searching the database
Your data matches 17 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000505
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> 1
{{1,2}}
=> 2
{{1},{2}}
=> 1
{{1,2,3}}
=> 3
{{1,2},{3}}
=> 2
{{1,3},{2}}
=> 3
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 1
{{1,2,3,4}}
=> 4
{{1,2,3},{4}}
=> 3
{{1,2,4},{3}}
=> 4
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 2
{{1,3,4},{2}}
=> 4
{{1,3},{2,4}}
=> 3
{{1,3},{2},{4}}
=> 3
{{1,4},{2,3}}
=> 4
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 4
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 1
{{1},{2},{3},{4}}
=> 1
{{1,2,3,4,5}}
=> 5
{{1,2,3,4},{5}}
=> 4
{{1,2,3,5},{4}}
=> 5
{{1,2,3},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> 3
{{1,2,4,5},{3}}
=> 5
{{1,2,4},{3,5}}
=> 4
{{1,2,4},{3},{5}}
=> 4
{{1,2,5},{3,4}}
=> 5
{{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> 5
{{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> 2
{{1,2},{3},{4},{5}}
=> 2
{{1,3,4,5},{2}}
=> 5
{{1,3,4},{2,5}}
=> 4
{{1,3,4},{2},{5}}
=> 4
{{1,3,5},{2,4}}
=> 5
{{1,3},{2,4,5}}
=> 3
{{1,3},{2,4},{5}}
=> 3
{{1,3,5},{2},{4}}
=> 5
{{1,3},{2,5},{4}}
=> 3
{{1,3},{2},{4,5}}
=> 3
{{1,3},{2},{4},{5}}
=> 3
{{1,4,5},{2,3}}
=> 5
{{1,4},{2,3,5}}
=> 4
Description
The biggest entry in the block containing the 1.
Matching statistic: St000439
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 91%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 91%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 2 = 1 + 1
{{1,2}}
=> [2,1] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
{{1},{2}}
=> [1,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
{{1,2},{3,4},{5,6},{7,8},{9,10}}
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2 + 1
{{1,4},{2,3},{5,6},{7,8},{9,10}}
=> ? => ? => ?
=> ? = 4 + 1
{{1,6},{2,3},{4,5},{7,8},{9,10}}
=> ? => ? => ?
=> ? = 6 + 1
{{1,8},{2,3},{4,5},{6,7},{9,10}}
=> ? => ? => ?
=> ? = 8 + 1
{{1,10},{2,3},{4,5},{6,7},{8,9}}
=> ? => ? => ?
=> ? = 10 + 1
{{1,2},{3,6},{4,5},{7,8},{9,10}}
=> ? => ? => ?
=> ? = 2 + 1
{{1,6},{2,5},{3,4},{7,8},{9,10}}
=> ? => ? => ?
=> ? = 6 + 1
{{1,8},{2,5},{3,4},{6,7},{9,10}}
=> ? => ? => ?
=> ? = 8 + 1
{{1,10},{2,5},{3,4},{6,7},{8,9}}
=> ? => ? => ?
=> ? = 10 + 1
{{1,2},{3,8},{4,5},{6,7},{9,10}}
=> ? => ? => ?
=> ? = 2 + 1
{{1,8},{2,7},{3,4},{5,6},{9,10}}
=> ? => ? => ?
=> ? = 8 + 1
{{1,10},{2,7},{3,4},{5,6},{8,9}}
=> ? => ? => ?
=> ? = 10 + 1
{{1,2},{3,10},{4,5},{6,7},{8,9}}
=> ? => ? => ?
=> ? = 2 + 1
{{1,10},{2,9},{3,4},{5,6},{7,8}}
=> ? => ? => ?
=> ? = 10 + 1
{{1,2},{3,4},{5,8},{6,7},{9,10}}
=> ? => ? => ?
=> ? = 2 + 1
{{1,4},{2,3},{5,8},{6,7},{9,10}}
=> ? => ? => ?
=> ? = 4 + 1
{{1,8},{2,3},{4,7},{5,6},{9,10}}
=> ? => ? => ?
=> ? = 8 + 1
{{1,10},{2,3},{4,7},{5,6},{8,9}}
=> ? => ? => ?
=> ? = 10 + 1
{{1,2},{3,8},{4,7},{5,6},{9,10}}
=> ? => ? => ?
=> ? = 2 + 1
{{1,8},{2,7},{3,6},{4,5},{9,10}}
=> ? => ? => ?
=> ? = 8 + 1
{{1,10},{2,7},{3,6},{4,5},{8,9}}
=> ? => ? => ?
=> ? = 10 + 1
{{1,2},{3,10},{4,7},{5,6},{8,9}}
=> ? => ? => ?
=> ? = 2 + 1
{{1,10},{2,9},{3,6},{4,5},{7,8}}
=> ? => ? => ?
=> ? = 10 + 1
{{1,2},{3,4},{5,10},{6,7},{8,9}}
=> ? => ? => ?
=> ? = 2 + 1
{{1,4},{2,3},{5,10},{6,7},{8,9}}
=> ? => ? => ?
=> ? = 4 + 1
{{1,10},{2,3},{4,9},{5,6},{7,8}}
=> ? => ? => ?
=> ? = 10 + 1
{{1,2},{3,10},{4,9},{5,6},{7,8}}
=> ? => ? => ?
=> ? = 2 + 1
{{1,10},{2,9},{3,8},{4,5},{6,7}}
=> ? => ? => ?
=> ? = 10 + 1
{{1,2},{3,4},{5,6},{7,10},{8,9}}
=> ? => ? => ?
=> ? = 2 + 1
{{1,4},{2,3},{5,6},{7,10},{8,9}}
=> ? => ? => ?
=> ? = 4 + 1
{{1,6},{2,3},{4,5},{7,10},{8,9}}
=> ? => ? => ?
=> ? = 6 + 1
{{1,10},{2,3},{4,5},{6,9},{7,8}}
=> ? => ? => ?
=> ? = 10 + 1
{{1,2},{3,6},{4,5},{7,10},{8,9}}
=> ? => ? => ?
=> ? = 2 + 1
{{1,6},{2,5},{3,4},{7,10},{8,9}}
=> ? => ? => ?
=> ? = 6 + 1
{{1,10},{2,5},{3,4},{6,9},{7,8}}
=> ? => ? => ?
