] => [3,1,1]
 => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,4,5,3] => [3,1,1]
 => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [3,2,1,5,4] => [2,2,1]
 => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [3,2,4,1,5] => [3,1,1]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [3,2,4,5,1] => [4,1]
 => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,3,2,4,5] => [2,1,1,1]
 => 3
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,1,2,4,3] => [4,1]
 => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [5,3,2,4,1] => [2,2,1]
 => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,5,2,3,4] => [4,1]
 => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [4,1,2,3,5] => [4,1]
 => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,5,2,3,1] => [5]
 => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,2,5,4] => [2,2,1]
 => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,1,2,5,3] => [5]
 => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [4,3,2,1,5] => [2,2,1]
 => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [4,3,2,5,1] => [3,2]
 => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,4,3,2,5] => [2,1,1,1]
 => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,1,3,2,4] => [4,1]
 => 1
{{1},{2},{3},{4},{5,7},{6},{8}}
 => ? => ? => ?
 => ? = 0
{{1},{2},{3},{4},{5,6,7},{8}}
 => ? => ? => ?
 => ? = 2
{{1,2},{3,8},{4,5,6,7}}
 => ? => ? => ?
 => ? = 4
{{1,7},{2},{3},{4,5,6},{8}}
 => [7,2,3,5,6,4,1,8] => [8,3,4,7,5,6,2,1] => ?
 => ? = 2
{{1,8},{2,3},{4,5},{6},{7}}
 => ? => ? => ?
 => ? = 3
{{1,2,7,8},{3},{4},{5},{6}}
 => ? => ? => ?
 => ? = 3
{{1,3,5,7},{2,8},{4},{6}}
 => [3,8,5,4,7,6,1,2] => [8,1,2,5,4,7,6,3] => ?
 => ? = 0
{{1,4,6},{2,5,8},{3,7}}
 => [4,5,7,6,8,1,3,2] => [7,1,8,2,3,5,4,6] => ?
 => ? = 0
{{1,3,7},{2,8},{4},{5,6}}
 => [3,8,7,4,6,5,1,2] => [8,1,2,5,7,6,4,3] => ?
 => ? = 1
{{1,4},{2,5},{3,6,7,8}}
 => [4,5,6,1,2,7,8,3] => [5,6,1,2,3,4,7,8] => ?
 => ? = 2
{{1,5,7},{2},{3,6,8},{4}}
 => [5,2,6,4,7,8,1,3] => ? => ?
 => ? = 0
{{1,8},{2,5,7},{3},{4,6}}
 => [8,5,3,6,7,4,2,1] => [1,8,4,7,3,5,6,2] => ?
 => ? = 1
{{1,6},{2,7},{3,4,5},{8}}
 => [6,7,4,5,3,1,2,8] => ? => ?
 => ? = 2
{{1,5},{2,4},{3,7},{6},{8}}
 => [5,4,7,2,1,6,3,8] => [6,5,8,3,2,7,4,1] => ?
 => ? = 0
{{1,5},{2,3,4,6},{7},{8}}
 => [5,3,4,6,1,2,7,8] => [6,7,3,4,2,5,8,1] => ?
 => ? = 2
{{1,4,8},{2,3,5},{6},{7}}
 => [4,3,5,8,2,6,7,1] => ? => ?
 => ? = 2
{{1},{2,4,5,7,8},{3,6}}
 => [1,4,6,5,7,3,8,2] => [2,1,7,3,5,4,6,8] => ?
 => ? = 2
{{1,4},{2,6,7,8},{3},{5}}
 => [4,6,3,1,5,7,8,2] => [5,1,4,2,6,3,7,8] => ?
 => ? = 2
{{1},{2,4,5,6},{3,7},{8}}
 => [1,4,7,5,6,2,3,8] => [2,7,8,3,5,6,4,1] => ?
 => ? = 2
{{1,4,5},{2,3,6,7,8}}
 => [4,3,6,5,1,7,8,2] => [6,1,3,2,5,4,7,8] => ?
 => ? = 4
{{1,3,6},{2,4,5,7,8}}
 => [3,4,6,5,7,1,8,2] => [7,1,2,3,5,4,6,8] => ?
 => ? = 2
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000445
Mapping
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => [1] => [1,0]
 => 1
{{1,2}}
 => {{1},{2}}
 => [1,1] => [1,0,1,0]
 => 2
{{1},{2}}
 => {{1,2}}
 => [2] => [1,1,0,0]
 => 0
{{1,2,3}}
 => {{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => 1
{{1},{2},{3}}
 => {{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 0
{{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
{{1,2,3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
{{1,2},{3,4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1,2},{3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
{{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
{{1,3},{2},{4}}
 => {{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
{{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
{{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1},{2,3},{4}}
 => {{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
{{1,4},{2},{3}}
 => {{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2,4},{3}}
 => {{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
{{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
{{1},{2},{3},{4}}
 => {{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
{{1,2,3,4,5}}
 => {{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
{{1,2,3,4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
{{1,2,3,5},{4}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
{{1,2,3},{4,5}}
 => {{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
{{1,2,4,5},{3}}
 => {{1},{2},{3,4},{5}}
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
{{1,2,4},{3,5}}
 => {{1,3},{2,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,2,5},{3,4}}
 => {{1},{2,4},{3},{5}}
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
{{1,2},{3,4,5}}
 => {{1,4},{2},{3},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
{{1,2,5},{3},{4}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
{{1,2},{3,5},{4}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,2},{3},{4,5}}
 => {{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
{{1,2},{3},{4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
{{1,3,4,5},{2}}
 => {{1},{2},{3},{4,5}}
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 3
{{1,3,4},{2,5}}
 => {{1,4},{2,5},{3}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,3,4},{2},{5}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
{{1,3,5},{2,4}}
 => {{1},{2,4},{3,5}}
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3},{2,4,5}}
 => {{1,4},{2},{3,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
{{1,3},{2,4},{5}}
 => {{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
{{1,3,5},{2},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3},{2,5},{4}}
 => {{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
{{1,3},{2},{4,5}}
 => {{1,3},{2},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
{{1,3},{2},{4},{5}}
 => {{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
{{1,4,5},{2,3}}
 => {{1},{2},{3,5},{4}}
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
{{1,4},{2,3,5}}
 => {{1,3},{2,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,4},{2,3},{5,6},{7,8}}
 => {{1,3,5},{2},{4},{6,8},{7}}
 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
{{1,2},{3,6},{4,5},{7,8}}
 => {{1,3,7},{2},{4,6},{5},{8}}
 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
{{1,2},{3,4},{5,8},{6,7}}
 => {{1,5,7},{2,4},{3},{6},{8}}
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
{{1},{2,4,6,7,8},{3},{5}}
 => {{1,8},{2},{3},{4,5},{6,7}}
 => ? => ?
 => ? = 2
{{1,2},{3,8},{4,5,6,7}}
 => {{1,7},{2,6},{3},{4},{5},{8}}
 => ? => ?
 => ? = 4
{{1,2,4,6,7,8},{3},{5}}
 => {{1},{2},{3},{4,5},{6,7},{8}}
 => ? => ?
 => ? = 4
{{1,2,4,5,6,7,8},{3}}
 => {{1},{2},{3},{4},{5},{6,7},{8}}
 => ? => ?
 => ? = 6
{{1,2,3,4,6,7,8},{5}}
 => {{1},{2},{3},{4,5},{6},{7},{8}}
 => ? => ?
 => ? = 6
{{1,2,8},{3,7},{4,5,6}}
 => {{1},{2,7},{3,6},{4},{5},{8}}
 => ? => ?
 => ? = 4
{{1,2,3,4,5,7,8},{6}}
 => {{1},{2},{3,4},{5},{6},{7},{8}}
 => ? => ?
 => ? = 6
{{1,2,4,6,7,8},{3,5}}
 => {{1},{2},{3},{4,6},{5,7},{8}}
 => ? => ?
 => ? = 4
{{1,2,4,5,7,8},{3,6}}
 => {{1},{2},{3,6},{4,7},{5},{8}}
 => ? => ?
 => ? = 4
{{1,3,5,7},{2,4,8},{6}}
 => {{1,5,7},{2,6,8},{3,4}}
 => ? => ?
 => ? = 0
{{1,4,6},{2,5,8},{3,7}}
 => {{1,4,7},{2,6},{3,5,8}}
 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0
{{1,2,3,4,5,6,7,8,9,10},{11}}
 => {{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
 => [2,1,1,1,1,1,1,1,1,1] => ?
 => ? = 9
{{1,3,5,7,9},{2,4,6,8,10}}
 => {{1,3,5,7,9},{2,4,6,8,10}}
 => ? => ?
 => ? = 0
{{1,4},{2,5},{3},{6,8},{7}}
 => {{1,4,6,7},{2,3},{5,8}}
 => ? => ?
 => ? = 0
{{1,5},{2,4},{3,7},{6},{8}}
 => {{1,2,6},{3,4,8},{5,7}}
 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0
{{1},{2,4,5,7,8},{3,6}}
 => {{1,8},{2},{3,6},{4,7},{5}}
 => ? => ?
