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Your data matches 29 different statistics following compositions of up to 3 maps.
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Matching statistic: St000993
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> {{1},{2}}
=> [1,1]
=> [2]
=> 1
[1,1,0,0]
=> {{1,2}}
=> [2]
=> [1,1]
=> 2
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [1,1,1]
=> [3]
=> 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> [2,1]
=> [2,1]
=> 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> [2,1]
=> [2,1]
=> 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1]
=> [2,1]
=> 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> [3]
=> [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> 1
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> 1
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 1
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> 1
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 1
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [4,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [4,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> [3,1,1]
=> [3,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> [2,2,1]
=> [3,2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [4,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [4,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> [3,1,1]
=> [3,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [3,1,1]
=> [3,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> [2,2,1]
=> [3,2]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [3,1,1]
=> [3,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> [4,1]
=> [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [4,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [4,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> [3,1,1]
=> [3,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [2,2,1]
=> [3,2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,1,1]
=> 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000657
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 90%
Mp00128: Set partitions —to composition⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 90%
Values
[1,0,1,0]
=> {{1},{2}}
=> [1,1] => 1
[1,1,0,0]
=> {{1,2}}
=> [2] => 2
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [1,1,1] => 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> [1,2] => 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> [2,1] => 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1] => 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> [3] => 3
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> [1,1,2] => 1
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [1,2,1] => 1
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [1,2,1] => 1
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [1,3] => 1
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [2,1,1] => 1
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [2,2] => 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [2,1,1] => 1
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1] => 1
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1] => 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [3,1] => 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2] => 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1] => 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> [1,1,1,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> [1,1,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> [1,1,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> [1,1,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> [1,2,1,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> [1,2,2] => 1
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [1,2,1,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> [1,2,1,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> [1,3,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [1,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> [1,2,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [1,3,1] => 1
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> [1,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> [2,1,2] => 1
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [2,2,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [2,2,1] => 1
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [2,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> [2,1,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> [3,1,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [3,1,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [2,1,2] => 1
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [3,1,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [4,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> [3,1,1] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9,10}}
=> [1,9] => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8,9,10},{11}}
=> [10,1] => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,9},{8},{10}}
=> [8,1,1] => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,10},{9}}
=> [1,8,1] => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9,10,11}}
=> [1,10] => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,10},{8,9}}
=> [1,7,2] => ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,10},{8},{9}}
=> [1,7,1,1] => ? = 1
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,10},{7},{8},{9}}
=> [1,6,1,1,1] => ? = 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,3,4,5,6,7,8,9},{2},{10}}
=> [8,1,1] => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> {{1,2,3,4,5,6,7,8},{9,10}}
=> [8,2] => ? = 2
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,2},{3,4,5,6,7,8,9,10}}
=> [2,8] => ? = 2
[1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,4,5,6,7,8,9,10},{3}}
=> [1,8,1] => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9},{10}}
=> [8,1,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,1,1,1,1,1,1,1,1,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
=> [1,1,1,1,1,1,1,1,2] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8,10},{9}}
=> [1,1,1,1,1,1,1,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7,10},{8},{9}}
=> [1,1,1,1,1,1,2,1,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,10},{7},{8},{9}}
=> [1,1,1,1,1,2,1,1,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4},{5,10},{6},{7},{8},{9}}
=> [1,1,1,1,2,1,1,1,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,10},{5},{6},{7},{8},{9}}
=> [1,1,1,2,1,1,1,1,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3,10},{4},{5},{6},{7},{8},{9}}
=> [1,1,2,1,1,1,1,1,1] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,10},{3},{4},{5},{6},{7},{8},{9}}
=> [1,2,1,1,1,1,1,1,1] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}
=> [1,1,1,1,1,1,1,1,1,2] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9,11},{10}}
=> [1,1,1,1,1,1,1,1,2,1] => ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,10},{2,3},{4},{5},{6},{7},{8},{9}}
=> [2,2,1,1,1,1,1,1] => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,11},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1,1] => ? = 1
[1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> {{1,10},{2,6},{3},{4},{5},{7},{8},{9}}
=> [2,2,1,1,1,1,1,1] => ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1},{2,3,10},{4},{5},{6},{7},{8},{9}}
=> [1,3,1,1,1,1,1,1] => ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> {{1,10},{2,9},{3},{4},{5},{6},{7},{8}}
=> [2,2,1,1,1,1,1,1] => ? = 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> {{1},{2,3,4,10},{5},{6},{7},{8},{9}}