=> ? = 10 + 1
{{1,2},{3,10},{4,5},{6,9},{7,8}}
=> ? => ? => ?
=> ? = 2 + 1
{{1,10},{2,9},{3,4},{5,8},{6,7}}
=> ? => ? => ?
=> ? = 10 + 1
{{1,2},{3,4},{5,10},{6,9},{7,8}}
=> ? => ? => ?
=> ? = 2 + 1
{{1,4},{2,3},{5,10},{6,9},{7,8}}
=> ? => ? => ?
=> ? = 4 + 1
{{1,10},{2,3},{4,9},{5,8},{6,7}}
=> ? => ? => ?
=> ? = 10 + 1
{{1,2},{3,10},{4,9},{5,8},{6,7}}
=> ? => ? => ?
=> ? = 2 + 1
{{1,10},{2,9},{3,8},{4,7},{5,6}}
=> ? => ? => ?
=> ? = 10 + 1
{{1},{2,3,4,5,6,7},{8}}
=> ? => ? => ?
=> ? = 1 + 1
{{1},{2,3,4,8},{5,6,7}}
=> ? => ? => ?
=> ? = 1 + 1
{{1,3,5,6,7,8},{2},{4}}
=> [3,2,5,4,6,7,8,1] => [8,2,1,4,3,5,6,7] => ?
=> ? = 8 + 1
{{1,2,3,7},{4,5,6},{8}}
=> ? => ? => ?
=> ? = 7 + 1
{{1,2,3,4,6,7},{5,8}}
=> [2,3,4,6,8,7,1,5] => [7,1,2,3,8,4,6,5] => ?
=> ? = 7 + 1
{{1,2,3,4,5,8},{6,7},{9}}
=> [2,3,4,5,8,7,6,1,9] => [8,1,2,3,4,7,6,5,9] => ?
=> ? = 8 + 1
{{1,2,3,4,5,7,8},{6},{9}}
=> [2,3,4,5,7,6,8,1,9] => ? => ?
=> ? = 8 + 1
{{1},{2,3,4,5,6,9},{7,8}}
=> ? => ? => ?
=> ? = 1 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000025
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 77%●distinct values known / distinct values provided: 64%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 77%●distinct values known / distinct values provided: 64%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 1
{{1,2}}
=> [2,1] => [2,1] => [1,1,0,0]
=> 2
{{1},{2}}
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 4
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 4
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
{{1,2},{3,4},{5,6},{7,8}}
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1,3},{2,4},{5,6},{7,8}}
=> [3,4,1,2,6,5,8,7] => [3,4,1,2,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
{{1,4},{2,3},{5,6},{7,8}}
=> [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
{{1,5},{2,3},{4,6},{7,8}}
=> [5,3,2,6,1,4,8,7] => [5,3,2,6,1,4,8,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> ? = 5
{{1,6},{2,3},{4,5},{7,8}}
=> [6,3,2,5,4,1,8,7] => [6,3,2,5,4,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
{{1,7},{2,3},{4,5},{6,8}}
=> [7,3,2,5,4,8,1,6] => [7,3,2,5,4,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 7
{{1,8},{2,3},{4,5},{6,7}}
=> [8,3,2,5,4,7,6,1] => [8,3,2,5,4,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
{{1,8},{2,4},{3,5},{6,7}}
=> [8,4,5,2,3,7,6,1] => [8,4,5,2,3,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
{{1,7},{2,4},{3,5},{6,8}}
=> [7,4,5,2,3,8,1,6] => [7,4,5,2,3,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 7
{{1,6},{2,4},{3,5},{7,8}}
=> [6,4,5,2,3,1,8,7] => [6,4,5,2,3,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
{{1,5},{2,4},{3,6},{7,8}}
=> [5,4,6,2,1,3,8,7] => [5,4,6,2,1,3,8,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> ? = 5
{{1,4},{2,5},{3,6},{7,8}}
=> [4,5,6,1,2,3,8,7] => [4,5,6,1,2,3,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> ? = 4
{{1,3},{2,5},{4,6},{7,8}}
=> [3,5,1,6,2,4,8,7] => [3,5,1,6,2,4,8,7] => [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 3
{{1,2},{3,5},{4,6},{7,8}}
=> [2,1,5,6,3,4,8,7] => [2,1,5,6,3,4,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 2
{{1,2},{3,6},{4,5},{7,8}}
=> [2,1,6,5,4,3,8,7] => [2,1,6,5,4,3,8,7] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
{{1,3},{2,6},{4,5},{7,8}}
=> [3,6,1,5,4,2,8,7] => [3,6,1,5,4,2,8,7] => [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 3
{{1,4},{2,6},{3,5},{7,8}}
=> [4,6,5,1,3,2,8,7] => [4,6,5,1,3,2,8,7] => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 4
{{1,5},{2,6},{3,4},{7,8}}
=> [5,6,4,3,1,2,8,7] => [5,6,4,3,1,2,8,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> ? = 5
{{1,6},{2,5},{3,4},{7,8}}
=> [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
{{1,7},{2,5},{3,4},{6,8}}
=> [7,5,4,3,2,8,1,6] => [7,5,4,3,2,8,1,6] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 7
{{1,8},{2,5},{3,4},{6,7}}
=> [8,5,4,3,2,7,6,1] => [8,5,4,3,2,7,6,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
{{1,8},{2,6},{3,4},{5,7}}
=> [8,6,4,3,7,2,5,1] => [8,6,4,3,7,2,5,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
{{1,7},{2,6},{3,4},{5,8}}
=> [7,6,4,3,8,2,1,5] => [7,6,4,3,8,2,1,5] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 7
{{1,6},{2,7},{3,4},{5,8}}
=> [6,7,4,3,8,1,2,5] => [6,7,4,3,8,1,2,5] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 6
{{1,5},{2,7},{3,4},{6,8}}
=> [5,7,4,3,1,8,2,6] => [5,7,4,3,1,8,2,6] => [1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0]
=> ? = 5
{{1,4},{2,7},{3,5},{6,8}}
=> [4,7,5,1,3,8,2,6] => [4,7,5,1,3,8,2,6] => [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 4
{{1,3},{2,7},{4,5},{6,8}}
=> [3,7,1,5,4,8,2,6] => [3,7,1,5,4,8,2,6] => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 3
{{1,2},{3,7},{4,5},{6,8}}
=> [2,1,7,5,4,8,3,6] => [2,1,7,5,4,8,3,6] => [1,1,0,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 2
{{1,2},{3,8},{4,5},{6,7}}
