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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000730
St000730: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 1
{{1},{2}}
=> 0
{{1,2,3}}
=> 1
{{1,2},{3}}
=> 1
{{1,3},{2}}
=> 2
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 1
{{1,2,3},{4}}
=> 1
{{1,2,4},{3}}
=> 2
{{1,2},{3,4}}
=> 1
{{1,2},{3},{4}}
=> 1
{{1,3,4},{2}}
=> 2
{{1,3},{2,4}}
=> 2
{{1,3},{2},{4}}
=> 2
{{1,4},{2,3}}
=> 3
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 3
{{1},{2,4},{3}}
=> 2
{{1},{2},{3,4}}
=> 1
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 1
{{1,2,3,4},{5}}
=> 1
{{1,2,3,5},{4}}
=> 2
{{1,2,3},{4,5}}
=> 1
{{1,2,3},{4},{5}}
=> 1
{{1,2,4,5},{3}}
=> 2
{{1,2,4},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> 2
{{1,2,5},{3,4}}
=> 3
{{1,2},{3,4,5}}
=> 1
{{1,2},{3,4},{5}}
=> 1
{{1,2,5},{3},{4}}
=> 3
{{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> 1
{{1,2},{3},{4},{5}}
=> 1
{{1,3,4,5},{2}}
=> 2
{{1,3,4},{2,5}}
=> 3
{{1,3,4},{2},{5}}
=> 2
{{1,3,5},{2,4}}
=> 2
{{1,3},{2,4,5}}
=> 2
{{1,3},{2,4},{5}}
=> 2
{{1,3,5},{2},{4}}
=> 2
{{1,3},{2,5},{4}}
=> 3
{{1,3},{2},{4,5}}
=> 2
{{1,3},{2},{4},{5}}
=> 2
{{1,4,5},{2,3}}
=> 3
{{1,4},{2,3,5}}
=> 3
{{1,4},{2,3},{5}}
=> 3
Description
The maximal arc length of a set partition.
The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the maximal arc length is $0$.
Matching statistic: St000442
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(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 0
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [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]
=> [1,1,0,1,0,0,1,0]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
{{1},{2,4,5,6,7,8},{3}}
=> [1,4,3,5,6,7,8,2] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
{{1},{2,3,5,6,7,8},{4}}
=> [1,3,5,4,6,7,8,2] => [1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 2
{{1},{2,3,4,6,7,8},{5}}
=> [1,3,4,6,5,7,8,2] => [1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2
{{1},{2,3,4,5,7,8},{6}}
=> [1,3,4,5,7,6,8,2] => [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2
{{1},{2,3,4,5,6,7},{8}}
=> ? => ?
=> ?
=> ? = 1
{{1},{2,3,4,5,6,8},{7}}
=> [1,3,4,5,6,8,7,2] => [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
{{1,4,5,6,7,8},{2},{3}}
=> [4,2,3,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
{{1,3,4,5,6,7,8},{2}}
=> [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
{{1,4,5,6,7,8},{2,3}}
=> [4,3,2,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
{{1,2,4,5,6,7,8},{3}}
=> [2,4,3,5,6,7,8,1] => [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
{{1,2,5,6,7,8},{3,4}}
=> [2,5,4,3,6,7,8,1] => [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3
{{1,2,3,5,6,7,8},{4}}
=> [2,3,5,4,6,7,8,1] => [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 2
{{1,2,3,6,7,8},{4,5}}
=> [2,3,6,5,4,7,8,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 3
{{1,2,3,4,6,7,8},{5}}
=> [2,3,4,6,5,7,8,1] => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 2
{{1,2,3,4,7,8},{5,6}}
=> [2,3,4,7,6,5,8,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
{{1,2,3,4,5,7,8},{6}}
=> [2,3,4,5,7,6,8,1] => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 2
{{1,8},{2,3,4,5,6,7}}
=> [8,3,4,5,6,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
{{1,2,3,4,5,8},{6,7}}
=> [2,3,4,5,8,7,6,1] => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 3
{{1,2,3,4,5,6,8},{7}}
=> [2,3,4,5,6,8,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2
{{1,3,4,6,7,8},{2,5}}
=> [3,5,4,6,2,7,8,1] => [1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 3
{{1,3,4,5,7,8},{2,6}}
=> [3,6,4,5,7,2,8,1] => [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 4
{{1,2,4,5,7,8},{3,6}}
=> [2,4,6,5,7,3,8,1] => [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 3
{{1,3,4,5,6,8},{2,7}}
=> [3,7,4,5,6,8,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 5
{{1,2,4,5,6,8},{3,7}}
=> [2,4,7,5,6,8,3,1] => [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 4
{{1,2,3,5,6,8},{4,7}}
=> [2,3,5,7,6,8,4,1] => [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? = 3
{{1,3,4,5,6,7},{2,8}}
=> [3,8,4,5,6,7,1,2] => ?