=> [1,4,1,1,1,1,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6,7},{8},{9},{10}}
=> [1,1,1,1,1,2,1,1,1] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5},{6},{7},{8},{9,10}}
=> [2,1,1,1,1,1,1,2] => ? = 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8},{9},{10}}
=> [3,1,1,1,1,1,1,1] => ? = 1
[1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1,2,3},{4,5,6,7,8,9,10}}
=> [3,7] => ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8},{9},{10}}
=> [4,1,1,1,1,1,1] => ? = 1
[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8,9,10}}
=> [4,6] => ? = 4
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8},{9},{10}}
=> [5,1,1,1,1,1] => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5},{6,7,8,9,10}}
=> [5,5] => ? = 5
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8},{9},{10}}
=> [6,1,1,1,1] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4,5,6},{7,8,9,10}}
=> [6,4] => ? = 4
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5,6,7},{8},{9},{10}}
=> [7,1,1,1] => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]
=> {{1,2,3,4,5,6,7},{8,9},{10}}
=> [7,2,1] => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,0,0,0]
=> {{1,2,3,4,5,6,7},{8,9,10}}
=> [7,3] => ? = 3
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9,10}}
=> [10] => ? = 10
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,3},{2},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,4},{2},{3},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,5},{2},{3},{4},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 1
Description
The smallest part of an integer composition.
Matching statistic: St001038
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 81%●distinct values known / distinct values provided: 80%
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 81%●distinct values known / distinct values provided: 80%
Values
[1,0,1,0]
=> {{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> {{1,2}}
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9}}
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9}}
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,8},{7},{9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,9},{8}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9,10}}
=> [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8,9,10},{11}}
=> [10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,9},{8},{10}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7,8},{6},{9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,8,9},{7}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,10},{9}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,8,9,10,11}}
=> [10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,9},{7},{8}}
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,10},{8,9}}
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7,10},{8},{9}}
=> [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,3,4,5,6,7,8},{2},{9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,9},{6},{7},{8}}
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2,4,5,6,7,8,9},{3}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6,7},{8},{9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,10},{7},{8},{9}}
=> [6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> {{1},{2},{3,4,5,6,7,8,9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,3,4,5,6,7,8,9},{2},{10}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> {{1,2,3,4,5,6,7,8},{9,10}}
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,2},{3,4,5,6,7,8,9,10}}
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1},{2,4,5,6,7,8,9,10},{3}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9},{10}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6},{7,8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> {{1,7,8,9},{2},{3},{4},{5},{6}}
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6,7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> {{1,9},{2},{3},{4},{5},{6,8},{7}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> {{1,6,9},{2},{3},{4},{5},{7},{8}}
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> {{1,6,8,9},{2},{3},{4},{5},{7}}
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> {{1,6,7,9},{2},{3},{4},{5},{8}}
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> {{1,5,9},{2},{3},{4},{6},{7},{8}}
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> {{1,5,6,9},{2},{3},{4},{7},{8}}
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> {{1,4,5,9},{2},{3},{6},{7},{8}}
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2},{3,4},{5},{6},{7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1,3,9},{2},{4},{5},{6},{7},{8}}
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> {{1,3,4,9},{2},{5},{6},{7},{8}}
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> {{1,3,4,5,6,7,9},{2},{8}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,3,4,5,6,7,8,9},{2}}
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2,3},{4},{5},{6},{7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1,9},{2,4},{3},{5},{6},{7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> {{1,9},{2,8},{3},{4},{5},{6},{7}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> {{1,2,9},{3},{4},{5},{6},{7},{8}}
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> {{1,2,4,8,9},{3},{5},{6},{7}}
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> {{1,2,4,5,6,7,8,9},{3}}
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000655
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,3,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,7,4,3,5,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,4,5,3,6,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,7,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,3,2,4,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,2,4,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,2,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,2,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [8,5,3,4,6,2,7,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,7,2,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,2,3,4,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,2,3,5,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,2,3,7,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,2,3,7,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,2,3,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [7,8,5,3,2,4,6,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,2,4,7,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,2,4,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,2,4,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,2,7,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,2,5,7,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [7,6,3,4,2,5,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,2,6,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [8,4,3,5,2,6,7,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,2,7,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [7,3,4,5,2,6,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,2,7,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [6,5,7,2,3,4,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [6,7,4,2,3,5,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [5,4,6,2,3,7,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [5,6,3,2,4,7,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [7,3,4,2,5,6,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,5,7,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [7,4,2,3,5,6,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [6,4,2,3,5,7,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [6,3,2,4,5,7,8,1] => ?