=> [2,1,8,5,4,7,6,3] => [2,1,8,5,4,7,6,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
{{1,3},{2,8},{4,5},{6,7}}
=> [3,8,1,5,4,7,6,2] => [3,8,1,5,4,7,6,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3
{{1,4},{2,8},{3,5},{6,7}}
=> [4,8,5,1,3,7,6,2] => [4,8,5,1,3,7,6,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4
{{1,5},{2,8},{3,4},{6,7}}
=> [5,8,4,3,1,7,6,2] => [5,8,4,3,1,7,6,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 5
{{1,6},{2,8},{3,4},{5,7}}
=> [6,8,4,3,7,1,5,2] => [6,8,4,3,7,1,5,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 6
{{1,7},{2,8},{3,4},{5,6}}
=> [7,8,4,3,6,5,1,2] => [7,8,4,3,6,5,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
{{1,8},{2,7},{3,4},{5,6}}
=> [8,7,4,3,6,5,2,1] => [8,7,4,3,6,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
{{1,8},{2,7},{3,5},{4,6}}
=> [8,7,5,6,3,4,2,1] => [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
{{1,7},{2,8},{3,5},{4,6}}
=> [7,8,5,6,3,4,1,2] => [7,8,5,6,3,4,1,2] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
{{1,6},{2,8},{3,5},{4,7}}
=> [6,8,5,7,3,1,4,2] => [6,8,5,7,3,1,4,2] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 6
{{1,5},{2,8},{3,6},{4,7}}
=> [5,8,6,7,1,3,4,2] => [5,8,6,7,1,3,4,2] => [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 5
{{1,4},{2,8},{3,6},{5,7}}
=> [4,8,6,1,7,3,5,2] => [4,8,6,1,7,3,5,2] => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4
{{1,3},{2,8},{4,6},{5,7}}
=> [3,8,1,6,7,4,5,2] => [3,8,1,6,7,4,5,2] => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3
{{1,2},{3,8},{4,6},{5,7}}
=> [2,1,8,6,7,4,5,3] => [2,1,8,6,7,4,5,3] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
{{1,2},{3,7},{4,6},{5,8}}
=> [2,1,7,6,8,4,3,5] => [2,1,7,6,8,4,3,5] => [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2
{{1,3},{2,7},{4,6},{5,8}}
=> [3,7,1,6,8,4,2,5] => [3,7,1,6,8,4,2,5] => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 3
{{1,4},{2,7},{3,6},{5,8}}
=> [4,7,6,1,8,3,2,5] => [4,7,6,1,8,3,2,5] => [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 4
{{1,5},{2,7},{3,6},{4,8}}
=> [5,7,6,8,1,3,2,4] => [5,7,6,8,1,3,2,4] => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> ? = 5
{{1,6},{2,7},{3,5},{4,8}}
=> [6,7,5,8,3,1,2,4] => [6,7,5,8,3,1,2,4] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 6
{{1,7},{2,6},{3,5},{4,8}}
=> [7,6,5,8,3,2,1,4] => [7,6,5,8,3,2,1,4] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 7
{{1,8},{2,6},{3,5},{4,7}}
=> [8,6,5,7,3,2,4,1] => [8,6,5,7,3,2,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
{{1,8},{2,5},{3,6},{4,7}}
=> [8,5,6,7,2,3,4,1] => [8,5,6,7,2,3,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 77%●distinct values known / distinct values provided: 64%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 77%●distinct values known / distinct values provided: 64%
Values
{{1}}
=> [1] => [.,.]
=> [1,0]
=> 1
{{1,2}}
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
{{1,2,3}}
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3
{{1,2},{3}}
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
{{1},{2,3}}
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 4
{{1,2,3},{4}}
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
{{1,2,4},{3}}
=> [2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 4
{{1,2},{3,4}}
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
{{1,3,4},{2}}
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 4
{{1,3},{2,4}}
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3
{{1,4},{2,3}}
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 4
{{1},{2,3,4}}
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 4
{{1},{2,4},{3}}
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 5
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 5
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
{{1,2},{3,4},{5,6},{7,8}}
=> [2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
{{1,3},{2,4},{5,6},{7,8}}
=> [3,4,1,2,6,5,8,7] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
{{1,4},{2,3},{5,6},{7,8}}
=> [4,3,2,1,6,5,8,7] => [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
{{1,5},{2,3},{4,6},{7,8}}
=> [5,3,2,6,1,4,8,7] => [[[[.,.],.],[.,.]],[.,[[.,.],.]]]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
=> ? = 5
{{1,6},{2,3},{4,5},{7,8}}
=> [6,3,2,5,4,1,8,7] => [[[[.,.],.],[[.,.],.]],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 6
{{1,7},{2,3},{4,5},{6,8}}
=> [7,3,2,5,4,8,1,6] => [[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 7
{{1,8},{2,3},{4,5},{6,7}}
=> [8,3,2,5,4,7,6,1] => [[[[.,.],.],[[.,.],[[.,.],.]]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 8
{{1,8},{2,4},{3,5},{6,7}}
=> [8,4,5,2,3,7,6,1] => [[[[.,.],[.,.]],[.,[[.,.],.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 8
{{1,7},{2,4},{3,5},{6,8}}
=> [7,4,5,2,3,8,1,6] => [[[[.,.],[.,.]],[.,[.,.]]],[.,.]]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 7
{{1,6},{2,4},{3,5},{7,8}}