 => ? = 2
{{1},{2,4,6,8},{3,5,7}}
 => {{1,8},{2,4,6},{3,5,7}}
 => ? => ?
 => ? = 0
{{1,4},{2,6,7,8},{3},{5}}
 => {{1,6,7},{2},{3},{4,5,8}}
 => ? => ?
 => ? = 2
{{1,3,5},{2,4,6,7,8}}
 => {{1,5,7},{2},{3},{4,6,8}}
 => ? => ?
 => ? = 2
{{1},{2,5},{3},{4,6,8},{7}}
 => {{1,5,8},{2,3},{4,6,7}}
 => ? => ?
 => ? = 0
{{1,4,5},{2,3,6,7,8}}
 => {{1,6},{2},{3},{4,8},{5},{7}}
 => ? => ?
 => ? = 4
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000674
Mapping
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 86%●distinct values known / distinct values provided: 82%
Values
{{1}}
 => {{1}}
 => [1] => [1,0]
 => ? = 1
{{1,2}}
 => {{1},{2}}
 => [1,1] => [1,0,1,0]
 => 2
{{1},{2}}
 => {{1,2}}
 => [2] => [1,1,0,0]
 => 0
{{1,2,3}}
 => {{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 3
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => 1
{{1},{2},{3}}
 => {{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 0
{{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
{{1,2,3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
{{1,2},{3,4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1,2},{3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
{{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
{{1,3},{2},{4}}
 => {{1,2},{3,4}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
{{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
{{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1},{2,3},{4}}
 => {{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
{{1,4},{2},{3}}
 => {{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2,4},{3}}
 => {{1,4},{2,3}}
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 0
{{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
{{1},{2},{3},{4}}
 => {{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
{{1,2,3,4,5}}
 => {{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
{{1,2,3,4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
{{1,2,3,5},{4}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
{{1,2,3},{4,5}}
 => {{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
{{1,2,4,5},{3}}
 => {{1},{2},{3,4},{5}}
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
{{1,2,4},{3,5}}
 => {{1,3},{2,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,2,5},{3,4}}
 => {{1},{2,4},{3},{5}}
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
{{1,2},{3,4,5}}
 => {{1,4},{2},{3},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
{{1,2,5},{3},{4}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
{{1,2},{3,5},{4}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,2},{3},{4,5}}
 => {{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
{{1,2},{3},{4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
{{1,3,4,5},{2}}
 => {{1},{2},{3},{4,5}}
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 3
{{1,3,4},{2,5}}
 => {{1,4},{2,5},{3}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,3,4},{2},{5}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
{{1,3,5},{2,4}}
 => {{1},{2,4},{3,5}}
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3},{2,4,5}}
 => {{1,4},{2},{3,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
{{1,3},{2,4},{5}}
 => {{1,2,4},{3,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
{{1,3,5},{2},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
{{1,3},{2,5},{4}}
 => {{1,4},{2,3,5}}
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 0
{{1,3},{2},{4,5}}
 => {{1,3},{2},{4,5}}
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1
{{1,3},{2},{4},{5}}
 => {{1,2,3},{4,5}}
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 0
{{1,4,5},{2,3}}
 => {{1},{2},{3,5},{4}}
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
{{1,4},{2,3,5}}
 => {{1,3},{2,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,4},{2,3},{5}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
{{1,2},{3,4},{5,6},{7,8}}
 => {{1,3,5,7},{2},{4},{6},{8}}
 => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
{{1,3},{2,4},{5,6},{7,8}}
 => {{1,3,5,7},{2},{4},{6,8}}
 => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 2
{{1,4},{2,3},{5,6},{7,8}}
 => {{1,3,5},{2},{4},{6,8},{7}}
 => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
{{1,5},{2,3},{4,6},{7,8}}
 => {{1,3,5},{2},{4,6,8},{7}}
 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
{{1,5},{2,4},{3,6},{7,8}}
 => {{1,3,6},{2},{4,8},{5,7}}
 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
{{1,4},{2,5},{3,6},{7,8}}
 => {{1,3,6},{2},{4,7},{5,8}}
 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
{{1,3},{2,5},{4,6},{7,8}}
 => {{1,3,5},{2},{4,7},{6,8}}
 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
{{1,2},{3,5},{4,6},{7,8}}
 => {{1,3,5,7},{2},{4,6},{8}}
 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
{{1,2},{3,6},{4,5},{7,8}}
 => {{1,3,7},{2},{4,6},{5},{8}}
 => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
{{1,3},{2,6},{4,5},{7,8}}
 => {{1,3,7},{2},{4,6,8},{5}}
 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
{{1,4},{2,6},{3,5},{7,8}}
 => {{1,3,7},{2},{4,6},{5,8}}
 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 1
{{1,5},{2,6},{3,4},{7,8}}
 => {{1,3,5,7},{2},{4,8},{6}}
 => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
{{1,2},{3,7},{4,5},{6,8}}
 => {{1,3,7},{2,4,6},{5},{8}}
 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2
{{1,5},{2,8},{3,4},{6,7}}
 => {{1,5,7},{2,4,8},{3},{6}}
 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2
{{1,6},{2,8},{3,4},{5,7}}
 => {{1,5,7},{2,4},{3,8},{6}}
 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
{{1,7},{2,8},{3,4},{5,6}}
 => {{1,3,5,7},{2,8},{4},{6}}
 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 2
{{1,2},{3,7},{4,6},{5,8}}
 => {{1,4,7},{2,6},{3,5},{8}}
 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
{{1,2},{3,6},{4,7},{5,8}}
 => {{1,4,7},{2,5},{3,6},{8}}
 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
{{1,2},{3,5},{4,7},{6,8}}
 => {{1,3,7},{2,5},{4,6},{8}}
 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
{{1,4},{2,3},{5,7},{6,8}}
 => {{1,3,5},{2,4},{6,8},{7}}
 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
{{1,2},{3,4},{5,7},{6,8}}
 => {{1,3,5,7},{2,4},{6},{8}}
 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 2
{{1,2},{3,4},{5,8},{6,7}}
 => {{1,5,7},{2,4},{3},{6},{8}}
 => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
{{1,3},{2,4},{5,8},{6,7}}
 => {{1,5,7},{2,4},{3},{6,8}}
 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
{{1,7},{2,3},{4,8},{5,6}}
 => {{1,3,5},{2,6,8},{4},{7}}
 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2
{{1,2},{3,5},{4,8},{6,7}}
 => {{1,5,7},{2,4,6},{3},{8}}
 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2
{{1,2},{3,6},{4,8},{5,7}}
 => {{1,5,7},{2,4},{3,6},{8}}
 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
{{1,7},{2,6},{3,8},{4,5}}
 => {{1,4,6},{2,8},{3,7},{5}}
 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
{{1,6},{2,7},{3,8},{4,5}}
 => {{1,4,6},{2,7},{3,8},{5}}
 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
{{1,3},{2,7},{4,8},{5,6}}
 => {{1,3,5},{2,7},{4},{6,8}}
 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
{{1,2},{3,7},{4,8},{5,6}}
 => {{1,3,5,7},{2,6},{4},{8}}
 => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 2
{{1,7},{2,8},{3,6},{4,5}}
 => {{1,3,7},{2,8},{4,6},{5}}
 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
{{1,6},{2,7},{3,8},{4,9},{5,10}}
 => {{1,6},{2,7},{3,8},{4,9},{5,10}}
 => [2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 0
{{1},{2},{3},{4},{5},{6},{7},{8}}
 => {{1,2,3,4,5,6,7,8}}
 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
{{1},{2},{3},{4},{5,7},{6},{8}}
 => {{1,2,5,6,7,8},{3,4}}
 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 0
{{1},{2},{3},{4},{5,6,7},{8}}
 => {{1,2,5,6,7,8},{3},{4}}
 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 2
{{1},{2,4},{3},{5,8},{6,7}}
 => {{1,5,8},{2,4},{3},{6,7}}
 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 1
{{1},{2,4,6,7,8},{3},{5}}
 => {{1,8},{2},{3},{4,5},{6,7}}
 => ? => ?
 => ? = 2
{{1,2},{3,8},{4,5,6,7}}
 => {{1,7},{2,6},{3},{4},{5},{8}}
 => ? => ?
 => ? = 4
{{1,2,4,6,7,8},{3},{5}}
 => {{1},{2},{3},{4,5},{6,7},{8}}
 => ? => ?
 => ? = 4
{{1,2,4,5,6,7,8},{3}}
 => {{1},{2},{3},{4},{5},{6,7},{8}}
 => ? => ?
 => ? = 6
{{1,2,3,4,6,7,8},{5}}
 => {{1},{2},{3},{4,5},{6},{7},{8}}
 => ? => ?
 => ? = 6
{{1,2,8},{3,7},{4,5,6}}
 => {{1},{2,7},{3,6},{4},{5},{8}}
 => ? => ?
 => ? = 4
{{1,2,3,4,5,7,8},{6}}
 => {{1},{2},{3,4},{5},{6},{7},{8}}
 => ? => ?