=> ?
=> ? = 2
{{1,2,4,5,6,7},{3,8}}
=> [2,4,8,5,6,7,1,3] => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2
{{1,2,3,5,6,7},{4,8}}
=> [2,3,5,8,6,7,1,4] => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 2
{{1,2,3,4,6,7},{5,8}}
=> [2,3,4,6,8,7,1,5] => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
{{1,4},{2,3,5,6,7,8}}
=> [4,3,5,1,6,7,8,2] => [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
{{1,5},{2,3,4,6,7,8}}
=> [5,3,4,6,1,7,8,2] => ?
=> ?
=> ? = 4
{{1,6},{2,3,4,5,7,8}}
=> [6,3,4,5,7,1,8,2] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 5
{{1,7},{2,3,4,5,6,8}}
=> [7,3,4,5,6,8,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 6
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St001203
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 2 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
{{1},{2},{3,4,5,6,7,8}}
=> [1,2,4,5,6,7,8,3] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
{{1},{2,4,5,6,7,8},{3}}
=> [1,4,3,5,6,7,8,2] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> ? = 2 + 1
{{1},{2,3,5,6,7,8},{4}}
=> [1,3,5,4,6,7,8,2] => [1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2 + 1
{{1},{2,3,4,6,7,8},{5}}
=> [1,3,4,6,5,7,8,2] => [1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2 + 1
{{1},{2,3,4,5,7,8},{6}}
=> [1,3,4,5,7,6,8,2] => [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2 + 1
{{1},{2,3,4,5,6,7},{8}}
=> ? => ?
=> ?
=> ? = 1 + 1
{{1},{2,3,4,5,6,8},{7}}
=> [1,3,4,5,6,8,7,2] => [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2 + 1
{{1},{2,3,4,5,6,7,8}}
=> [1,3,4,5,6,7,8,2] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
{{1,2},{3,4,5,6,7,8}}
=> [2,1,4,5,6,7,8,3] => [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 1 + 1
{{1,4,5,6,7,8},{2},{3}}
=> [4,2,3,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 3 + 1
{{1,3,5,6,7,8},{2},{4}}
=> [3,2,5,4,6,7,8,1] => [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,3,4,5,6,7,8},{2}}
=> [3,2,4,5,6,7,8,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,4,5,6,7,8},{2,3}}
=> [4,3,2,5,6,7,8,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 3 + 1
{{1,2,4,5,6,7,8},{3}}
=> [2,4,3,5,6,7,8,1] => [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,2,5,6,7,8},{3,4}}
=> [2,5,4,3,6,7,8,1] => [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 3 + 1
{{1,2,3,5,6,7,8},{4}}
=> [2,3,5,4,6,7,8,1] => [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,2,3,6,7,8},{4,5}}
=> [2,3,6,5,4,7,8,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 3 + 1
{{1,2,3,4,6,7,8},{5}}
=> [2,3,4,6,5,7,8,1] => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,2,3,4,5,6},{7,8}}
=> [2,3,4,5,6,1,8,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
{{1,2,3,4,7,8},{5,6}}
=> [2,3,4,7,6,5,8,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 3 + 1
{{1,2,3,4,5,7,8},{6}}
=> [2,3,4,5,7,6,8,1] => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,2,3,4,5,6,7},{8}}
=> [2,3,4,5,6,7,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 1
{{1,8},{2,3,4,5,6,7}}
=> [8,3,4,5,6,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
{{1,2,3,4,5,8},{6,7}}
=> [2,3,4,5,8,7,6,1] => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 3 + 1
{{1,2,3,4,5,6,8},{7}}
=> [2,3,4,5,6,8,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,2,3,4,5,6,7,8}}
=> [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1 + 1
{{1,3,5,6,7,8},{2,4}}
=> [3,4,5,2,6,7,8,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,3,4,6,7,8},{2,5}}
=> [3,5,4,6,2,7,8,1] => [1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 3 + 1
{{1,2,4,6,7,8},{3,5}}
=> [2,4,5,6,3,7,8,1] => [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,3,4,5,7,8},{2,6}}
=> [3,6,4,5,7,2,8,1] => [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 4 + 1
{{1,2,4,5,7,8},{3,6}}
=> [2,4,6,5,7,3,8,1] => [1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> ? = 3 + 1
{{1,2,3,5,7,8},{4,6}}
=> [2,3,5,6,7,4,8,1] => [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,3,4,5,6,8},{2,7}}
=> [3,7,4,5,6,8,2,1] => [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 5 + 1
{{1,2,4,5,6,8},{3,7}}
=> [2,4,7,5,6,8,3,1] => [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 4 + 1
{{1,2,3,5,6,8},{4,7}}
=> [2,3,5,7,6,8,4,1] => [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
{{1,2,3,4,6,8},{5,7}}
=> [2,3,4,6,7,8,5,1] => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,3,4,5,6,7},{2,8}}
=> [3,8,4,5,6,7,1,2] => ?