=> [6,4,5,2,3,1,8,7] => [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 6
{{1,5},{2,4},{3,6},{7,8}}
=> [5,4,6,2,1,3,8,7] => [[[[.,.],[.,.]],.],[.,[[.,.],.]]]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 5
{{1,4},{2,5},{3,6},{7,8}}
=> [4,5,6,1,2,3,8,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
{{1,3},{2,5},{4,6},{7,8}}
=> [3,5,1,6,2,4,8,7] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
{{1,2},{3,5},{4,6},{7,8}}
=> [2,1,5,6,3,4,8,7] => [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 2
{{1,2},{3,6},{4,5},{7,8}}
=> [2,1,6,5,4,3,8,7] => [[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
{{1,3},{2,6},{4,5},{7,8}}
=> [3,6,1,5,4,2,8,7] => [[.,[.,.]],[[[.,.],.],[[.,.],.]]]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
{{1,4},{2,6},{3,5},{7,8}}
=> [4,6,5,1,3,2,8,7] => [[.,[[.,.],.]],[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
{{1,5},{2,6},{3,4},{7,8}}
=> [5,6,4,3,1,2,8,7] => [[[[.,[.,.]],.],.],[.,[[.,.],.]]]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
{{1,6},{2,5},{3,4},{7,8}}
=> [6,5,4,3,2,1,8,7] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
{{1,7},{2,5},{3,4},{6,8}}
=> [7,5,4,3,2,8,1,6] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 7
{{1,8},{2,5},{3,4},{6,7}}
=> [8,5,4,3,2,7,6,1] => [[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 8
{{1,8},{2,6},{3,4},{5,7}}
=> [8,6,4,3,7,2,5,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 8
{{1,7},{2,6},{3,4},{5,8}}
=> [7,6,4,3,8,2,1,5] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 7
{{1,6},{2,7},{3,4},{5,8}}
=> [6,7,4,3,8,1,2,5] => [[[[.,[.,.]],.],[.,.]],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 6
{{1,5},{2,7},{3,4},{6,8}}
=> [5,7,4,3,1,8,2,6] => [[[[.,[.,.]],.],.],[[.,.],[.,.]]]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 5
{{1,4},{2,7},{3,5},{6,8}}
=> [4,7,5,1,3,8,2,6] => [[.,[[.,.],.]],[[.,[.,.]],[.,.]]]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> ? = 4
{{1,3},{2,7},{4,5},{6,8}}
=> [3,7,1,5,4,8,2,6] => [[.,[.,.]],[[[.,.],[.,.]],[.,.]]]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 3
{{1,2},{3,7},{4,5},{6,8}}
=> [2,1,7,5,4,8,3,6] => [[.,.],[[[[.,.],.],[.,.]],[.,.]]]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 2
{{1,2},{3,8},{4,5},{6,7}}
=> [2,1,8,5,4,7,6,3] => [[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 2
{{1,3},{2,8},{4,5},{6,7}}
=> [3,8,1,5,4,7,6,2] => [[.,[.,.]],[[[.,.],[[.,.],.]],.]]
=> [1,1,0,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 3
{{1,4},{2,8},{3,5},{6,7}}
=> [4,8,5,1,3,7,6,2] => [[.,[[.,.],.]],[[.,[[.,.],.]],.]]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 4
{{1,5},{2,8},{3,4},{6,7}}
=> [5,8,4,3,1,7,6,2] => [[[[.,[.,.]],.],.],[[[.,.],.],.]]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
{{1,6},{2,8},{3,4},{5,7}}
=> [6,8,4,3,7,1,5,2] => [[[[.,[.,.]],.],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 6
{{1,7},{2,8},{3,4},{5,6}}
=> [7,8,4,3,6,5,1,2] => [[[[.,[.,.]],.],[[.,.],.]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
=> ? = 7
{{1,8},{2,7},{3,4},{5,6}}
=> [8,7,4,3,6,5,2,1] => [[[[[[.,.],.],.],[[.,.],.]],.],.]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 8
{{1,8},{2,7},{3,5},{4,6}}
=> [8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 8
{{1,7},{2,8},{3,5},{4,6}}
=> [7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 7
{{1,6},{2,8},{3,5},{4,7}}
=> [6,8,5,7,3,1,4,2] => [[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 6
{{1,5},{2,8},{3,6},{4,7}}
=> [5,8,6,7,1,3,4,2] => [[.,[[.,.],[.,.]]],[[.,[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 5
{{1,4},{2,8},{3,6},{5,7}}
=> [4,8,6,1,7,3,5,2] => [[.,[[.,.],.]],[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 4
{{1,3},{2,8},{4,6},{5,7}}
=> [3,8,1,6,7,4,5,2] => [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 3
{{1,2},{3,8},{4,6},{5,7}}
=> [2,1,8,6,7,4,5,3] => [[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 2
{{1,2},{3,7},{4,6},{5,8}}
=> [2,1,7,6,8,4,3,5] => [[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2
{{1,3},{2,7},{4,6},{5,8}}
=> [3,7,1,6,8,4,2,5] => [[.,[.,.]],[[[.,[.,.]],.],[.,.]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3
{{1,4},{2,7},{3,6},{5,8}}
=> [4,7,6,1,8,3,2,5] => [[.,[[.,.],.]],[[[.,.],.],[.,.]]]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4
{{1,5},{2,7},{3,6},{4,8}}
=> [5,7,6,8,1,3,2,4] => [[.,[[.,.],[.,.]]],[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 5
{{1,6},{2,7},{3,5},{4,8}}
=> [6,7,5,8,3,1,2,4] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 6
{{1,7},{2,6},{3,5},{4,8}}
=> [7,6,5,8,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 7
{{1,8},{2,6},{3,5},{4,7}}
=> [8,6,5,7,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> ? = 8
{{1,8},{2,5},{3,6},{4,7}}
=> [8,5,6,7,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 8
Description
The position of the first return of a Dyck path.
