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Your data matches 13 different statistics following compositions of up to 3 maps.
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Matching statistic: St000993
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00186: Skew partitions —dominating partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00186: Skew partitions —dominating partition⟶ 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]
=> [[1,1],[]]
=> [1,1]
=> 2
[1,1,0,0]
=> [[2],[]]
=> [2]
=> 1
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1]
=> 1
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [2,1]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> [3]
=> 1
[1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2]
=> 2
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [3,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4]
=> 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [3,2]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [2,2,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2]
=> 1
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2]
=> 3
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [2,2,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [2,2,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [3,2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [3,2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [3,3]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [2,2,2]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [3,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [4,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [3,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [3,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [4,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [4,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> [5]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [4,2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [3,2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [4,2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [3,2,2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [4,3]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [2,2,1,1]
=> 2
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000700
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 70% ●values known / values provided: 94%●distinct values known / distinct values provided: 70%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 70% ●values known / values provided: 94%●distinct values known / distinct values provided: 70%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[[]]]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [[],[]]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[[]],[]]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[[]],[[]]]]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[[]],[]],[]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[[],[]]],[]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[],[[[[[[]]]]]]]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[],[[[[[]]]]]]]]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[],[],[[[[[]]]]]]]
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[]]
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[],[],[],[],[],[],[[],[]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [[[[[[[[[[]]]]]]]]]]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[],[],[[[[[[[]]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[],[],[],[[[[[[]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[],[],[],[[[[[]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[],[],[],[],[],[[[[]]]]]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[],[],[],[],[],[],[[[]]]]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[],[],[],[[]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [[[[[[[[[[[]]]]]]]]]]]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[],[[[[[[[]]]]]]]]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[],[],[[[[[[[[]]]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[],[[],[[[[[[]]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[],[],[],[[[[[[[]]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[],[],[[],[[[[[]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[],[],[],[],[[[[[[]]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [[],[],[],[[],[[[[]]]]]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[],[],[],[],[[[[[]]]]]]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [[],[],[],[],[[],[[[]]]]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [[],[],[],[],[],[],[[[[]]]]]
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [[],[],[],[],[],[[],[[]]]]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[],[],[],[],[],[],[],[[[]]]]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[],[],[],[],[[]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[],[],[],[]]
=> ? = 1
Description
The protection number of an ordered tree.
This is the minimal distance from the root to a leaf.
Matching statistic: St001038
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St001038: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 93%●distinct values known / distinct values provided: 80%
Values
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001829
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 88%●distinct values known / distinct values provided: 70%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 88%●distinct values known / distinct values provided: 70%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => ([(0,5),(0,6),(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
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,5,6,1] => ([(0,5),(0,6),(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
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1,4,5] => ([(0,5),(0,6),(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
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [8,7,1,2,3,4,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,5,6,8] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [8,7,6,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [8,6,1,2,3,4,5,7] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [6,1,2,3,4,5,7,8] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5,8] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [8,7,5,1,2,3,4,6] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [8,5,1,2,3,4,6,7] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [8,6,5,1,2,3,4,7] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,1,2,3,5] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [8,7,4,1,2,3,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,1,2,3,6] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,3,1,2,4] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [8,7,6,3,1,2,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,3,1,2,5] => ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,2,1,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,3,2,1,4] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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] => ([],9)
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [9,8,1,2,3,4,5,6,7] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [9,8,7,1,2,3,4,5,6] => ([(0,6),(0,7),(0,8),(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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] => ([],10)
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1,2,3,4,5,6,7,9] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [10,9,1,2,3,4,5,6,7,8] => ([(0,8),(0,9),(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [9,7,1,2,3,4,5,6,8] => ([(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [10,9,8,1,2,3,4,5,6,7] => ([(0,7),(0,8),(0,9),(1,7),(1,8),(1,9),(2,7),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
=> [9,8,6,1,2,3,4,5,7] => ([(0,7),(0,8),(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [10,9,8,7,1,2,3,4,5,6] => ([(0,6),(0,7),(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> [9,8,7,5,1,2,3,4,6] => ([(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [10,9,8,7,6,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,4,1,2,3,5] => ([(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
Description
The common independence number of a graph.