=> ?
=> ? = 2 + 1
{{1,2,4,5,6,7},{3,8}}
=> [2,4,8,5,6,7,1,3] => [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
{{1,2,3,5,6,7},{4,8}}
=> [2,3,5,8,6,7,1,4] => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2 + 1
{{1,2,3,4,6,7},{5,8}}
=> [2,3,4,6,8,7,1,5] => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,2,3,4,5,7},{6,8}}
=> [2,3,4,5,7,8,1,6] => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,3},{2,4,5,6,7,8}}
=> [3,4,1,5,6,7,8,2] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 2 + 1
{{1,4},{2,3,5,6,7,8}}
=> [4,3,5,1,6,7,8,2] => [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 3 + 1
{{1,5},{2,3,4,6,7,8}}
=> [5,3,4,6,1,7,8,2] => ?
=> ?
=> ? = 4 + 1
{{1,6},{2,3,4,5,7,8}}
=> [6,3,4,5,7,1,8,2] => [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 5 + 1
{{1,7},{2,3,4,5,6,8}}
=> [7,3,4,5,6,8,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 6 + 1
Description
We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
In the list $L$ delete the first entry $c_0$ and substract from all other entries $n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra.
Example:
[5,6,6,6,6] goes into [2,2,2,2,1].
Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra.
The statistic return the global dimension of the CNakayama algebra divided by 2.
Matching statistic: St000306
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 86%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 86%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 2
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 2
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 1
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 2
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 1
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 2
{{1,2,4,5,7},{3,6}}
=> [2,4,6,5,7,3,1] => [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,2,4,5},{3,6,7}}
=> [2,4,6,5,1,7,3] => [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 3
{{1,2,4,5},{3,6},{7}}
=> [2,4,6,5,1,3,7] => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 3
{{1,2,4,5},{3},{6},{7}}
=> [2,4,3,5,1,6,7] => [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 2
{{1,2,4,6},{3,5},{7}}
=> [2,4,5,6,3,1,7] => [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 2
{{1,2,4,7},{3,5,6}}
=> [2,4,5,7,6,3,1] => [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,2,4},{3,5,6},{7}}
=> [2,4,5,1,6,3,7] => [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 2
{{1,2,4,7},{3,5},{6}}
=> [2,4,5,7,3,6,1] => [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,2,4},{3,5},{6,7}}
=> [2,4,5,1,3,7,6] => [1,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 2
{{1,2,4,6},{3},{5,7}}
=> [2,4,3,6,7,1,5] => [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 2
{{1,2,4,6},{3},{5},{7}}
=> [2,4,3,6,5,1,7] => [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 2
{{1,2,4,7},{3,6},{5}}
=> [2,4,6,7,5,3,1] => [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,2,4},{3,6,7},{5}}
=> [2,4,6,1,5,7,3] => [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 3
{{1,2,4},{3,6},{5,7}}
=> [2,4,6,1,7,3,5] => [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,2,4},{3,6},{5},{7}}
=> [2,4,6,1,5,3,7] => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 3
{{1,2,4,7},{3},{5,6}}
=> [2,4,3,7,6,5,1] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 3
{{1,2,4},{3},{5,6,7}}
=> [2,4,3,1,6,7,5] => [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 2
{{1,2,4},{3},{5,6},{7}}
=> [2,4,3,1,6,5,7] => [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 2
{{1,2,4,7},{3},{5},{6}}
=> [2,4,3,7,5,6,1] => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 3
{{1,2,4},{3},{5},{6,7}}
=> [2,4,3,1,5,7,6] => [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 2
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 3
{{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,2,5,6},{3,4},{7}}
=> [2,5,4,3,6,1,7] => [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 3
{{1,2,5,7},{3,4,6}}
=> [2,5,4,6,7,3,1] => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,2,5},{3,4,6,7}}
=> [2,5,4,6,1,7,3] => [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 3
{{1,2,5},{3,4,6},{7}}
=> [2,5,4,6,1,3,7] => [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 3
{{1,2,5},{3,4,7},{6}}
=> [2,5,4,7,1,6,3] => [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 3
{{1,2,5},{3,4},{6,7}}
=> [2,5,4,3,1,7,6] => [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 3
{{1,2,6},{3,4,5,7}}
=> [2,6,4,5,7,1,3] => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 4
{{1,2},{3,4,5,6},{7}}
=> [2,1,4,5,6,3,7] => [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 1
{{1,2},{3,4,5,7},{6}}
=> [2,1,4,5,7,6,3] => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
{{1,2},{3,4,5},{6,7}}
=> [2,1,4,5,3,7,6] => [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 1
{{1,2},{3,4,5},{6},{7}}
=> [2,1,4,5,3,6,7] => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 1
{{1,2,6},{3,4,7},{5}}
=> [2,6,4,7,5,1,3] => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
{{1,2,6},{3,4},{5,7}}
=> [2,6,4,3,7,1,5] => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 4
{{1,2},{3,4,6,7},{5}}