Matching statistic: St000054
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 73%
Mp00066: Permutations —inverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 73%
Values
{{1}}
=> [1] => [1] => [1] => 1
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 2
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [3,1,2] => 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 2
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [3,2,1] => 3
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,3,2] => 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [4,1,3,2] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [3,1,4,2] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => [4,1,3,2] => 4
{{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] => [2,1,4,3] => 2
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => [4,2,1,3] => 4
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [3,4,1,2] => 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,4,3,2] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [4,2,3,1] => 4
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,4,3,2] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,4,3,2] => 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,4,3,2] => 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,5,3,2] => 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => [5,1,4,3,2] => 5
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,1,5,4,2] => 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,5,4,2] => 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => [5,1,4,3,2] => 5
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => [4,1,5,3,2] => 4
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => [4,1,5,3,2] => 4
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => [5,1,4,3,2] => 5
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,5,4,3] => 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => [5,1,4,3,2] => 5
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,5,4,3] => 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,5,4,3] => 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => [5,2,1,4,3] => 5
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => [4,5,1,3,2] => 4
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => [4,2,1,5,3] => 4
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => [5,4,1,3,2] => 5
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => [3,5,1,4,2] => 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => [3,5,1,4,2] => 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => [5,2,1,4,3] => 5
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [3,5,1,4,2] => 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,5,4] => 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => [5,3,2,1,4] => 5
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => [4,5,2,1,3] => 4
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [6,1,2,3,4,5,7] => [6,1,7,5,4,3,2] => ? = 6
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [5,1,2,3,4,7,6] => [5,1,7,6,4,3,2] => ? = 5
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [5,1,2,3,4,6,7] => [5,1,7,6,4,3,2] => ? = 5
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [7,1,2,3,5,4,6] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [6,1,2,3,7,4,5] => [6,1,7,5,4,3,2] => ? = 6
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [6,1,2,3,5,4,7] => [6,1,7,5,4,3,2] => ? = 6
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [4,1,2,3,7,5,6] => [4,1,7,6,5,3,2] => ? = 4
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [4,1,2,3,6,5,7] => [4,1,7,6,5,3,2] => ? = 4
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [7,1,2,3,5,6,4] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [4,1,2,3,7,6,5] => [4,1,7,6,5,3,2] => ? = 4
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [4,1,2,3,5,7,6] => [4,1,7,6,5,3,2] => ? = 4
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [4,1,2,3,5,6,7] => [4,1,7,6,5,3,2] => ? = 4
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [7,1,2,4,3,5,6] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [6,1,2,7,3,5,4] => [6,1,7,5,4,3,2] => ? = 6
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [6,1,2,4,3,5,7] => [6,1,7,5,4,3,2] => ? = 6
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [5,1,2,7,3,4,6] => [5,1,7,6,4,3,2] => ? = 5
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [5,1,2,6,3,4,7] => [5,1,7,6,4,3,2] => ? = 5
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [7,1,2,4,3,6,5] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [5,1,2,7,3,6,4] => [5,1,7,6,4,3,2] => ? = 5
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [5,1,2,4,3,7,6] => [5,1,7,6,4,3,2] => ? = 5
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [5,1,2,4,3,6,7] => [5,1,7,6,4,3,2] => ? = 5
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [7,1,2,5,4,3,6] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [6,1,2,7,4,3,5] => [6,1,7,5,4,3,2] => ? = 6
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [6,1,2,5,4,3,7] => [6,1,7,5,4,3,2] => ? = 6
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [7,1,2,6,4,5,3] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [3,1,2,7,4,5,6] => [3,1,7,6,5,4,2] => ? = 3
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [3,1,2,6,4,5,7] => [3,1,7,6,5,4,2] => ? = 3
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [7,1,2,5,4,6,3] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [3,1,2,7,4,6,5] => [3,1,7,6,5,4,2] => ? = 3
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [3,1,2,5,4,7,6] => [3,1,7,6,5,4,2] => ? = 3
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [3,1,2,5,4,6,7] => [3,1,7,6,5,4,2] => ? = 3
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [7,1,2,4,5,3,6] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [6,1,2,7,5,3,4] => [6,1,7,5,4,3,2] => ? = 6
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [6,1,2,4,7,3,5] => [6,1,7,5,4,3,2] => ? = 6
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [6,1,2,4,5,3,7] => [6,1,7,5,4,3,2] => ? = 6
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [7,1,2,6,5,4,3] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [3,1,2,7,5,4,6] => [3,1,7,6,5,4,2] => ? = 3
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [3,1,2,6,7,4,5] => [3,1,7,6,5,4,2] => ? = 3
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [3,1,2,6,5,4,7] => [3,1,7,6,5,4,2] => ? = 3
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [7,1,2,4,6,5,3] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [3,1,2,7,6,5,4] => [3,1,7,6,5,4,2] => ? = 3
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [3,1,2,4,7,5,6] => [3,1,7,6,5,4,2] => ? = 3
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [3,1,2,4,6,5,7] => [3,1,7,6,5,4,2] => ? = 3
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [7,1,2,4,5,6,3] => [7,1,6,5,4,3,2] => ? = 7
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [3,1,2,7,5,6,4] => [3,1,7,6,5,4,2] => ? = 3
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [3,1,2,4,7,6,5] => [3,1,7,6,5,4,2] => ? = 3
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000740
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 64%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 64%
Values
{{1}}
=> [1] => [.,.]
=> [1] => 1
{{1,2}}
=> [2,1] => [[.,.],.]
=> [1,2] => 2
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> [2,1] => 1
{{1,2,3}}
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 3
{{1,2},{3}}
=> [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 2
{{1,3},{2}}
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 3
{{1},{2,3}}
=> [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
{{1},{2},{3}}
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 1
{{1,2,3,4}}
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 4
{{1,2},{3,4}}
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2
{{1,3,4},{2}}
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 4
{{1,3},{2,4}}
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 3
{{1,4},{2,3}}
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 4
{{1},{2,3,4}}
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 4
{{1},{2,4},{3}}
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 5
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => 5
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 4
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 4
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 5
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => 5
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 5
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 4
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 4
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 5
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => 5
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 5
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 4
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => ? = 7
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ? = 6
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [5,6,4,3,2,1,7] => ? = 7
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [4,3,2,1,6,7,5] => ? = 5
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [4,6,5,3,2,1,7] => ? = 7
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ? = 6
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [[.,[.,[.,[[.,.],.]]]],[.,.]]
=> [4,5,3,2,1,7,6] => ? = 6
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [[.,[.,[.,[[[.,.],.],.]]]],.]
=> [4,5,6,3,2,1,7] => ? = 7
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [3,2,1,6,5,7,4] => ? = 4
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [3,2,1,5,7,6,4] => ? = 4
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [4,6,5,3,2,1,7] => ? = 7
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [3,2,1,5,6,7,4] => ? = 4
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [3,2,1,6,7,5,4] => ? = 4
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 4
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [3,6,5,4,2,1,7] => ? = 7
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [[.,[.,[.,[[.,.],.]]]],[.,.]]
=> [4,5,3,2,1,7,6] => ? = 6
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [[.,[.,[[.,.],[.,.]]]],[.,.]]
=> [3,5,4,2,1,7,6] => ? = 6
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [[.,[.,[[.,[.,[.,.]]],.]]],.]
=> [5,4,3,6,2,1,7] => ? = 7
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [4,3,2,1,6,7,5] => ? = 5
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ? = 5
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [3,5,6,4,2,1,7] => ? = 7
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [4,3,2,1,6,7,5] => ? = 5
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [[.,[.,[[.,.],.]]],[[.,.],.]]