=> ?
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,3,1,2] => ?
=> ?
=> ? = 1
[1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,3,1,2] => ?
=> ?
=> ? = 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,3,8,1,2] => ?
=> ?
=> ? = 1
[1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,3,8,1,2] => ?
=> ?
=> ? = 1
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,8,1,2] => ?
=> ?
=> ? = 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,4,8,1,2] => ?
=> ?
=> ? = 2
Description
The length of the minimal rise of a Dyck path.
For the length of a maximal rise, see [[St000444]].
Matching statistic: St000700
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 70%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 70%
Values
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [[],[]]
=> 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 2
[1,0,1,0,1,0]
=> [3,2,1] => [[[.,.],.],.]
=> [[],[],[]]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> [[],[[]]]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [[[]],[]]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [[],[],[],[]]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [[],[],[],[],[]]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [[],[],[[]],[]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> [[],[[]],[],[]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [[],[],[[]],[]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [[],[[[]]],[]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [[],[[[[]]]]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> [[[]],[],[],[]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [[[]],[],[[]]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [[[]],[[]],[]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [[[]],[],[[]]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [[[]],[[[]]]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [[],[[]],[],[]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [[],[],[[]],[]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
=> [[],[[[]]],[]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [[],[[[[]]]]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [[[[]]],[],[]]
=> 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,3,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,7,4,3,5,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,4,5,3,6,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,8,2,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,7,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,8,2,3,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,3,2,4,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,2,4,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,2,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,2,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [8,5,3,4,6,2,7,1] => ?
=> ?
=> ? = 1
[1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,7,2,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,2,3,4,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,2,3,5,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,2,3,7,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,2,3,7,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,2,3,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [7,8,5,3,2,4,6,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,2,4,7,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,2,4,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,2,4,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,2,7,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,2,5,7,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [7,6,3,4,2,5,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,2,6,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [8,4,3,5,2,6,7,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,2,7,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [7,3,4,5,2,6,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,2,7,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [6,5,7,2,3,4,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [6,7,4,2,3,5,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [5,4,6,2,3,7,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [5,6,3,2,4,7,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [7,3,4,2,5,6,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [6,3,4,2,5,7,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [7,4,2,3,5,6,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [6,4,2,3,5,7,8,1] => ?
=> ?
=> ? = 1
[1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [6,3,2,4,5,7,8,1] => ?
=> ?
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,3,1,2] => ?
=> ?
=> ? = 1
[1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,3,1,2] => ?
=> ?
=> ? = 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,3,8,1,2] => ?
=> ?
=> ? = 1
[1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,3,8,1,2] => ?
=> ?
=> ? = 1
[1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,8,1,2] => ?
=> ?
=> ? = 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,4,8,1,2] => ?
=> ?
=> ? = 2
Description
The protection number of an ordered tree.
This is the minimal distance from the root to a leaf.