=> [2,1,4,6,5,7,3] => [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
{{1,2},{3,4,6},{5,7}}
=> [2,1,4,6,7,3,5] => [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 2
{{1,2},{3,4,6},{5},{7}}
=> [2,1,4,6,5,3,7] => [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 2
{{1,2},{3,4,7},{5,6}}
=> [2,1,4,7,6,5,3] => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 3
{{1,2},{3,4,7},{5},{6}}
=> [2,1,4,7,5,6,3] => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 3
{{1,2},{3,4},{5,7},{6}}
=> [2,1,4,3,7,6,5] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
{{1,2},{3,4},{5},{6,7}}
=> [2,1,4,3,5,7,6] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
Description
The bounce count of a Dyck path.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St001046
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001046: Perfect matchings ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 86%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001046: Perfect matchings ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 86%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 2
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 3
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> 3
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 3
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> ? = 1
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11),(13,14)]
=> ? = 1
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,13),(11,12)]
=> ? = 2
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,14),(12,13)]
=> ? = 1
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12),(13,14)]
=> ? = 1
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,11),(9,10),(12,13)]
=> ? = 2
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,13),(9,10),(11,12)]
=> ? = 2
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,11),(9,10),(13,14)]
=> ? = 2
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,13),(9,12),(10,11)]
=> ? = 3
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,14),(10,11),(12,13)]
=> ? = 1
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11),(13,14)]
=> ? = 1
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,13),(9,12),(10,11)]
=> ? = 3
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,14),(10,13),(11,12)]
=> ? = 2
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,14),(12,13)]
=> ? = 1
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12),(13,14)]
=> ? = 1
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,9),(7,8),(10,11),(12,13)]
=> ? = 2
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,8),(9,12),(10,11)]
=> ? = 3
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [(1,12),(2,3),(4,5),(6,9),(7,8),(10,11),(13,14)]
=> ? = 2
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,8),(9,10),(11,12)]
=> ? = 2
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,11),(7,8),(9,10),(12,13)]
=> ? = 2
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10),(13,14)]
=> ? = 2
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [(1,14),(2,3),(4,5),(6,9),(7,8),(10,13),(11,12)]
=> ? = 2
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,8),(9,12),(10,11)]
=> ? = 3
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12),(13,14)]
=> ? = 2
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,11),(7,10),(8,9),(12,13)]
=> ? = 3
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,10),(8,9),(11,12)]
=> ? = 3
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9),(13,14)]
=> ? = 3
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ? = 4
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,9),(10,11),(12,13)]
=> ? = 1
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11),(13,14)]
=> ? = 1
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ? = 4
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,9),(10,13),(11,12)]
=> ? = 2
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,14),(12,13)]
=> ? = 1
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,11),(7,10),(8,9),(12,13)]
=> ? = 3
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,12),(8,9),(10,11)]
=> ? = 3
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,10),(8,9),(11,12)]
=> ? = 3
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9),(13,14)]
=> ? = 3
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ? = 4
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,11),(9,10),(12,13)]
=> ? = 2
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,13),(9,10),(11,12)]
=> ? = 2
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10),(13,14)]
=> ? = 2
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ? = 4
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 3
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,14),(10,11),(12,13)]
=> ? = 1
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11),(13,14)]
=> ? = 1
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ? = 4
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 3
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,14),(10,13),(11,12)]
=> ? = 2
Description
The maximal number of arcs nesting a given arc of a perfect matching.
This is also the largest weight of a down step in the histoire d'Hermite corresponding to the perfect matching.