=> [3,4,2,1,6,7,5] => ? = 5
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [[.,[.,[[.,.],.]]],[.,[.,.]]]
=> [3,4,2,1,7,6,5] => ? = 5
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [[.,[.,[[[.,.],.],[.,.]]]],.]
=> [3,4,6,5,2,1,7] => ? = 7
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [[.,[.,[[.,.],[.,.]]]],[.,.]]
=> [3,5,4,2,1,7,6] => ? = 6
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [[.,[.,[[[.,.],.],.]]],[.,.]]
=> [3,4,5,2,1,7,6] => ? = 6
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [[.,[.,[[[.,.],[.,.]],.]]],.]
=> [3,5,4,6,2,1,7] => ? = 7
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [[.,[.,.]],[[.,[.,[.,.]]],.]]
=> [2,1,6,5,4,7,3] => ? = 3
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [2,1,5,4,7,6,3] => ? = 3
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [[.,[.,[[[.,.],.],[.,.]]]],.]
=> [3,4,6,5,2,1,7] => ? = 7
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [[.,[.,.]],[[.,[[.,.],.]],.]]
=> [2,1,5,6,4,7,3] => ? = 3
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [2,1,4,6,7,5,3] => ? = 3
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,7,6,5,3] => ? = 3
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [3,6,5,4,2,1,7] => ? = 7
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [[.,[.,[[.,[.,.]],.]]],[.,.]]
=> [4,3,5,2,1,7,6] => ? = 6
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [[.,[.,[[.,.],[.,.]]]],[.,.]]
=> [3,5,4,2,1,7,6] => ? = 6
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [[.,[.,[[.,.],[.,.]]]],[.,.]]
=> [3,5,4,2,1,7,6] => ? = 6
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [[.,[.,[[[[.,.],.],.],.]]],.]
=> [3,4,5,6,2,1,7] => ? = 7
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [[.,[.,.]],[[[.,.],[.,.]],.]]
=> [2,1,4,6,5,7,3] => ? = 3
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [2,1,5,4,7,6,3] => ? = 3
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [[.,[.,.]],[[[.,.],.],[.,.]]]
=> [2,1,4,5,7,6,3] => ? = 3
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [3,5,6,4,2,1,7] => ? = 7
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [[.,[.,.]],[[[[.,.],.],.],.]]
=> [2,1,4,5,6,7,3] => ? = 3
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [2,1,6,5,7,4,3] => ? = 3
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [2,1,5,7,6,4,3] => ? = 3
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [3,6,5,4,2,1,7] => ? = 7
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [[.,[.,.]],[[[.,.],[.,.]],.]]
=> [2,1,4,6,5,7,3] => ? = 3
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [[.,[.,.]],[.,[[[.,.],.],.]]]
=> [2,1,5,6,7,4,3] => ? = 3
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St000066
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 73%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 73%
Values
{{1}}
=> [1] => [1] => [[1]]
=> 1
{{1,2}}
=> [2,1] => [2,1] => [[0,1],[1,0]]
=> 2
{{1},{2}}
=> [1,2] => [1,2] => [[1,0],[0,1]]
=> 1
{{1,2,3}}
=> [2,3,1] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 2
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 4
{{1,2,3},{4}}
=> [2,3,1,4] => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3
{{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 4
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 4
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,5,4,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 5
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,5,1,4,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,5,1,4,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0]]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 5
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,5,4,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> 4
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,5,4,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> 4
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 5
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [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]]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [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]]
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 5
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [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]]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [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]]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [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]]
=> 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 5
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,5,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 4
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,5,1,4] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 4
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,5,4,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 5
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [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]]
=> 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [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]]
=> 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 5
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [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]]
=> 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [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]]
=> 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [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]]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> 5
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [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]]
=> 4
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [2,6,5,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 5
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [2,6,5,1,4,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 4
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [2,6,5,1,4,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 4
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [2,6,5,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 5
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [2,6,5,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 5
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [2,6,1,5,4,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 3
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [2,6,1,5,4,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 3
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [2,6,1,5,4,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 3
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [2,6,1,5,4,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 3
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [2,6,1,5,4,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 3
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [2,6,5,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 5
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [2,6,5,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 5
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [2,6,5,1,4,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 4
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [2,6,5,1,4,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 4
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [2,6,5,1,4,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 4
{{1,2,4},{3},{5,6}}
=> [2,4,3,1,6,5] => [2,6,5,1,4,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 4
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [2,6,5,1,4,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 4
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [2,6,5,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 5
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [2,6,5,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 5
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [2,6,5,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 5
{{1,2,5},{3},{4,6}}
=> [2,5,3,6,1,4] => [2,6,5,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 5
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [2,6,5,4,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 5
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [3,2,6,5,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 5
{{1,3,4},{2,5,6}}
=> [3,5,4,1,6,2] => [3,6,5,1,4,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 4
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [3,6,5,1,4,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 4
{{1,3,4},{2,6},{5}}
=> [3,6,4,1,5,2] => [3,6,5,1,4,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 4
{{1,3,4},{2},{5,6}}
=> [3,2,4,1,6,5] => [3,2,6,1,5,4] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 4
{{1,3,4},{2},{5},{6}}
=> [3,2,4,1,5,6] => [3,2,6,1,5,4] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 4
{{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => [3,6,5,2,4,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 6
{{1,3,5},{2,4},{6}}
=> [3,4,5,2,1,6] => [3,6,5,2,1,4] => [[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 5
{{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => [3,6,5,2,4,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 6
{{1,3,5},{2},{4,6}}
=> [3,2,5,6,1,4] => [3,2,6,5,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 5
{{1,3,5},{2},{4},{6}}
=> [3,2,5,4,1,6] => [3,2,6,5,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 5
{{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [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]]
=> ? = 5
{{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [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]]
=> ? = 4
{{1,4},{2,3,5},{6}}
=> [4,3,5,1,2,6] => [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]]
=> ? = 4
{{1,4},{2,3,6},{5}}
=> [4,3,6,1,5,2] => [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]]
=> ? = 4
{{1,5,6},{2,3,4}}
=> [5,3,4,2,6,1] => [5,3,6,2,4,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 6
{{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => [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]]
=> ? = 5
{{1,5},{2,3,4},{6}}
=> [5,3,4,2,1,6] => [5,3,6,2,1,4] => [[0,0,0,0,1,0],[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]]
=> ? = 5
{{1,6},{2,3,4},{5}}
=> [6,3,4,2,5,1] => [6,3,5,2,4,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 6
{{1,5,6},{2,3},{4}}
=> [5,3,2,4,6,1] => [5,3,2,6,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 6
{{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => [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]]
=> ? = 5
{{1,4,5,6},{2},{3}}
=> [4,2,3,5,6,1] => [4,2,6,5,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 6
{{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => [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]]
=> ? = 5
{{1,4,5},{2},{3},{6}}
=> [4,2,3,5,1,6] => [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]]
=> ? = 5
{{1,4,6},{2,5},{3}}
=> [4,5,3,6,2,1] => [4,6,3,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 6
{{1,4},{2,5,6},{3}}
=> [4,5,3,1,6,2] => [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]]
=> ? = 4
{{1,4},{2,5},{3},{6}}
=> [4,5,3,1,2,6] => [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]]
=> ? = 4
{{1,4,6},{2},{3,5}}
=> [4,2,5,6,3,1] => [4,2,6,5,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 6
Description
The column of the unique '1' in the first row of the alternating sign matrix.