 => ? = 6
{{1,2,4,6,7,8},{3,5}}
 => {{1},{2},{3},{4,6},{5,7},{8}}
 => ? => ?
 => ? = 4
{{1,2,4,5,7,8},{3,6}}
 => {{1},{2},{3,6},{4,7},{5},{8}}
 => ? => ?
 => ? = 4
{{1,3,5,7},{2,4,6,8}}
 => {{1,3,5,7},{2,4,6,8}}
 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 0
{{1,3,5,7},{2,4,8},{6}}
 => {{1,5,7},{2,6,8},{3,4}}
 => ? => ?
 => ? = 0
{{1,4,6},{2,5,8},{3,7}}
 => {{1,4,7},{2,6},{3,5,8}}
 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0
{{1,3,4},{2,5,7},{6},{8}}
 => {{1,2,7},{3,4},{5,8},{6}}
 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
Description
The number of hills of a Dyck path. A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000215
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 1
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 2
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [3,2,1] => 3
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [2,3,1] => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 4
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 5
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => [4,3,2,1,5] => 3
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [5,1,2,4,3] => [4,5,3,2,1] => 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 3
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [5,1,3,2,4] => [3,5,4,2,1] => 3
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [4,1,5,2,3] => [4,2,1,5,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [4,1,3,2,5] => [3,4,2,1,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [5,1,4,3,2] => [4,3,5,2,1] => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 3
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [5,1,3,4,2] => [3,4,5,2,1] => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [5,2,1,3,4] => [2,5,4,3,1] => 3
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,5,1,3,2] => [4,3,1,5,2] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [4,2,1,3,5] => [2,4,3,1,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [5,4,1,2,3] => [4,2,5,3,1] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,5,1,2,4] => [3,1,5,4,2] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,4,1,2,5] => [3,1,4,2,5] => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [5,2,1,4,3] => [2,4,5,3,1] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,5,1,4,2] => [3,1,4,5,2] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,3,2,1,4] => [3,2,5,4,1] => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,5,2,1,3] => [4,1,5,3,2] => 1
{{1,2,3,4,6,7},{5}}
 => [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => [5,7,6,4,3,2,1] => ? = 5
{{1,2,3,4,7},{5},{6}}
 => [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => [5,6,7,4,3,2,1] => ? = 4
{{1,2,3,5,6,7},{4}}
 => [2,3,5,4,6,7,1] => [7,1,2,4,3,5,6] => [4,7,6,5,3,2,1] => ? = 5
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => [6,4,7,5,3,2,1] => ? = 3
{{1,2,3},{4,7},{5,6}}
 => [2,3,1,7,6,5,4] => [3,1,2,7,6,5,4] => [3,2,1,6,5,7,4] => ? = 3
{{1,2,3,7},{4},{5},{6}}
 => [2,3,7,4,5,6,1] => [7,1,2,4,5,6,3] => [4,5,6,7,3,2,1] => ? = 3
{{1,2,4,6},{3,5},{7}}
 => [2,4,5,6,3,1,7] => [6,1,5,2,3,4,7] => [5,3,6,4,2,1,7] => ? = 1
{{1,2,5},{3,4},{6,7}}
 => [2,5,4,3,1,7,6] => [5,1,4,3,2,7,6] => [4,3,5,2,1,7,6] => ? = 3
{{1,2},{3,4,7},{5,6}}
 => [2,1,4,7,6,5,3] => [2,1,7,3,6,5,4] => [2,1,6,5,7,4,3] => ? = 3
{{1,2,5},{3},{4},{6,7}}
 => [2,5,3,4,1,7,6] => [5,1,3,4,2,7,6] => [3,4,5,2,1,7,6] => ? = 2
{{1,2},{3,7},{4,5,6}}
 => [2,1,7,5,6,4,3] => [2,1,7,6,4,5,3] => [2,1,6,5,4,7,3] => ? = 3
{{1,2},{3,7},{4,6},{5}}
 => [2,1,7,6,5,4,3] => [2,1,7,6,5,4,3] => [2,1,5,6,4,7,3] => ? = 1
{{1,2},{3,7},{4},{5,6}}
 => [2,1,7,4,6,5,3] => [2,1,7,4,6,5,3] => [2,1,4,6,5,7,3] => ? = 2
{{1,2},{3},{4,7},{5,6}}
 => [2,1,3,7,6,5,4] => [2,1,3,7,6,5,4] => [2,1,3,6,5,7,4] => ? = 2
{{1,2,7},{3},{4},{5},{6}}
 => [2,7,3,4,5,6,1] => [7,1,3,4,5,6,2] => [3,4,5,6,7,2,1] => ? = 2
{{1,3,4,5,6},{2,7}}
 => [3,7,4,5,6,1,2] => [6,7,1,3,4,5,2] => [6,5,4,3,1,7,2] => ? = 3
{{1,3,4,6},{2,7},{5}}
 => [3,7,4,6,5,1,2] => [6,7,1,3,5,4,2] => [5,6,4,3,1,7,2] => ? = 1
{{1,3,5,6},{2,4,7}}
 => [3,4,5,7,6,1,2] => [6,7,1,2,3,5,4] => [6,5,3,1,7,4,2] => ? = 1
{{1,3,5,6},{2,4},{7}}
 => [3,4,5,2,6,1,7] => [6,4,1,2,3,5,7] => [4,2,6,5,3,1,7] => ? = 1
{{1,3,5,7},{2,4,6}}
 => [3,4,5,6,7,2,1] => [7,6,1,2,3,4,5] => [6,4,2,7,5,3,1] => ? = 1
{{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => [5,7,1,2,3,4,6] => [5,3,1,7,6,4,2] => ? = 1
{{1,3,5},{2,4,6},{7}}
 => [3,4,5,6,1,2,7] => [5,6,1,2,3,4,7] => [5,3,1,6,4,2,7] => ? = 0
{{1,3},{2,4,5,6},{7}}
 => [3,4,1,5,6,2,7] => [3,6,1,2,4,5,7] => [3,1,6,5,4,2,7] => ? = 2
{{1,3,6,7},{2,4},{5}}
 => [3,4,6,2,5,7,1] => [7,4,1,2,5,3,6] => [4,2,5,7,6,3,1] => ? = 2
{{1,3,6},{2,4,7},{5}}
 => [3,4,6,7,5,1,2] => [6,7,1,2,5,3,4] => [5,6,3,1,7,4,2] => ? = 0
{{1,3},{2,4,6,7},{5}}
 => [3,4,1,6,5,7,2] => [3,7,1,2,5,4,6] => [3,1,5,7,6,4,2] => ? = 1
{{1,3,5},{2},{4},{6,7}}
 => [3,2,5,4,1,7,6] => [5,2,1,4,3,7,6] => [2,4,5,3,1,7,6] => ? = 1
{{1,3},{2},{4,7},{5,6}}
 => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => [2,3,1,6,5,7,4] => ? = 1
{{1,3},{2,7},{4},{5},{6}}
 => [3,7,1,4,5,6,2] => [3,7,1,4,5,6,2] => [3,1,4,5,6,7,2] => ? = 0
{{1,4,5},{2,3},{6,7}}
 => [4,3,2,5,1,7,6] => [5,3,2,1,4,7,6] => [3,2,5,4,1,7,6] => ? = 3
{{1,4,7},{2,3},{5,6}}
 => [4,3,2,7,6,5,1] => [7,3,2,1,6,5,4] => [3,2,6,5,7,4,1] => ? = 3
{{1,4},{2,3},{5,6},{7}}
 => [4,3,2,1,6,5,7] => [4,3,2,1,6,5,7] => [3,2,4,1,6,5,7] => ? = 2
{{1,4},{2,3},{5,7},{6}}
 => [4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => [3,2,4,1,6,7,5] => ? = 1
{{1,4},{2,3},{5},{6,7}}
 => [4,3,2,1,5,7,6] => [4,3,2,1,5,7,6] => [3,2,4,1,5,7,6] => ? = 2
{{1,5},{2,3,4},{6,7}}
 => [5,3,4,2,1,7,6] => [5,4,2,3,1,7,6] => [4,3,2,5,1,7,6] => ? = 3
{{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => [7,6,2,3,4,5,1] => [6,5,4,3,2,7,1] => ? = 5
{{1},{2,3,4,5,6},{7}}
 => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 4
{{1,5},{2,3},{4},{6,7}}
 => [5,3,2,4,1,7,6] => [5,3,2,4,1,7,6] => [3,2,4,5,1,7,6] => ? = 2
{{1},{2,3,5,7},{4,6}}
 => [1,3,5,6,7,4,2] => [1,7,2,6,3,4,5] => [1,6,4,7,5,3,2] => ? = 1
{{1},{2,3},{4,7},{5,6}}
 => [1,3,2,7,6,5,4] => [1,3,2,7,6,5,4] => [1,3,2,6,5,7,4] => ? = 2
{{1},{2,3,7},{4},{5},{6}}
 => [1,3,7,4,5,6,2] => [1,7,2,4,5,6,3] => [1,4,5,6,7,3,2] => ? = 1
{{1,4,5},{2},{3},{6,7}}
 => [4,2,3,5,1,7,6] => [5,2,3,1,4,7,6] => [2,3,5,4,1,7,6] => ? = 2
{{1,4},{2,5,6},{3,7}}
 => [4,5,7,1,6,2,3] => [4,6,7,1,2,5,3] => [4,1,6,5,2,7,3] => ? = 1
{{1,4},{2,5},{3,6,7}}
 => [4,5,6,1,2,7,3] => [4,5,7,1,2,3,6] => [4,1,5,2,7,6,3] => ? = 1
{{1,4},{2,5},{3,6},{7}}
 => [4,5,6,1,2,3,7] => [4,5,6,1,2,3,7] => [4,1,5,2,6,3,7] => ? = 0
{{1,4},{2,6},{3,7},{5}}
 => [4,6,7,1,5,2,3] => [4,6,7,1,5,2,3] => [4,1,5,6,2,7,3] => ? = 0
{{1,4,7},{2},{3},{5,6}}
 => [4,2,3,7,6,5,1] => [7,2,3,1,6,5,4] => [2,3,6,5,7,4,1] => ? = 2
{{1,5},{2,4},{3},{6,7}}
 => [5,4,3,2,1,7,6] => [5,4,3,2,1,7,6] => [3,4,2,5,1,7,6] => ? = 1
{{1},{2,4,6},{3,5,7}}
 => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => [1,6,4,2,7,5,3] => ? = 0
{{1},{2,4,7},{3},{5,6}}
 => [1,4,3,7,6,5,2] => [1,7,3,2,6,5,4] => [1,3,6,5,7,4,2] => ? = 1
Description
The number of adjacencies of a permutation, zero appended. An adjacency is a descent of the form $(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic, $\operatorname{adj_0}$, counts adjacencies in the word with a zero appended. $(\operatorname{adj_0}, \operatorname{des})$ and $(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].