Matching statistic: St000720
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000720: Perfect matchings ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 86%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000720: Perfect matchings ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 86%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 2 = 1 + 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2 = 1 + 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2 = 1 + 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 3 = 2 + 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 2 = 1 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 3 = 2 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 3 = 2 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 3 = 2 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3 = 2 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 3 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 4 = 3 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 3 = 2 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 2 = 1 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> 3 = 2 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> 2 = 1 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> 3 = 2 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> 3 = 2 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> 4 = 3 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> 2 = 1 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> 4 = 3 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 3 = 2 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 4 = 3 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> 3 = 2 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> 3 = 2 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 4 = 3 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 3 = 2 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> 4 = 3 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> 4 = 3 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 4 = 3 + 1
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> ? = 1 + 1
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11),(13,14)]
=> ? = 1 + 1
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,9),(10,13),(11,12)]
=> ? = 2 + 1
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,14),(12,13)]
=> ? = 1 + 1
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12),(13,14)]
=> ? = 1 + 1
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,11),(9,10),(12,13)]
=> ? = 2 + 1
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,13),(9,10),(11,12)]
=> ? = 2 + 1
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,11),(9,10),(13,14)]
=> ? = 2 + 1
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,13),(9,12),(10,11)]
=> ? = 3 + 1
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,14),(10,11),(12,13)]
=> ? = 1 + 1
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11),(13,14)]
=> ? = 1 + 1
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,7),(8,13),(9,12),(10,11)]
=> ? = 3 + 1
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,14),(10,13),(11,12)]
=> ? = 2 + 1
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,14),(12,13)]
=> ? = 1 + 1
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12),(13,14)]
=> ? = 1 + 1
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,9),(7,8),(10,11),(12,13)]
=> ? = 2 + 1
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,8),(9,12),(10,11)]
=> ? = 3 + 1
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [(1,12),(2,3),(4,5),(6,9),(7,8),(10,11),(13,14)]
=> ? = 2 + 1
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,8),(9,10),(11,12)]
=> ? = 2 + 1
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,11),(7,8),(9,10),(12,13)]
=> ? = 2 + 1
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10),(13,14)]
=> ? = 2 + 1
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [(1,14),(2,3),(4,5),(6,9),(7,8),(10,13),(11,12)]
=> ? = 2 + 1
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,8),(9,12),(10,11)]
=> ? = 3 + 1
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,14),(12,13)]
=> ? = 2 + 1
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12),(13,14)]
=> ? = 2 + 1
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,11),(7,10),(8,9),(12,13)]
=> ? = 3 + 1
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,10),(8,9),(11,12)]
=> ? = 3 + 1
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9),(13,14)]
=> ? = 3 + 1
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ? = 4 + 1
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,9),(10,11),(12,13)]
=> ? = 1 + 1
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11),(13,14)]
=> ? = 1 + 1
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ? = 4 + 1
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,9),(10,13),(11,12)]
=> ? = 2 + 1
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,14),(12,13)]
=> ? = 1 + 1
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1 + 1
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [(1,14),(2,3),(4,5),(6,11),(7,10),(8,9),(12,13)]
=> ? = 3 + 1
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,12),(8,9),(10,11)]
=> ? = 3 + 1
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,10),(8,9),(11,12)]
=> ? = 3 + 1
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9),(13,14)]
=> ? = 3 + 1
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ? = 4 + 1
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,11),(9,10),(12,13)]
=> ? = 2 + 1
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,13),(9,10),(11,12)]
=> ? = 2 + 1
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10),(13,14)]
=> ? = 2 + 1
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ? = 4 + 1
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 3 + 1
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,14),(10,11),(12,13)]
=> ? = 1 + 1
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11),(13,14)]
=> ? = 1 + 1
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ? = 4 + 1
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,6),(2,3),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 3 + 1
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,14),(10,13),(11,12)]
=> ? = 2 + 1
Description
The size of the largest partition in the oscillating tableau corresponding to the perfect matching.
Equivalently, this is the maximal number of crosses in the corresponding triangular rook filling that can be covered by a rectangle.