The generating function of this statistic is given by
$$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$$
see [2].
Matching statistic: St001497
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001497: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 64%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001497: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 64%
Values
{{1}}
=> [1] => [1] => [1] => 1
{{1,2}}
=> [2,1] => [2,1] => [1,2] => 2
{{1},{2}}
=> [1,2] => [1,2] => [2,1] => 1
{{1,2,3}}
=> [2,3,1] => [2,3,1] => [2,1,3] => 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,3,1] => 2
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [1,2,3] => 3
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [3,1,2] => 1
{{1},{2},{3}}
=> [1,2,3] => [1,3,2] => [3,1,2] => 1
{{1,2,3,4}}
=> [2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 4
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 4
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 3
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 4
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,5,4,3,1] => [4,1,2,3,5] => 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,5,4,1,3] => [4,1,2,5,3] => 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,5,4,3,1] => [4,1,2,3,5] => 5
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,5,1,4,3] => [4,1,5,2,3] => 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,5,1,4,3] => [4,1,5,2,3] => 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,5,4,3,1] => [4,1,2,3,5] => 5
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,5,4,1,3] => [4,1,2,5,3] => 4
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,5,4,1,3] => [4,1,2,5,3] => 4
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,5,4,3,1] => [4,1,2,3,5] => 5
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => [4,5,1,2,3] => 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,5,4,3] => [4,5,1,2,3] => 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,4,3,1] => [4,1,2,3,5] => 5
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [4,5,1,2,3] => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,5,4,3] => [4,5,1,2,3] => 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,5,4,3] => [4,5,1,2,3] => 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,5,4,1] => [3,4,1,2,5] => 5
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,5,4,1,2] => [3,1,2,5,4] => 4
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,5,1,4] => [3,4,1,5,2] => 4
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,5,4,2,1] => [3,1,2,4,5] => 5
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,4,2] => [3,1,5,2,4] => 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,5,1,4,2] => [3,1,5,2,4] => 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,2,5,4,1] => [3,4,1,2,5] => 5
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [3,1,5,2,4] => 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => [3,4,5,1,2] => 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,5,4] => [3,4,5,1,2] => 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => [2,3,4,1,5] => 5
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,5,1,2] => [2,3,1,5,4] => 4
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [2,7,6,5,4,1,3] => [6,1,2,3,4,7,5] => ? = 6
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [2,7,6,5,1,4,3] => [6,1,2,3,7,4,5] => ? = 5
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [2,7,6,5,1,4,3] => [6,1,2,3,7,4,5] => ? = 5
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [2,7,6,5,4,1,3] => [6,1,2,3,4,7,5] => ? = 6
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [2,7,6,5,4,1,3] => [6,1,2,3,4,7,5] => ? = 6
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 4
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 4
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 4
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 4
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [2,7,6,1,5,4,3] => [6,1,2,7,3,4,5] => ? = 4
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [2,7,6,5,4,1,3] => [6,1,2,3,4,7,5] => ? = 6
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [2,7,6,5,4,1,3] => [6,1,2,3,4,7,5] => ? = 6
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [2,7,6,5,1,4,3] => [6,1,2,3,7,4,5] => ? = 5
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [2,7,6,5,1,4,3] => [6,1,2,3,7,4,5] => ? = 5
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [2,7,6,5,1,4,3] => [6,1,2,3,7,4,5] => ? = 5
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [2,7,6,5,1,4,3] => [6,1,2,3,7,4,5] => ? = 5
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [2,7,6,5,1,4,3] => [6,1,2,3,7,4,5] => ? = 5
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [2,7,6,5,4,1,3] => [6,1,2,3,4,7,5] => ? = 6
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [2,7,6,5,4,1,3] => [6,1,2,3,4,7,5] => ? = 6
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 3
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 3
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 3
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 3
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 3
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [2,7,6,5,4,1,3] => [6,1,2,3,4,7,5] => ? = 6
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [2,7,6,5,4,1,3] => [6,1,2,3,4,7,5] => ? = 6
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [2,7,6,5,4,1,3] => [6,1,2,3,4,7,5] => ? = 6
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 3
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 3
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 3
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 3
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 3
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 3
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [2,7,6,5,4,3,1] => [6,1,2,3,4,5,7] => ? = 7
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 3
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [2,7,1,6,5,4,3] => [6,1,7,2,3,4,5] => ? = 3
Description
The position of the largest weak excedence of a permutation.
Matching statistic: St000051
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 55%
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 55%
Values
{{1}}
=> [1] => [.,.]
=> 0 = 1 - 1
{{1,2}}
=> [2,1] => [[.,.],.]
=> 1 = 2 - 1
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [[.,[.,.]],.]
=> 2 = 3 - 1
{{1,2},{3}}
=> [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
{{1,3},{2}}
=> [3,2,1] => [[[.,.],.],.]
=> 2 = 3 - 1
{{1},{2,3}}
=> [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
{{1},{2},{3}}
=> [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 3 = 4 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [[.,[[.,.],.]],.]
=> 3 = 4 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> 3 = 4 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 2 = 3 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 3 = 4 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> 3 = 4 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0 = 1 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> 4 = 5 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 4 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> 4 = 5 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> 2 = 3 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 3 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> 4 = 5 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 4 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> 3 = 4 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> 4 = 5 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> 1 = 2 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 1 = 2 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> 4 = 5 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 1 = 2 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 1 = 2 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 2 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> 4 = 5 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [[.,[[.,.],.]],[.,.]]