Matching statistic: St000247
(load all 57 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 73%
Values
{{1}}
 => [1] => [1] => {{1}}
 => ? = 1
{{1,2}}
 => [2,1] => [1,2] => {{1},{2}}
 => 2
{{1},{2}}
 => [1,2] => [2,1] => {{1,2}}
 => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => {{1},{2},{3}}
 => 3
{{1,2},{3}}
 => [2,1,3] => [3,2,1] => {{1,3},{2}}
 => 1
{{1,3},{2}}
 => [3,2,1] => [1,3,2] => {{1},{2,3}}
 => 1
{{1},{2,3}}
 => [1,3,2] => [2,1,3] => {{1,2},{3}}
 => 1
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => {{1,2,3}}
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
 => 4
{{1,2,3},{4}}
 => [2,3,1,4] => [4,2,3,1] => {{1,4},{2},{3}}
 => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
 => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [3,2,1,4] => {{1,3},{2},{4}}
 => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [3,2,4,1] => {{1,3,4},{2}}
 => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,2,4] => {{1},{2,3},{4}}
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => {{1,2,3,4}}
 => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [4,3,2,1] => {{1,4},{2,3}}
 => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,3,2] => {{1},{2,4},{3}}
 => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [2,1,3,4] => {{1,2},{3},{4}}
 => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,4,3,1] => {{1,2,4},{3}}
 => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,3,4,2] => {{1},{2,3,4}}
 => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [2,1,4,3] => {{1,2},{3,4}}
 => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,3,1,4] => {{1,2,3},{4}}
 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => {{1,2,3,4}}
 => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 5
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
 => 3
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
 => 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
 => 3
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [4,2,3,5,1] => {{1,4,5},{2},{3}}
 => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
 => 3
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [5,2,1,3,4] => {{1,3,4,5},{2}}
 => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
 => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
 => 3
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [3,2,5,4,1] => {{1,3,5},{2},{4}}
 => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
 => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
 => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [3,2,4,1,5] => {{1,3,4},{2},{5}}
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [3,2,4,5,1] => {{1,3,4,5},{2}}
 => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
 => 3
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,1,2,4,3] => {{1,2,3,5},{4}}
 => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
 => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,5,2,3,4] => {{1},{2,3,4,5}}
 => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [4,1,2,3,5] => {{1,2,3,4},{5}}
 => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,5,2,3,1] => {{1,2,3,4,5}}
 => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
 => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,1,2,5,3] => {{1,2,3,4,5}}
 => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
 => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [4,3,2,5,1] => {{1,4,5},{2,3}}
 => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
 => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,1,3,2,4] => {{1,2,4,5},{3}}
 => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
 => 1
{{1,2},{3,4},{5,6},{7,8}}
 => [2,1,4,3,6,5,8,7] => [3,2,5,4,7,6,1,8] => {{1,3,5,7},{2},{4},{6},{8}}
 => ? = 4
{{1,3},{2,4},{5,6},{7,8}}
 => [3,4,1,2,6,5,8,7] => [4,5,2,3,7,6,1,8] => {{1,2,3,4,5,7},{6},{8}}
 => ? = 2
{{1,4},{2,3},{5,6},{7,8}}
 => [4,3,2,1,6,5,8,7] => [5,4,3,2,7,6,1,8] => {{1,5,7},{2,4},{3},{6},{8}}
 => ? = 3
{{1,5},{2,3},{4,6},{7,8}}
 => [5,3,2,6,1,4,8,7] => [6,4,3,7,2,5,1,8] => {{1,2,4,5,6,7},{3},{8}}
 => ? = 2
{{1,6},{2,3},{4,5},{7,8}}
 => [6,3,2,5,4,1,8,7] => [7,4,3,6,5,2,1,8] => {{1,7},{2,4,6},{3},{5},{8}}
 => ? = 3
{{1,7},{2,3},{4,5},{6,8}}
 => [7,3,2,5,4,8,1,6] => [8,4,3,6,5,1,2,7] => {{1,2,4,6,7,8},{3},{5}}
 => ? = 2
{{1,8},{2,3},{4,5},{6,7}}
 => [8,3,2,5,4,7,6,1] => [1,4,3,6,5,8,7,2] => {{1},{2,4,6,8},{3},{5},{7}}
 => ? = 4
{{1,6},{2,4},{3,5},{7,8}}
 => [6,4,5,2,3,1,8,7] => [7,5,6,3,4,2,1,8] => {{1,7},{2,3,4,5,6},{8}}
 => ? = 1