Matching statistic: St000141
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
St000141: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => 1
{{1},{2}}
=> [1,2] => 0
{{1,2,3}}
=> [2,3,1] => 1
{{1,2},{3}}
=> [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => 2
{{1},{2,3}}
=> [1,3,2] => 1
{{1},{2},{3}}
=> [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => 1
{{1,2,3},{4}}
=> [2,3,1,4] => 1
{{1,2,4},{3}}
=> [2,4,3,1] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => 1
{{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] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => 3
{{1},{2,3,4}}
=> [1,3,4,2] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => 3
{{1},{2,4},{3}}
=> [1,4,3,2] => 2
{{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] => 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => 3
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 3
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => 3
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => 3
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => ? = 2
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => ? = 2
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => ? = 2
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => ? = 3
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => ? = 1
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => ? = 1
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => ? = 3
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => ? = 2
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => ? = 2
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => ? = 3
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => ? = 2
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => ? = 2
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => ? = 2
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => ? = 2
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => ? = 2
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => ? = 3
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => ? = 2
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => ? = 2
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => ? = 3
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => ? = 3
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => ? = 3
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => ? = 4
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => ? = 1
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => ? = 1
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => ? = 4
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => ? = 2
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => ? = 1
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => ? = 1
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => ? = 3
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => ? = 3
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => ? = 3
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => ? = 3
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => ? = 4
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => ? = 2
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => ? = 2
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => ? = 2
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => ? = 4
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => ? = 3
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => ? = 1
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => ? = 1
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => ? = 4
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => ? = 3
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => ? = 2
{{1,2,3},{4},{5},{6,7}}
=> [2,3,1,4,5,7,6] => ? = 1
{{1,2,4,5,6,7},{3}}
=> [2,4,3,5,6,7,1] => ? = 2
{{1,2,4,5,6},{3,7}}
=> [2,4,7,5,6,1,3] => ? = 4
{{1,2,4,5,6},{3},{7}}
=> [2,4,3,5,6,1,7] => ? = 2
{{1,2,4,5,7},{3,6}}
=> [2,4,6,5,7,3,1] => ? = 3
{{1,2,4,5},{3,6,7}}
=> [2,4,6,5,1,7,3] => ? = 3
{{1,2,4,5},{3,6},{7}}
=> [2,4,6,5,1,3,7] => ? = 3
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000062
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 86%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 86%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [1,2] => 2 = 1 + 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [2,1] => 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 2 = 1 + 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2 = 1 + 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 3 = 2 + 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2 = 1 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2 = 1 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 3 = 2 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2 = 1 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3 = 2 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 3 = 2 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4 = 3 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2 = 1 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2 = 1 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4 = 3 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 2 = 1 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 2 = 1 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 3 = 2 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 2 = 1 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 2 = 1 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 3 = 2 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 3 = 2 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 4 = 3 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2 = 1 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 4 = 3 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 2 = 1 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 3 = 2 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 4 = 3 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 3 = 2 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => 3 = 2 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 4 = 3 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3 = 2 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => 4 = 3 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 4 = 3 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4 = 3 + 1
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => ? = 1 + 1
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,6] => ? = 1 + 1
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => ? = 2 + 1
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,1,5] => ? = 1 + 1
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1,5] => ? = 1 + 1
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,1,7] => ? = 2 + 1
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => ? = 2 + 1
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,1,6] => ? = 2 + 1
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => ? = 3 + 1
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,1,4] => ? = 1 + 1
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,2,1,4] => ? = 1 + 1
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,1,7] => ? = 3 + 1
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,1,4] => ? = 2 + 1
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,2,1,4] => ? = 1 + 1
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1,4] => ? = 1 + 1
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,1,7] => ? = 2 + 1
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,1,7] => ? = 3 + 1
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,1,6] => ? = 2 + 1
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,1,7] => ? = 2 + 1
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,1,7] => ? = 2 + 1
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,2,1,6] => ? = 2 + 1
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,1,7] => ? = 2 + 1
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,1,7] => ? = 3 + 1
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,1,5] => ? = 2 + 1
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,2,1,5] => ? = 2 + 1
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,1,7] => ? = 3 + 1
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,1,7] => ? = 3 + 1
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,2,1,6] => ? = 3 + 1
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,1,7] => ? = 4 + 1