Matching statistic: St001075
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001075: Set partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 80%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001075: Set partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 80%
Values
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => {{1},{2}}
=> 1
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => {{1,2}}
=> 2
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => {{1},{2},{3}}
=> 1
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => {{1},{2,3}}
=> 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => {{1,2},{3}}
=> 1
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => {{1},{2,3}}
=> 1
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => {{1,2,3}}
=> 3
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => {{1},{2,3,4}}
=> 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => {{1},{2,3,4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => {{1,2,3},{4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => {{1},{2,3,4}}
=> 1
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => {{1,2,3,4}}
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7,8] => {{1},{2},{3},{4},{5,6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => {{1},{2},{3},{4},{5,6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,7,5,6,8] => {{1},{2},{3},{4},{5,6,7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => {{1},{2},{3},{4},{5,6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,8,5,6,7] => {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,5,4,6,8,7] => {{1},{2},{3},{4,5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,5,4,6,8,7] => {{1},{2},{3},{4,5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => {{1},{2},{3},{4},{5,6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,6,4,5,8,7] => {{1},{2},{3},{4,5,6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,3,8,4,5,6,7] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,2,4,3,5,6,8,7] => {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,4,3,5,6,8,7] => {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,1,0,0],[0,0,0,1,0,-1,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,0,1,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,0,1,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,5,3,4,6,7,8] => {{1},{2},{3,4,5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,4,3,5,6,8,7] => {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,5,4,6,8,7] => {{1},{2},{3},{4,5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => {{1},{2},{3},{4},{5,6},{7,8}}
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,-1,0,1,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,5,3,4,6,8,7] => {{1},{2},{3,4,5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,-1,0,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,-1,1,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,8,5,6,7] => {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,7,3,4,5,6,8] => {{1},{2},{3,4,5,6,7},{8}}
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,4,3,8,5,6,7] => {{1},{2},{3,4},{5,6,7,8}}
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,1,0,0,0,-1,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]
=> [1,2,3,8,4,5,6,7] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 1
Description
The minimal size of a block of a set partition.
Matching statistic: St000685
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000685: Dyck paths ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 70%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000685: Dyck paths ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 70%
Values
[1,0,1,0]
=> [1,2] => [2] => [1,1,0,0]
=> 1
[1,1,0,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,4,6,5,7,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,4,6,5,8,7] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,4,6,7,5,8] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,4,6,8,7,5] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,4,7,6,5,8] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,6,8,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,6,5] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,7,6,5] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,3,5,4,6,8,7] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3,5,8,7,6,4] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,4] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,3,6,5,8,7,4] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,6,7,5,8,4] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,6,5,8,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,7,6,5,4] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,3,5,6,8,7] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,2,4,3,5,7,8,6] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,8,7,6,5] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,2,4,5,3,8,7,6] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,2,4,6,5,8,7,3] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,2,4,6,8,7,5,3] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,2,4,7,6,5,8,3] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,2,4,7,6,8,5,3] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,2,4,8,7,6,5,3] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,5,4,3,6,7,8] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,4,3,8,7,6] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,2,5,4,8,7,6,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,5,6,4,7,8,3] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,2,5,6,4,8,7,3] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,2,5,6,7,4,8,3] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,5,7,6,4,8,3] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,5,4,3,8,7] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,6,5,4,7,8,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,2,6,5,4,8,7,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,2,6,5,7,4,8,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,6,7,8,5,4,3] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,7,6,5,4,3,8] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,2,7,6,5,4,8,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,2,7,6,5,8,4,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,7,6,8,5,4,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,7,8,6,5,4,3] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,8,7,6,5,4,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
Description
The dominant dimension of the LNakayama algebra associated to a Dyck path.
To every Dyck path there is an LNakayama algebra associated as described in [[St000684]].