{{1,5},{2,4},{3,6},{7,8}}
 => [5,4,6,2,1,3,8,7] => [6,5,7,3,2,4,1,8] => {{1,3,4,6,7},{2,5},{8}}
 => ? = 1
{{1,4},{2,5},{3,6},{7,8}}
 => [4,5,6,1,2,3,8,7] => [5,6,7,2,3,4,1,8] => {{1,3,5,7},{2,4,6},{8}}
 => ? = 1
{{1,3},{2,5},{4,6},{7,8}}
 => [3,5,1,6,2,4,8,7] => [4,6,2,7,3,5,1,8] => {{1,4,7},{2,3,5,6},{8}}
 => ? = 1
{{1,2},{3,5},{4,6},{7,8}}
 => [2,1,5,6,3,4,8,7] => [3,2,6,7,4,5,1,8] => {{1,3,4,5,6,7},{2},{8}}
 => ? = 2
{{1,2},{3,6},{4,5},{7,8}}
 => [2,1,6,5,4,3,8,7] => [3,2,7,6,5,4,1,8] => {{1,3,7},{2},{4,6},{5},{8}}
 => ? = 3
{{1,3},{2,6},{4,5},{7,8}}
 => [3,6,1,5,4,2,8,7] => [4,7,2,6,5,3,1,8] => {{1,2,3,4,6,7},{5},{8}}
 => ? = 2
{{1,4},{2,6},{3,5},{7,8}}
 => [4,6,5,1,3,2,8,7] => [5,7,6,2,4,3,1,8] => {{1,2,4,5,7},{3,6},{8}}
 => ? = 1
{{1,5},{2,6},{3,4},{7,8}}
 => [5,6,4,3,1,2,8,7] => [6,7,5,4,2,3,1,8] => {{1,2,3,5,6,7},{4},{8}}
 => ? = 2
{{1,6},{2,5},{3,4},{7,8}}
 => [6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,1,8] => {{1,7},{2,6},{3,5},{4},{8}}
 => ? = 2
{{1,7},{2,5},{3,4},{6,8}}
 => [7,5,4,3,2,8,1,6] => [8,6,5,4,3,1,2,7] => {{1,2,6,7,8},{3,5},{4}}
 => ? = 1
{{1,8},{2,5},{3,4},{6,7}}
 => [8,5,4,3,2,7,6,1] => [1,6,5,4,3,8,7,2] => {{1},{2,6,8},{3,5},{4},{7}}
 => ? = 3
{{1,7},{2,6},{3,4},{5,8}}
 => [7,6,4,3,8,2,1,5] => [8,7,5,4,1,3,2,6] => {{1,3,5,6,8},{2,7},{4}}
 => ? = 1
{{1,6},{2,7},{3,4},{5,8}}
 => [6,7,4,3,8,1,2,5] => [7,8,5,4,1,2,3,6] => {{1,3,5,7},{2,6,8},{4}}
 => ? = 1
{{1,5},{2,7},{3,4},{6,8}}
 => [5,7,4,3,1,8,2,6] => [6,8,5,4,2,1,3,7] => {{1,6},{2,3,5,7,8},{4}}
 => ? = 1
{{1,3},{2,7},{4,5},{6,8}}
 => [3,7,1,5,4,8,2,6] => [4,8,2,6,5,1,3,7] => {{1,4,6},{2,3,7,8},{5}}
 => ? = 1
{{1,2},{3,7},{4,5},{6,8}}
 => [2,1,7,5,4,8,3,6] => [3,2,8,6,5,1,4,7] => {{1,3,4,6,7,8},{2},{5}}
 => ? = 2
{{1,2},{3,8},{4,5},{6,7}}
 => [2,1,8,5,4,7,6,3] => [3,2,1,6,5,8,7,4] => {{1,3},{2},{4,6,8},{5},{7}}
 => ? = 3
{{1,3},{2,8},{4,5},{6,7}}
 => [3,8,1,5,4,7,6,2] => [4,1,2,6,5,8,7,3] => {{1,2,3,4,6,8},{5},{7}}
 => ? = 2
{{1,4},{2,8},{3,5},{6,7}}
 => [4,8,5,1,3,7,6,2] => [5,1,6,2,4,8,7,3] => {{1,2,4,5},{3,6,8},{7}}
 => ? = 1
{{1,5},{2,8},{3,4},{6,7}}
 => [5,8,4,3,1,7,6,2] => [6,1,5,4,2,8,7,3] => {{1,2,3,5,6,8},{4},{7}}
 => ? = 2
{{1,6},{2,8},{3,4},{5,7}}
 => [6,8,4,3,7,1,5,2] => [7,1,5,4,8,2,6,3] => {{1,2,6,7},{3,5,8},{4}}
 => ? = 1
{{1,7},{2,8},{3,4},{5,6}}
 => [7,8,4,3,6,5,1,2] => [8,1,5,4,7,6,2,3] => {{1,2,3,5,7,8},{4},{6}}
 => ? = 2
{{1,8},{2,7},{3,4},{5,6}}
 => [8,7,4,3,6,5,2,1] => [1,8,5,4,7,6,3,2] => {{1},{2,8},{3,5,7},{4},{6}}
 => ? = 3
{{1,8},{2,7},{3,5},{4,6}}
 => [8,7,5,6,3,4,2,1] => [1,8,6,7,4,5,3,2] => {{1},{2,8},{3,4,5,6,7}}
 => ? = 1
{{1,2},{3,8},{4,6},{5,7}}
 => [2,1,8,6,7,4,5,3] => [3,2,1,7,8,5,6,4] => {{1,3},{2},{4,5,6,7,8}}
 => ? = 1
{{1,2},{3,7},{4,6},{5,8}}
 => [2,1,7,6,8,4,3,5] => [3,2,8,7,1,5,4,6] => {{1,3,5,6,8},{2},{4,7}}
 => ? = 1
{{1,4},{2,7},{3,6},{5,8}}
 => [4,7,6,1,8,3,2,5] => [5,8,7,2,1,4,3,6] => {{1,5},{2,4,6,8},{3,7}}
 => ? = 0
{{1,5},{2,7},{3,6},{4,8}}
 => [5,7,6,8,1,3,2,4] => [6,8,7,1,2,4,3,5] => {{1,4,6},{2,5,8},{3,7}}
 => ? = 0
{{1,7},{2,6},{3,5},{4,8}}
 => [7,6,5,8,3,2,1,4] => [8,7,6,1,4,3,2,5] => {{1,4,5,8},{2,7},{3,6}}
 => ? = 0
{{1,8},{2,6},{3,5},{4,7}}
 => [8,6,5,7,3,2,4,1] => [1,7,6,8,4,3,5,2] => {{1},{2,4,5,7,8},{3,6}}
 => ? = 1
{{1,8},{2,5},{3,6},{4,7}}
 => [8,5,6,7,2,3,4,1] => [1,6,7,8,3,4,5,2] => {{1},{2,4,6,8},{3,5,7}}
 => ? = 1
{{1,6},{2,5},{3,7},{4,8}}
 => [6,5,7,8,2,1,3,4] => [7,6,8,1,3,2,4,5] => {{1,4,7},{2,6},{3,5,8}}
 => ? = 0
{{1,4},{2,6},{3,7},{5,8}}
 => [4,6,7,1,8,2,3,5] => [5,7,8,2,1,3,4,6] => {{1,5},{2,4,7},{3,6,8}}
 => ? = 0
{{1,2},{3,6},{4,7},{5,8}}
 => [2,1,6,7,8,3,4,5] => [3,2,7,8,1,4,5,6] => {{1,3,5,7},{2},{4,6,8}}
 => ? = 1
{{1,2},{3,5},{4,7},{6,8}}
 => [2,1,5,7,3,8,4,6] => [3,2,6,8,4,1,5,7] => {{1,3,6},{2},{4,5,7,8}}
 => ? = 1
{{1,5},{2,4},{3,7},{6,8}}
 => [5,4,7,2,1,8,3,6] => [6,5,8,3,2,1,4,7] => {{1,6},{2,5},{3,4,7,8}}
 => ? = 0
{{1,8},{2,4},{3,6},{5,7}}
 => [8,4,6,2,7,3,5,1] => [1,5,7,3,8,4,6,2] => {{1},{2,5,8},{3,4,6,7}}
 => ? = 1
{{1,7},{2,3},{4,6},{5,8}}
 => [7,3,2,6,8,4,1,5] => [8,4,3,7,1,5,2,6] => {{1,5,6,8},{2,4,7},{3}}
 => ? = 1
{{1,6},{2,3},{4,7},{5,8}}
 => [6,3,2,7,8,1,4,5] => [7,4,3,8,1,2,5,6] => {{1,5,7},{2,4,6,8},{3}}
 => ? = 1
{{1,5},{2,3},{4,7},{6,8}}
 => [5,3,2,7,1,8,4,6] => [6,4,3,8,2,1,5,7] => {{1,6},{2,4,5,7,8},{3}}
 => ? = 1
{{1,4},{2,3},{5,7},{6,8}}
 => [4,3,2,1,7,8,5,6] => [5,4,3,2,8,1,6,7] => {{1,5,6,7,8},{2,4},{3}}
 => ? = 1
Description
The number of singleton blocks of a set partition.