=> 3 = 4 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> 3 = 4 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> 4 = 5 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> 2 = 3 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 3 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> 4 = 5 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> 2 = 3 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> 2 = 3 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> 2 = 3 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> 4 = 5 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> 3 = 4 - 1
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> ? = 7 - 1
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> ? = 6 - 1
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> ? = 7 - 1
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> ? = 5 - 1
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> ? = 5 - 1
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> ? = 7 - 1
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> ? = 6 - 1
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [[.,[.,[.,[[.,.],.]]]],[.,.]]
=> ? = 6 - 1
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [[.,[.,[.,[[[.,.],.],.]]]],.]
=> ? = 7 - 1
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> ? = 4 - 1
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> ? = 4 - 1
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> ? = 7 - 1
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [[.,[.,[.,.]]],[[[.,.],.],.]]
=> ? = 4 - 1
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> ? = 4 - 1
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> ? = 4 - 1
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> ? = 7 - 1
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [[.,[.,[.,[[.,.],.]]]],[.,.]]
=> ? = 6 - 1
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [[.,[.,[[.,.],[.,.]]]],[.,.]]
=> ? = 6 - 1
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [[.,[.,[[.,[.,[.,.]]],.]]],.]
=> ? = 7 - 1
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> ? = 5 - 1
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> ? = 5 - 1
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [[.,[.,[[.,.],[[.,.],.]]]],.]
=> ? = 7 - 1
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> ? = 5 - 1
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [[.,[.,[[.,.],.]]],[[.,.],.]]
=> ? = 5 - 1
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [[.,[.,[[.,.],.]]],[.,[.,.]]]
=> ? = 5 - 1
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [[.,[.,[[[.,.],.],[.,.]]]],.]
=> ? = 7 - 1
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [[.,[.,[[.,.],[.,.]]]],[.,.]]
=> ? = 6 - 1
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [[.,[.,[[[.,.],.],.]]],[.,.]]
=> ? = 6 - 1
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [[.,[.,[[[.,.],[.,.]],.]]],.]
=> ? = 7 - 1
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [[.,[.,.]],[[.,[.,[.,.]]],.]]
=> ? = 3 - 1
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> ? = 3 - 1
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [[.,[.,[[[.,.],.],[.,.]]]],.]
=> ? = 7 - 1
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [[.,[.,.]],[[.,[[.,.],.]],.]]
=> ? = 3 - 1
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> ? = 3 - 1
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> ? = 3 - 1
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> ? = 7 - 1
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [[.,[.,[[.,[.,.]],.]]],[.,.]]
=> ? = 6 - 1
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [[.,[.,[[.,.],[.,.]]]],[.,.]]
=> ? = 6 - 1
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [[.,[.,[[.,.],[.,.]]]],[.,.]]
=> ? = 6 - 1
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [[.,[.,[[[[.,.],.],.],.]]],.]
=> ? = 7 - 1
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [[.,[.,.]],[[[.,.],[.,.]],.]]
=> ? = 3 - 1
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> ? = 3 - 1
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [[.,[.,.]],[[[.,.],.],[.,.]]]
=> ? = 3 - 1
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [[.,[.,[[.,.],[[.,.],.]]]],.]
=> ? = 7 - 1
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [[.,[.,.]],[[[[.,.],.],.],.]]
=> ? = 3 - 1
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [[.,[.,.]],[.,[[.,[.,.]],.]]]
=> ? = 3 - 1
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> ? = 3 - 1
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> ? = 7 - 1
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [[.,[.,.]],[[[.,.],[.,.]],.]]
=> ? = 3 - 1
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [[.,[.,.]],[.,[[[.,.],.],.]]]
=> ? = 3 - 1
Description
The size of the left subtree of a binary tree.
Matching statistic: St001291
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 55%
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 55%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 1
{{1,2}}
=> [2,1] => [1,2] => [1,0,1,0]
=> 2
{{1},{2}}
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
{{1,2,3}}
=> [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> 3
{{1,2},{3}}
=> [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3
{{1},{2,3}}
=> [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
{{1,2,4},{3}}
=> [2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 4
{{1,2},{3,4}}
=> [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
{{1,3,4},{2}}
=> [3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 4
{{1,3},{2,4}}
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
{{1,4},{2,3}}
=> [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 4
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 4
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 5
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 5
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 4
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 4
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 5
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 3
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 5
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 5
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 4
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 6
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [6,5,4,3,1,2,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [6,5,4,3,7,1,2] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 5
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [6,5,4,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 5
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [6,5,4,2,3,1,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [6,5,4,2,1,7,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 6
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [6,5,4,2,3,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 6
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [6,5,4,1,2,3,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [6,5,4,7,2,1,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 4
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [6,5,4,7,2,3,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 4
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [6,5,4,1,3,2,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [6,5,4,7,1,2,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 4
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [6,5,4,7,3,1,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 4
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [6,5,4,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 4
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [6,5,3,4,2,1,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [6,5,3,1,2,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 6
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [6,5,3,4,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 6
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [6,5,3,2,1,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [6,5,3,2,7,1,4] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 5
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [6,5,3,2,7,4,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 5
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [6,5,3,4,1,2,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [6,5,3,1,7,2,4] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 5
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [6,5,3,4,7,1,2] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 5
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [6,5,3,4,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 5
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [6,5,2,3,4,1,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [6,5,2,3,1,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 6
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [6,5,2,3,4,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 6
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [6,5,1,3,2,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [6,5,7,3,2,1,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [6,5,7,3,2,4,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [6,5,1,3,4,2,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [6,5,7,3,1,2,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [6,5,7,3,4,1,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [6,5,7,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [6,5,2,4,3,1,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [6,5,2,1,3,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 6
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [6,5,2,4,1,7,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 6
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [6,5,2,4,3,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 6
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [6,5,1,2,3,4,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [6,5,7,2,3,1,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [6,5,7,2,1,4,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [6,5,7,2,3,4,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [6,5,1,4,2,3,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [6,5,7,1,2,3,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [6,5,7,4,2,1,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [6,5,7,4,2,3,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [6,5,1,4,3,2,7] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 7
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [6,5,7,1,3,2,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [6,5,7,4,1,2,3] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000133The "bounce" of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!