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,1,3] => ? = 1 + 1
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,2,1,3] => ? = 1 + 1
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,1,7] => ? = 4 + 1
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,1,3] => ? = 2 + 1
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,1,3] => ? = 1 + 1
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,1,3] => ? = 1 + 1
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,2,1,7] => ? = 3 + 1
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,1,7] => ? = 3 + 1
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,1,7] => ? = 3 + 1
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,2,1,6] => ? = 3 + 1
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,1,7] => ? = 4 + 1
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,1,3] => ? = 2 + 1
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,1,3] => ? = 2 + 1
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,2,1,3] => ? = 2 + 1
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,1,7] => ? = 4 + 1
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,1,3] => ? = 3 + 1
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,1,3] => ? = 1 + 1
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,1,3] => ? = 1 + 1
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,1,7] => ? = 4 + 1
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,1,3] => ? = 3 + 1
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,1,3] => ? = 2 + 1
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000166
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000166: Ordered trees ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 86%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000166: Ordered trees ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 86%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [[[]]]
=> 2 = 1 + 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [[[],[]]]
=> 2 = 1 + 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 2 = 1 + 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 3 = 2 + 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [[],[[]]]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 2 = 1 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 3 = 2 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 3 = 2 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 3 = 2 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 3 = 2 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 4 = 3 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 4 = 3 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 3 = 2 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> 2 = 1 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 3 = 2 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> 2 = 1 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> 3 = 2 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> 3 = 2 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 4 = 3 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> 2 = 1 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 4 = 3 + 1
{{1,2},{3,5},{4}}
=> [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] => [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> 3 = 2 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> 4 = 3 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[[[]],[]],[]]
=> 3 = 2 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [[[[],[]]],[]]
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [[[[]],[[]]]]
=> 3 = 2 + 1
{{1,3},{2,5},{4}}
=> [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] => [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 3 = 2 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> 4 = 3 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> 4 = 3 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 4 = 3 + 1
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[],[]]]
=> ? = 1 + 1
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[[],[],[],[],[]],[]]
=> ? = 1 + 1
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[[],[],[],[],[[]]]]
=> ? = 2 + 1
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [[[],[],[],[]],[[]]]
=> ? = 1 + 1
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[[],[],[],[]],[],[]]
=> ? = 1 + 1
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[[],[],[],[[]],[]]]
=> ? = 2 + 1
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[[],[],[],[[],[]]]]
=> ? = 2 + 1
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [[[],[],[],[[]]],[]]
=> ? = 2 + 1
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[[],[],[],[[[]]]]]
=> ? = 3 + 1
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [[[],[],[]],[[],[]]]
=> ? = 1 + 1
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [[[],[],[]],[[]],[]]
=> ? = 1 + 1
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[[],[],[],[[[]]]]]
=> ? = 3 + 1
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [[[],[],[]],[[[]]]]
=> ? = 2 + 1
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [[[],[],[]],[],[[]]]
=> ? = 1 + 1
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[[],[],[]],[],[],[]]
=> ? = 1 + 1
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[[],[],[[]],[],[]]]
=> ? = 2 + 1
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[],[[],[[]]]]]
=> ? = 3 + 1
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [[[],[],[[]],[]],[]]
=> ? = 2 + 1
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[[],[],[[],[],[]]]]
=> ? = 2 + 1
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [[[],[],[[],[]],[]]]
=> ? = 2 + 1
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [[[],[],[[],[]]],[]]
=> ? = 2 + 1
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [[[],[],[[]],[[]]]]
=> ? = 2 + 1
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[],[[],[[]]]]]
=> ? = 3 + 1
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [[[],[],[[]]],[[]]]
=> ? = 2 + 1
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [[[],[],[[]]],[],[]]
=> ? = 2 + 1
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [[[],[],[[[]]],[]]]
=> ? = 3 + 1
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [[[],[],[[[]],[]]]]
=> ? = 3 + 1
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [[[],[],[[[]]]],[]]
=> ? = 3 + 1
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[],[[[[]]]]]]
=> ? = 4 + 1
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [[[],[]],[[],[],[]]]
=> ? = 1 + 1
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [[[],[]],[[],[]],[]]
=> ? = 1 + 1
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[],[[[[]]]]]]
=> ? = 4 + 1
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [[[],[]],[[],[[]]]]
=> ? = 2 + 1
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [[[],[]],[[]],[[]]]
=> ? = 1 + 1
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [[[],[]],[[]],[],[]]
=> ? = 1 + 1
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [[[],[],[[[]]],[]]]
=> ? = 3 + 1
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [[[],[],[[[],[]]]]]
=> ? = 3 + 1
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [[[],[],[[[]],[]]]]
=> ? = 3 + 1
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [[[],[],[[[]]]],[]]
=> ? = 3 + 1
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[],[[[[]]]]]]
=> ? = 4 + 1
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [[[],[]],[[[]],[]]]
=> ? = 2 + 1
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [[[],[]],[[[],[]]]]
=> ? = 2 + 1
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [[[],[]],[[[]]],[]]
=> ? = 2 + 1
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[],[[[[]]]]]]
=> ? = 4 + 1
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[[],[]],[[[[]]]]]
=> ? = 3 + 1
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [[[],[]],[],[[],[]]]
=> ? = 1 + 1
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [[[],[]],[],[[]],[]]
=> ? = 1 + 1
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[],[[[[]]]]]]
=> ? = 4 + 1
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[[],[]],[[[[]]]]]
=> ? = 3 + 1
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [[[],[]],[],[[[]]]]
=> ? = 2 + 1
Description
The depth minus 1 of an ordered tree.
The ordered trees of size $n$ are bijection with the Dyck paths of size $n-1$, and this statistic then corresponds to [[St000013]].