Matching statistic: St000248
(load all 18 compositions to match this statistic)
Mapping
St000248: Set partitions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 82%
Values
{{1}}
 => ? = 1
{{1,2}}
 => 2
{{1},{2}}
 => 0
{{1,2,3}}
 => 3
{{1,2},{3}}
 => 1
{{1,3},{2}}
 => 1
{{1},{2,3}}
 => 1
{{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => 4
{{1,2,3},{4}}
 => 2
{{1,2,4},{3}}
 => 2
{{1,2},{3,4}}
 => 2
{{1,2},{3},{4}}
 => 1
{{1,3,4},{2}}
 => 2
{{1,3},{2,4}}
 => 0
{{1,3},{2},{4}}
 => 0
{{1,4},{2,3}}
 => 2
{{1},{2,3,4}}
 => 2
{{1},{2,3},{4}}
 => 1
{{1,4},{2},{3}}
 => 1
{{1},{2,4},{3}}
 => 0
{{1},{2},{3,4}}
 => 1
{{1},{2},{3},{4}}
 => 0
{{1,2,3,4,5}}
 => 5
{{1,2,3,4},{5}}
 => 3
{{1,2,3,5},{4}}
 => 3
{{1,2,3},{4,5}}
 => 3
{{1,2,3},{4},{5}}
 => 2
{{1,2,4,5},{3}}
 => 3
{{1,2,4},{3,5}}
 => 1
{{1,2,4},{3},{5}}
 => 1
{{1,2,5},{3,4}}
 => 3
{{1,2},{3,4,5}}
 => 3
{{1,2},{3,4},{5}}
 => 2
{{1,2,5},{3},{4}}
 => 2
{{1,2},{3,5},{4}}
 => 1
{{1,2},{3},{4,5}}
 => 2
{{1,2},{3},{4},{5}}
 => 1
{{1,3,4,5},{2}}
 => 3
{{1,3,4},{2,5}}
 => 1
{{1,3,4},{2},{5}}
 => 1
{{1,3,5},{2,4}}
 => 1
{{1,3},{2,4,5}}
 => 1
{{1,3},{2,4},{5}}
 => 0
{{1,3,5},{2},{4}}
 => 1
{{1,3},{2,5},{4}}
 => 0
{{1,3},{2},{4,5}}
 => 1
{{1,3},{2},{4},{5}}
 => 0
{{1,4,5},{2,3}}
 => 3
{{1,4},{2,3,5}}
 => 1
{{1,4},{2,3},{5}}
 => 1
{{1,2},{3,4},{5,6},{7,8}}
 => ? = 4
{{1,3},{2,4},{5,6},{7,8}}
 => ? = 2
{{1,4},{2,3},{5,6},{7,8}}
 => ? = 3
{{1,5},{2,3},{4,6},{7,8}}
 => ? = 2
{{1,6},{2,3},{4,5},{7,8}}
 => ? = 3
{{1,7},{2,3},{4,5},{6,8}}
 => ? = 2
{{1,8},{2,3},{4,5},{6,7}}
 => ? = 4
{{1,8},{2,4},{3,5},{6,7}}
 => ? = 2
{{1,7},{2,4},{3,5},{6,8}}
 => ? = 0
{{1,6},{2,4},{3,5},{7,8}}
 => ? = 1
{{1,5},{2,4},{3,6},{7,8}}
 => ? = 1
{{1,4},{2,5},{3,6},{7,8}}
 => ? = 1
{{1,3},{2,5},{4,6},{7,8}}
 => ? = 1
{{1,2},{3,5},{4,6},{7,8}}
 => ? = 2
{{1,2},{3,6},{4,5},{7,8}}
 => ? = 3
{{1,3},{2,6},{4,5},{7,8}}
 => ? = 2
{{1,4},{2,6},{3,5},{7,8}}
 => ? = 1
{{1,5},{2,6},{3,4},{7,8}}
 => ? = 2
{{1,6},{2,5},{3,4},{7,8}}
 => ? = 2
{{1,7},{2,5},{3,4},{6,8}}
 => ? = 1
{{1,8},{2,5},{3,4},{6,7}}
 => ? = 3
{{1,8},{2,6},{3,4},{5,7}}
 => ? = 2
{{1,7},{2,6},{3,4},{5,8}}
 => ? = 1
{{1,6},{2,7},{3,4},{5,8}}
 => ? = 1
{{1,5},{2,7},{3,4},{6,8}}
 => ? = 1
{{1,4},{2,7},{3,5},{6,8}}
 => ? = 0
{{1,3},{2,7},{4,5},{6,8}}
 => ? = 1
{{1,2},{3,7},{4,5},{6,8}}
 => ? = 2
{{1,2},{3,8},{4,5},{6,7}}
 => ? = 3
{{1,3},{2,8},{4,5},{6,7}}
 => ? = 2
{{1,4},{2,8},{3,5},{6,7}}
 => ? = 1
{{1,5},{2,8},{3,4},{6,7}}
 => ? = 2
{{1,6},{2,8},{3,4},{5,7}}
 => ? = 1
{{1,7},{2,8},{3,4},{5,6}}
 => ? = 2
{{1,8},{2,7},{3,4},{5,6}}
 => ? = 3
{{1,8},{2,7},{3,5},{4,6}}
 => ? = 1
{{1,7},{2,8},{3,5},{4,6}}
 => ? = 0
{{1,6},{2,8},{3,5},{4,7}}
 => ? = 0
{{1,5},{2,8},{3,6},{4,7}}
 => ? = 0
{{1,4},{2,8},{3,6},{5,7}}
 => ? = 0
{{1,3},{2,8},{4,6},{5,7}}
 => ? = 0
{{1,2},{3,8},{4,6},{5,7}}
 => ? = 1
{{1,2},{3,7},{4,6},{5,8}}
 => ? = 1
{{1,3},{2,7},{4,6},{5,8}}
 => ? = 0
{{1,4},{2,7},{3,6},{5,8}}
 => ? = 0
{{1,5},{2,7},{3,6},{4,8}}
 => ? = 0
{{1,6},{2,7},{3,5},{4,8}}
 => ? = 0
{{1,7},{2,6},{3,5},{4,8}}
 => ? = 0
{{1,8},{2,6},{3,5},{4,7}}
 => ? = 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: St000895
(load all 3 compositions to match this statistic)
Mapping
Mp00221: Set partitions —conjugate⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 64%
Values
{{1}}
 => {{1}}
 => [1] => [[1]]
 => 1
{{1,2}}
 => {{1},{2}}
 => [1,2] => [[1,0],[0,1]]
 => 2
{{1},{2}}
 => {{1,2}}
 => [2,1] => [[0,1],[1,0]]
 => 0
{{1,2,3}}
 => {{1},{2},{3}}
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => 1
{{1},{2},{3}}
 => {{1,2,3}}
 => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 0
{{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
{{1,2,3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 2
{{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 2
{{1,2},{3,4}}
 => {{1,3},{2},{4}}
 => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => 2
{{1,2},{3},{4}}
 => {{1,2,3},{4}}
 => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 1
{{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 2
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => 0
{{1,3},{2},{4}}
 => {{1,2},{3,4}}
 => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 0
{{1,4},{2,3}}
 => {{1},{2,4},{3}}
 => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => 2
{{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 2
{{1},{2,3},{4}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 1
{{1,4},{2},{3}}
 => {{1},{2,3,4}}
 => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 1
{{1},{2,4},{3}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 0
{{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => 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
{{1,2,3,4,5}}
 => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 5
{{1,2,3,4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 3
{{1,2,3,5},{4}}
 => {{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 3
{{1,2,3},{4,5}}
 => {{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 3
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [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
{{1,2,4,5},{3}}
 => {{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 3
{{1,2,4},{3,5}}
 => {{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 1
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 1
{{1,2,5},{3,4}}
 => {{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 3
{{1,2},{3,4,5}}
 => {{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => 3
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 2
{{1,2,5},{3},{4}}
 => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 2
{{1,2},{3,5},{4}}
 => {{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => 1
{{1,2},{3},{4,5}}
 => {{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 2
{{1,2},{3},{4},{5}}
 => {{1,2,3,4},{5}}
 => [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
{{1,3,4,5},{2}}
 => {{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 3
{{1,3,4},{2,5}}
 => {{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
 => 1
{{1,3,4},{2},{5}}
 => {{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 1
{{1,3,5},{2,4}}
 => {{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
 => 1
{{1,3},{2,4,5}}
 => {{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
 => 1
{{1,3},{2,4},{5}}
 => {{1,2,4},{3,5}}
 => [2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
 => 0
{{1,3,5},{2},{4}}
 => {{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 1
{{1,3},{2,5},{4}}
 => {{1,4},{2,3,5}}
 => [4,3,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
 => 0
{{1,3},{2},{4,5}}
 => {{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 1
{{1,3},{2},{4},{5}}
 => {{1,2,3},{4,5}}
 => [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 0
{{1,4,5},{2,3}}
 => {{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => 3
{{1,4},{2,3,5}}
 => {{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
 => 1
{{1,3,4,6},{2,5}}
 => {{1},{2,5},{3,6},{4}}
 => [1,5,6,4,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 2
{{1,3,4},{2,5,6}}
 => {{1,5},{2},{3,6},{4}}
 => [5,2,6,4,1,3] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 2
{{1,3,4},{2,5},{6}}
 => {{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 1
{{1,3,4},{2,6},{5}}
 => {{1,5},{2,3,6},{4}}
 => [5,3,6,4,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 1
{{1,3,6},{2,4,5}}
 => {{1},{2,5},{3},{4,6}}
 => [1,5,3,6,2,4] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 2
{{1,3},{2,4,5,6}}
 => {{1,5},{2},{3},{4,6}}
 => [5,2,3,6,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 2
{{1,3},{2,4,5},{6}}
 => {{1,2,5},{3},{4,6}}
 => [2,5,3,6,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 1
{{1,3},{2,4},{5,6}}
 => {{1,3,5},{2},{4,6}}
 => [3,2,5,6,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 1
{{1,3,6},{2,5},{4}}
 => {{1},{2,5},{3,4,6}}
 => [1,5,4,6,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 1
{{1,3},{2,5,6},{4}}
 => {{1,5},{2},{3,4,6}}
 => [5,2,4,6,1,3] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 1
{{1,3},{2,5},{4},{6}}
 => {{1,2,5},{3,4,6}}
 => [2,5,4,6,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
 => ? = 0
{{1,4,5},{2,3,6}}
 => {{1,4},{2,6},{3},{5}}