Matching statistic: St001119
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001119: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 70%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001119: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 70%
Values
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => ([],2)
=> 0 = 1 - 1
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => ([],3)
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => ([(1,2)],3)
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => ([(1,2)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => ([],4)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => ([(2,3)],4)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => ([(2,3)],4)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => ([(2,3)],4)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => ([(2,3)],4)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => ([(2,3)],4)
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => ([(2,3)],4)
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1 = 2 - 1
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => ([],5)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1 - 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,-1,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,-1,0,1],[1,0,0,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,-1,1],[1,0,0,0,-1,1,0],[0,0,0,0,1,0,0]]
=> [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,-1,1],[0,1,0,0,-1,1,0],[1,0,0,-1,1,0,0],[0,0,0,1,0,0,0]]
=> [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7,8] => ([(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,5,7,6,8] => ([(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,7,6,5,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,0,0]]
=> [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
=> [1,2,3,4,8,7,6,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
=> ? => ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,-1,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,6,5,4,8,7] => ([(3,4),(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0]]
=> [1,2,3,8,7,6,5,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
=> ? => ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,-1,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,1,0,0],[0,0,0,0,1,-1,0,1],[0,0,0,1,-1,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
=> ? => ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,1,0,-1,1,0],[0,0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0]]
=> ? => ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,5,4,3,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,-1,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
=> ? => ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,0,-1,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,1,-1,1,-1,1,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,0,1,0],[0,0,1,-1,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,5,4,3,6,8,7] => ([(3,4),(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,-1,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1 - 1
Description
The length of a shortest maximal path in a graph.
Matching statistic: St001316
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001316: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 60%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001316: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 60%
Values
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => ([],2)
=> 1
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => ([(0,1)],2)
=> 2
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => ([],3)
=> 1
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => ([],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => ([(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => ([(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => ([],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[0,0,1,0,0,0,0],[0,1,-1,0,0,0,1],[1,-1,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[0,0,0,1,0,0,0],[0,0,1,-1,0,0,1],[0,1,-1,0,0,1,0],[1,-1,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,1,-1,0,1,0],[0,1,-1,0,1,0,0],[1,-1,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,1,-1,1,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7,8] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,5,7,6,8] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1]]
=> [1,2,3,4,7,6,5,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,0,0]]
=> [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
=> [1,2,3,4,8,7,6,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,-1,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,6,5,4,8,7] => ([(3,4),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0]]
=> [1,2,3,8,7,6,5,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,-1,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,1,0,0],[0,0,0,0,1,-1,0,1],[0,0,0,1,-1,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,1,0,-1,1,0],[0,0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,-1,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> [1,2,5,4,3,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,-1,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,0,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,0,-1,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,1,-1,1,-1,1,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,0,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,-1,0,1,0],[0,0,1,-1,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,-1,1],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> ? => ?
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> [1,2,5,4,3,6,8,7] => ([(3,4),(5,6),(5,7),(6,7)],8)
=> ? = 1
Description
The domatic number of a graph.
This is the maximal size of a partition of the vertices into dominating sets.
Matching statistic: St000908
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 70%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 70%
Values
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => ([],2)
=> 2
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => ([],3)
=> 3
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => ([],4)
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,2,3,4,5,7,6] => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,3,4,5,7,6] => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> [1,2,3,4,7,6,5] => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,3,4,5,7,6] => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> [1,2,3,4,7,6,5] => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> [1,2,3,4,7,6,5] => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> [1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,3,4,5,7,6] => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> [1,2,3,4,7,6,5] => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> [1,2,3,4,7,6,5] => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> [1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> [1,2,3,4,7,6,5] => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,1,0,-1,1,0],[0,0,0,0,1,0,0]]
=> [1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> [1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,3,4,5,7,6] => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> [1,2,3,4,7,6,5] => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> [1,2,3,4,7,6,5] => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> [1,2,5,4,3,6,7] => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,-1,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> [1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,0,1,0]]
=> [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,1,-1,1,-1,1],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0]]
=> [1,2,3,4,7,6,5] => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> ? = 1
Description
The length of the shortest maximal antichain in a poset.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000487The length of the shortest cycle of a permutation. St000210Minimum over maximum difference of elements in cycles. St000260The radius of a connected graph. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St000264The girth of a graph, which is not a tree. St001498The normalised height of a Nakayama algebra with magnitude 1. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000310The minimal degree of a vertex of a graph. St001545The second Elser number of a connected graph. St000259The diameter of a connected graph. St001198The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001206The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St001200The number of simple modules in eAe with projective dimension at most 2 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001330The hat guessing number of a graph.
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