Matching statistic: St000094
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000094: Ordered trees ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 86%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000094: Ordered trees ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 86%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [[[]]]
=> 3 = 1 + 2
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [[],[]]
=> 2 = 0 + 2
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [[[],[]]]
=> 3 = 1 + 2
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 3 = 1 + 2
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 4 = 2 + 2
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [[],[[]]]
=> 3 = 1 + 2
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 2 = 0 + 2
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 3 = 1 + 2
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 3 = 1 + 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 4 = 2 + 2
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 3 = 1 + 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3 = 1 + 2
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 4 = 2 + 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 4 = 2 + 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 4 = 2 + 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 5 = 3 + 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 3 = 1 + 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3 = 1 + 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 5 = 3 + 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4 = 2 + 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3 = 1 + 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 2 = 0 + 2
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> 3 = 1 + 2
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> 3 = 1 + 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 4 = 2 + 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> 3 = 1 + 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> 3 = 1 + 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> 4 = 2 + 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> 4 = 2 + 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> 4 = 2 + 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 5 = 3 + 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> 3 = 1 + 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3 = 1 + 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 5 = 3 + 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 4 = 2 + 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 3 = 1 + 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 3 = 1 + 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> 4 = 2 + 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> 5 = 3 + 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[[[]],[]],[]]
=> 4 = 2 + 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> 4 = 2 + 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> 4 = 2 + 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [[[[],[]]],[]]
=> 4 = 2 + 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [[[[]],[[]]]]
=> 4 = 2 + 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> 5 = 3 + 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 4 = 2 + 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 4 = 2 + 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> 5 = 3 + 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> 5 = 3 + 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 5 = 3 + 2
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[],[]]]
=> ? = 1 + 2
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[[],[],[],[],[]],[]]
=> ? = 1 + 2
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[[],[],[],[],[[]]]]
=> ? = 2 + 2
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [[[],[],[],[]],[[]]]
=> ? = 1 + 2
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[[],[],[],[]],[],[]]
=> ? = 1 + 2
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[[],[],[],[[]],[]]]
=> ? = 2 + 2
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[[],[],[],[[],[]]]]
=> ? = 2 + 2
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [[[],[],[],[[]]],[]]
=> ? = 2 + 2
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[[],[],[],[[[]]]]]
=> ? = 3 + 2
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [[[],[],[]],[[],[]]]
=> ? = 1 + 2
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [[[],[],[]],[[]],[]]
=> ? = 1 + 2
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[[],[],[],[[[]]]]]
=> ? = 3 + 2
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [[[],[],[]],[[[]]]]
=> ? = 2 + 2
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [[[],[],[]],[],[[]]]
=> ? = 1 + 2
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[[],[],[]],[],[],[]]
=> ? = 1 + 2
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[[],[],[[]],[],[]]]
=> ? = 2 + 2
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[],[[],[[]]]]]
=> ? = 3 + 2
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [[[],[],[[]],[]],[]]
=> ? = 2 + 2
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[[],[],[[],[],[]]]]
=> ? = 2 + 2
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [[[],[],[[],[]],[]]]
=> ? = 2 + 2
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [[[],[],[[],[]]],[]]
=> ? = 2 + 2
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [[[],[],[[]],[[]]]]
=> ? = 2 + 2
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[],[[],[[]]]]]
=> ? = 3 + 2
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [[[],[],[[]]],[[]]]
=> ? = 2 + 2
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [[[],[],[[]]],[],[]]
=> ? = 2 + 2
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [[[],[],[[[]]],[]]]
=> ? = 3 + 2
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [[[],[],[[[]],[]]]]
=> ? = 3 + 2
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [[[],[],[[[]]]],[]]
=> ? = 3 + 2
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[],[[[[]]]]]]
=> ? = 4 + 2
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [[[],[]],[[],[],[]]]
=> ? = 1 + 2
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [[[],[]],[[],[]],[]]
=> ? = 1 + 2
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[],[[[[]]]]]]
=> ? = 4 + 2
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [[[],[]],[[],[[]]]]
=> ? = 2 + 2
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [[[],[]],[[]],[[]]]
=> ? = 1 + 2
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [[[],[]],[[]],[],[]]
=> ? = 1 + 2
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [[[],[],[[[]]],[]]]
=> ? = 3 + 2
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [[[],[],[[[],[]]]]]
=> ? = 3 + 2
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [[[],[],[[[]],[]]]]
=> ? = 3 + 2
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [[[],[],[[[]]]],[]]
=> ? = 3 + 2
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[],[[[[]]]]]]
=> ? = 4 + 2
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [[[],[]],[[[]],[]]]
=> ? = 2 + 2
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [[[],[]],[[[],[]]]]
=> ? = 2 + 2
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [[[],[]],[[[]]],[]]
=> ? = 2 + 2
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[],[[[[]]]]]]
=> ? = 4 + 2
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[[],[]],[[[[]]]]]
=> ? = 3 + 2
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [[[],[]],[],[[],[]]]
=> ? = 1 + 2
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [[[],[]],[],[[]],[]]
=> ? = 1 + 2
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[],[[[[]]]]]]
=> ? = 4 + 2
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[[],[]],[[[[]]]]]
=> ? = 3 + 2
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [[[],[]],[],[[[]]]]
=> ? = 2 + 2
Description
The depth of an ordered tree.
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