 => [4,6,3,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 2
{{1,4,6},{2,3,5}}
 => {{1},{2,4},{3,6},{5}}
 => [1,4,6,2,5,3] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 2
{{1,4},{2,3,5,6}}
 => {{1,4},{2},{3,6},{5}}
 => [4,2,6,1,5,3] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 2
{{1,4},{2,3,5},{6}}
 => {{1,2,4},{3,6},{5}}
 => [2,4,6,1,5,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 1
{{1,4},{2,3,6},{5}}
 => {{1,4},{2,3,6},{5}}
 => [4,3,6,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 1
{{1,5},{2,3,4,6}}
 => {{1,3},{2,6},{4},{5}}
 => [3,6,1,4,5,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 2
{{1,5},{2,3,6},{4}}
 => {{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 1
{{1,5},{2,3},{4,6}}
 => {{1,3},{2,4,6},{5}}
 => [3,4,1,6,5,2] => [[0,0,1,0,0,0],[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]]
 => ? = 1
{{1},{2,3,5},{4,6}}
 => {{1,3,6},{2,4},{5}}
 => [3,4,6,2,5,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[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
{{1,4,5},{2},{3,6}}
 => {{1,4},{2,5,6},{3}}
 => [4,5,3,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
{{1,4,6},{2,5},{3}}
 => {{1},{2,4,5},{3,6}}
 => [1,4,6,5,2,3] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 1
{{1,4},{2,5,6},{3}}
 => {{1,4,5},{2},{3,6}}
 => [4,2,6,5,1,3] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 1
{{1,4},{2,5},{3},{6}}
 => {{1,2,4,5},{3,6}}
 => [2,4,6,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 0
{{1,4},{2},{3,5,6}}
 => {{1,4},{2},{3,5,6}}
 => [4,2,5,1,6,3] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
{{1,4},{2,6},{3},{5}}
 => {{1,4,5},{2,3,6}}
 => [4,3,6,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 0
{{1,4},{2},{3,6},{5}}
 => {{1,4},{2,3,5,6}}
 => [4,3,5,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 0
{{1},{2,4},{3,5,6}}
 => {{1,4,6},{2},{3,5}}
 => [4,2,5,6,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
 => ? = 1
{{1,5},{2},{3,4,6}}
 => {{1,3},{2,5,6},{4}}
 => [3,5,1,4,6,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
{{1,5},{2,6},{3},{4}}
 => {{1,3,4,5},{2,6}}
 => [3,6,4,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 0
{{1,5},{2},{3,6},{4}}
 => {{1,3,4},{2,5,6}}
 => [3,5,4,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 0
{{1,2,3,4,5,6,7}}
 => {{1},{2},{3},{4},{5},{6},{7}}
 => [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => ? = 7
{{1,2,3,4,5,6},{7}}
 => {{1,2},{3},{4},{5},{6},{7}}
 => [2,1,3,4,5,6,7] => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => ? = 5
{{1,2,3,4,6,7},{5}}
 => {{1},{2},{3,4},{5},{6},{7}}
 => [1,2,4,3,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => ? = 5
{{1,2,3,4,7},{5},{6}}
 => {{1},{2,3,4},{5},{6},{7}}
 => [1,3,4,2,5,6,7] => [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => ? = 4
{{1,2,3,5,6,7},{4}}
 => {{1},{2},{3},{4,5},{6},{7}}
 => [1,2,3,5,4,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => ? = 5
{{1,2,3,5,7},{4,6}}
 => {{1},{2,4},{3,5},{6},{7}}
 => [1,4,5,2,3,6,7] => [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => ? = 3
{{1,2,3},{4,7},{5,6}}
 => {{1,5},{2,4},{3},{6},{7}}
 => [5,4,3,2,1,6,7] => [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => ? = 3
{{1,2,3,7},{4},{5},{6}}
 => {{1},{2,3,4,5},{6},{7}}
 => [1,3,4,5,2,6,7] => [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
 => ? = 3
{{1,2,4,6},{3,5},{7}}
 => {{1,2},{3,5},{4,6},{7}}
 => [2,1,5,6,3,4,7] => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
 => ? = 1
{{1,2,5},{3,4},{6,7}}
 => {{1,3},{2},{4,6},{5},{7}}
 => [3,2,1,6,5,4,7] => [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
 => ? = 3
{{1,2},{3,4,7},{5,6}}
 => {{1,6},{2,4},{3},{5},{7}}
 => [6,4,3,2,5,1,7] => [[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1]]
 => ? = 3
{{1,2,5},{3},{4},{6,7}}
 => {{1,3},{2},{4,5,6},{7}}
 => [3,2,1,5,6,4,7] => [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => ? = 2
{{1,2},{3,7},{4,5,6}}
 => {{1,6},{2,5},{3},{4},{7}}
 => [6,5,3,4,2,1,7] => [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1]]
 => ? = 3
{{1,2},{3,7},{4},{5,6}}
 => {{1,6},{2,4,5},{3},{7}}
 => [6,4,3,5,2,1,7] => [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1]]
 => ? = 2
{{1,2},{3},{4,7},{5,6}}
 => {{1,5,6},{2,4},{3},{7}}
 => [5,4,3,2,6,1,7] => [[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => ? = 2
{{1,2,7},{3},{4},{5},{6}}
 => {{1},{2,3,4,5,6},{7}}
 => [1,3,4,5,6,2,7] => [[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
 => ? = 2
{{1,3,4,5,6},{2,7}}
 => {{1,6},{2,7},{3},{4},{5}}
 => [6,7,3,4,5,1,2] => [[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0]]
 => ? = 3
{{1,3,4,6},{2,7},{5}}
 => {{1,6},{2,7},{3,4},{5}}
 => [6,7,4,3,5,1,2] => [[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0]]
 => ? = 1
{{1,3,5,6},{2,4,7}}
 => {{1,4,6},{2,5,7},{3}}
 => [4,5,3,6,7,1,2] => [[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0]]
 => ? = 1
Description
The number of ones on the main diagonal of an alternating sign matrix.
Matching statistic: St000241
(load all 10 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 64%
Values
{{1}}
 => [1] => 1
{{1,2}}
 => [2,1] => 2
{{1},{2}}
 => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => 3
{{1,2},{3}}
 => [2,1,3] => 1
{{1,3},{2}}
 => [3,2,1] => 1
{{1},{2,3}}
 => [1,3,2] => 1
{{1},{2},{3}}
 => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => 4
{{1,2,3},{4}}
 => [2,3,1,4] => 2
{{1,2,4},{3}}
 => [2,4,3,1] => 2
{{1,2},{3,4}}
 => [2,1,4,3] => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => 2
{{1},{2,3,4}}
 => [1,3,4,2] => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => 5
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => 3
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => 3
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => 3
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => 3
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => 3
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => ? = 7
{{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => ? = 5
{{1,2,3,4,6,7},{5}}
 => [2,3,4,6,5,7,1] => ? = 5
{{1,2,3,4,7},{5},{6}}
 => [2,3,4,7,5,6,1] => ? = 4
{{1,2,3,5,6,7},{4}}
 => [2,3,5,4,6,7,1] => ? = 5
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => ? = 3
{{1,2,3},{4,7},{5,6}}
 => [2,3,1,7,6,5,4] => ? = 3
{{1,2,3,7},{4},{5},{6}}
 => [2,3,7,4,5,6,1] => ? = 3
{{1,2,4,6},{3,5},{7}}
 => [2,4,5,6,3,1,7] => ? = 1
{{1,2,5},{3,4},{6,7}}
 => [2,5,4,3,1,7,6] => ? = 3
{{1,2},{3,4,7},{5,6}}
 => [2,1,4,7,6,5,3] => ? = 3
{{1,2,5},{3},{4},{6,7}}
 => [2,5,3,4,1,7,6] => ? = 2
{{1,2},{3,7},{4,5,6}}
 => [2,1,7,5,6,4,3] => ? = 3
{{1,2},{3,7},{4,6},{5}}
 => [2,1,7,6,5,4,3] => ? = 1
{{1,2},{3,7},{4},{5,6}}
 => [2,1,7,4,6,5,3] => ? = 2
{{1,2},{3},{4,7},{5,6}}
 => [2,1,3,7,6,5,4] => ? = 2
{{1,2,7},{3},{4},{5},{6}}
 => [2,7,3,4,5,6,1] => ? = 2
{{1,3,4,5,6},{2,7}}
 => [3,7,4,5,6,1,2] => ? = 3
{{1,3,4,6},{2,7},{5}}
 => [3,7,4,6,5,1,2] => ? = 1
{{1,3,5,6},{2,4,7}}
 => [3,4,5,7,6,1,2] => ? = 1
{{1,3,5,6},{2,4},{7}}
 => [3,4,5,2,6,1,7] => ? = 1
{{1,3,5,7},{2,4,6}}
 => [3,4,5,6,7,2,1] => ? = 1
{{1,3,5},{2,4,6,7}}
 => [3,4,5,6,1,7,2] => ? = 1
{{1,3,5},{2,4,6},{7}}
 => [3,4,5,6,1,2,7] => ? = 0
{{1,3},{2,4,5,6},{7}}
 => [3,4,1,5,6,2,7] => ? = 2
{{1,3,6,7},{2,4},{5}}
 => [3,4,6,2,5,7,1] => ? = 2
{{1,3,6},{2,4,7},{5}}
 => [3,4,6,7,5,1,2] => ? = 0
{{1,3},{2,4,6,7},{5}}
 => [3,4,1,6,5,7,2] => ? = 1
{{1,3,5},{2},{4},{6,7}}
 => [3,2,5,4,1,7,6] => ? = 1
{{1,3},{2},{4,7},{5,6}}
 => [3,2,1,7,6,5,4] => ? = 1
{{1,3},{2,7},{4},{5},{6}}
 => [3,7,1,4,5,6,2] => ? = 0
{{1,4,5},{2,3},{6,7}}
 => [4,3,2,5,1,7,6] => ? = 3
{{1,4,7},{2,3},{5,6}}
 => [4,3,2,7,6,5,1] => ? = 3
{{1,4},{2,3},{5,6},{7}}
 => [4,3,2,1,6,5,7] => ? = 2
{{1,4},{2,3},{5,7},{6}}
 => [4,3,2,1,7,6,5] => ? = 1
{{1,4},{2,3},{5},{6,7}}
 => [4,3,2,1,5,7,6] => ? = 2
{{1,5},{2,3,4},{6,7}}
 => [5,3,4,2,1,7,6] => ? = 3
{{1,7},{2,3,4,5,6}}
 => [7,3,4,5,6,2,1] => ? = 5
{{1},{2,3,4,5,6,7}}
 => [1,3,4,5,6,7,2] => ? = 5
{{1},{2,3,4,5,6},{7}}
 => [1,3,4,5,6,2,7] => ? = 4
{{1,5},{2,3},{4},{6,7}}
 => [5,3,2,4,1,7,6] => ? = 2
{{1},{2,3,5,7},{4,6}}
 => [1,3,5,6,7,4,2] => ? = 1
{{1},{2,3},{4,7},{5,6}}
 => [1,3,2,7,6,5,4] => ? = 2
{{1},{2,3,7},{4},{5},{6}}
 => [1,3,7,4,5,6,2] => ? = 1
{{1,4,5},{2},{3},{6,7}}
 => [4,2,3,5,1,7,6] => ? = 2
{{1,4},{2,5,6},{3,7}}
 => [4,5,7,1,6,2,3] => ? = 1
{{1,4},{2,5},{3,6,7}}
 => [4,5,6,1,2,7,3] => ? = 1
{{1,4},{2,5},{3,6},{7}}
 => [4,5,6,1,2,3,7] => ? = 0
{{1,4},{2,6},{3,7},{5}}
 => [4,6,7,1,5,2,3] => ? = 0
{{1,4,7},{2},{3},{5,6}}
 => [4,2,3,7,6,5,1] => ? = 2
Description
The number of cyclical small excedances. A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
Matching statistic: St000894
(load all 3 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000894: Alternating sign matrices ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 64%
Values
{{1}}
 => [1] => [1] => [[1]]
 => ? = 1
{{1,2}}
 => [2,1] => [1,2] => [[1,0],[0,1]]
 => 2
{{1},{2}}
 => [1,2] => [2,1] => [[0,1],[1,0]]
 => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
{{1,2},{3}}
 => [2,1,3] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => 1
{{1,3},{2}}
 => [3,2,1] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 1
{{1},{2,3}}
 => [1,3,2] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => 1
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 2
{{1,2,4},{3}}
 => [2,4,3,1] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => 2
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => 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
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 5
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 3
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => 3
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 3
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,5,2,3,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
 => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
 => 3
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
 => 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 3
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,1,3,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0]]
 => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,1,2,3,5] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
 => 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [5,1,2,4,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
 => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,1,2,5,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
 => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,1,4,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
 => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
 => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [1,2,3,6,5,4] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 4
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [1,2,3,5,6,4] => [[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],[0,0,0,0,1,0]]
 => ? = 3
{{1,2,3,5},{4,6}}
 => [2,3,5,6,1,4] => [1,2,6,3,4,5] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
 => ? = 2
{{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => [1,2,6,4,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 4
{{1,2,3},{4,5},{6}}
 => [2,3,1,5,4,6] => [1,2,5,4,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 3
{{1,2,3},{4,6},{5}}
 => [2,3,1,6,5,4] => [1,2,6,5,4,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 2
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => [1,2,4,6,5,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 3
{{1,2,3},{4},{5},{6}}
 => [2,3,1,4,5,6] => [1,2,4,5,6,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 2
{{1,2,4,5},{3,6}}
 => [2,4,6,5,1,3] => [1,6,2,4,3,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
 => ? = 2
{{1,2,4},{3,5,6}}
 => [2,4,5,1,6,3] => [1,6,2,3,5,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 2
{{1,2,4},{3,5},{6}}
 => [2,4,5,1,3,6] => [1,5,2,3,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
{{1,2,4},{3,6},{5}}
 => [2,4,6,1,5,3] => [1,6,2,5,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 1
{{1,2,4},{3},{5,6}}
 => [2,4,3,1,6,5] => [1,3,2,6,5,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 2
{{1,2,4},{3},{5},{6}}
 => [2,4,3,1,5,6] => [1,3,2,5,6,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 1
{{1,2,5},{3,4,6}}
 => [2,5,4,6,1,3] => [1,6,3,2,4,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
 => ? = 2
{{1,2},{3,4,5,6}}
 => [2,1,4,5,6,3] => [1,6,3,4,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 4
{{1,2},{3,4,5},{6}}
 => [2,1,4,5,3,6] => [1,5,3,4,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 3
{{1,2},{3,4,6},{5}}
 => [2,1,4,6,5,3] => [1,6,3,5,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 2
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => [1,4,3,6,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 3
{{1,2},{3,4},{5},{6}}
 => [2,1,4,3,5,6] => [1,4,3,5,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 2
{{1,2,5},{3,6},{4}}
 => [2,5,6,4,1,3] => [1,6,4,2,3,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
 => ? = 1
{{1,2,5},{3},{4,6}}
 => [2,5,3,6,1,4] => [1,3,6,2,4,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
 => ? = 1
{{1,2},{3,5,6},{4}}
 => [2,1,5,4,6,3] => [1,6,4,3,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 2
{{1,2},{3,5},{4,6}}
 => [2,1,5,6,3,4] => [1,5,6,3,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 1
{{1,2},{3,5},{4},{6}}
 => [2,1,5,4,3,6] => [1,5,4,3,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
{{1,2},{3,6},{4,5}}
 => [2,1,6,5,4,3] => [1,6,5,4,3,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
 => ? = 2
{{1,2},{3},{4,5,6}}
 => [2,1,3,5,6,4] => [1,3,6,4,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 3
{{1,2},{3},{4,5},{6}}
 => [2,1,3,5,4,6] => [1,3,5,4,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
 => ? = 2
{{1,2},{3,6},{4},{5}}
 => [2,1,6,4,5,3] => [1,6,4,5,3,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 1
{{1,2},{3},{4,6},{5}}
 => [2,1,3,6,5,4] => [1,3,6,5,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
 => ? = 1
{{1,2},{3},{4},{5,6}}
 => [2,1,3,4,6,5] => [1,3,4,6,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 2
{{1,2},{3},{4},{5},{6}}
 => [2,1,3,4,5,6] => [1,3,4,5,6,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 1
{{1,3,4,5},{2,6}}
 => [3,6,4,5,1,2] => [6,1,3,4,2,5] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
 => ? = 2
{{1,3,4},{2,5,6}}
 => [3,5,4,1,6,2] => [6,1,3,2,5,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 2
{{1,3,4},{2,5},{6}}
 => [3,5,4,1,2,6] => [5,1,3,2,6,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
{{1,3,4},{2,6},{5}}
 => [3,6,4,1,5,2] => [6,1,3,5,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 1
{{1,3,4},{2},{5,6}}
 => [3,2,4,1,6,5] => [2,1,3,6,5,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 2
{{1,3,4},{2},{5},{6}}
 => [3,2,4,1,5,6] => [2,1,3,5,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 1
{{1,3,5},{2,4,6}}
 => [3,4,5,6,1,2] => [6,1,2,3,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
 => ? = 0
{{1,3},{2,4,5,6}}
 => [3,4,1,5,6,2] => [6,1,2,4,5,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 2
{{1,3},{2,4,5},{6}}
 => [3,4,1,5,2,6] => [5,1,2,4,6,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 1
{{1,3},{2,4,6},{5}}
 => [3,4,1,6,5,2] => [6,1,2,5,4,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
 => ? = 0
{{1,3},{2,4},{5,6}}
 => [3,4,1,2,6,5] => [4,1,2,6,5,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
 => ? = 1
{{1,3},{2,4},{5},{6}}
 => [3,4,1,2,5,6] => [4,1,2,5,6,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
 => ? = 0
{{1,3,5},{2,6},{4}}
 => [3,6,5,4,1,2] => [6,1,4,3,2,5] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
 => ? = 0
{{1,3,5},{2},{4,6}}
 => [3,2,5,6,1,4] => [2,1,6,3,4,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
 => ? = 0
{{1,3},{2,5,6},{4}}
 => [3,5,1,4,6,2] => [6,1,4,2,5,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 1
{{1,3},{2,5},{4,6}}
 => [3,5,1,6,2,4] => [5,1,6,2,4,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
 => ? = 0
{{1,3},{2,5},{4},{6}}
 => [3,5,1,4,2,6] => [5,1,4,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
 => ? = 0
Description
The trace of an alternating sign matrix.
The following 1 statistic also match your data. Click on any of them to see the details.
St001903The number of fixed points of a parking function.