The common independence number of a graph $G$ is the greatest integer $r$ such that every vertex of $G$ belongs to some independent set $X$ of vertices of cardinality at least $r$.
Matching statistic: St001316
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001316: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 87%●distinct values known / distinct values provided: 60%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001316: Graphs ⟶ ℤResult quality: 60% ●values known / values provided: 87%●distinct values known / distinct values provided: 60%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [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
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [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
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ([(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)
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [7,5,4,3,2,6,1] => ([(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)
=> ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [7,5,4,3,6,2,1] => ([(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)
=> ? = 4
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [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
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [7,6,4,5,3,2,1] => ([(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)
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,6,5,4,3,2,8,1] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [6,5,4,3,2,7,1,8] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [8,6,5,4,3,2,7,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [6,5,4,3,2,7,8,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [5,4,3,2,6,1,7,8] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,5,4,3,2,6,1,8] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [5,4,3,2,6,7,1,8] => ([(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [4,3,2,5,1,6,7,8] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [6,4,3,2,5,1,7,8] => ([(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [4,3,2,5,6,1,7,8] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => ([(6,7)],8)
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,2,4,1,5,6,7,8] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [5,3,2,4,1,6,7,8] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,2,4,5,1,6,7,8] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8] => ([(5,7),(6,7)],8)
=> ? = 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [4,2,3,1,5,6,7,8] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7,8] => ([(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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] => ([],9)
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8,9] => ([(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 9
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9] => ([(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9] => ([(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9] => ([(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9] => ([(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9] => ([(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [10,9,8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,7,6,5,4,3,2,9,1] => ([(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1,9,10] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [7,6,5,4,3,2,8,1,9] => ([(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9,10] => ([(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
=> [6,5,4,3,2,7,1,8,9] => ([(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9,10] => ([(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,2,6,1,7,8,9] => ([(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
Description
The domatic number of a graph.
This is the maximal size of a partition of the vertices into dominating sets.
Matching statistic: St001322
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 80%●distinct values known / distinct values provided: 70%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 80%●distinct values known / distinct values provided: 70%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1,2] => ([(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)
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => ([(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)
=> ? = 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => ([(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)
=> ? = 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => ([(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)
=> ? = 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ([(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)
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ([(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)
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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)
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ([(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)
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => ([(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)
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => ([(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)
=> ? = 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => ([(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)
=> ? = 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => ([(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)
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => ([(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)
=> ? = 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,4,1] => ([(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)
=> ? = 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,5,1] => ([(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)
=> ? = 1
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,6,1] => ([(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)
=> ? = 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => ([(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)
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1,3] => ([(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)
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,1,3] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,2,1,3] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,1,3] => ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1,4] => ([(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)
=> ? = 1
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,1,4] => ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1,5] => ([(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)
=> ? = 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,6] => ([(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)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => ([],8)
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [8,7,1,2,3,4,5,6] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,5,6,8] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [8,7,6,1,2,3,4,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [8,6,1,2,3,4,5,7] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [6,1,2,3,4,5,7,8] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5,8] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [8,7,5,1,2,3,4,6] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
Description
The size of a minimal independent dominating set in a graph.
Matching statistic: St000906
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 77%●distinct values known / distinct values provided: 60%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 77%●distinct values known / distinct values provided: 60%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => ([],3)
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(3,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,5,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [7,6,4,1,2,3,5] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,2,3,5,6] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,2,3,5,6,7] => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [6,4,1,2,3,5,7] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,1,2] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,1,2] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,1,2] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,1,2] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,1,2] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,1,2] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,1,2] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,1,2,4] => ([(3,6),(4,5),(5,6)],7)
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [7,6,3,1,2,4,5] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [7,5,3,1,2,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,3,1,2,4,6,7] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 3
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [7,5,6,2,3,4,1] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [6,7,5,2,3,4,1] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,2,3,1] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => ([(1,6),(2,5),(3,4)],7)
=> ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 1
[1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 1
Description
The length of the shortest maximal chain in a poset.
Matching statistic: St000908
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 70%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 70%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => ([],2)
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([],3)
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(2,3)],4)
=> 3
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(4,3),(5,3),(6,3)],7)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,4,3,7,6] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,4,3,6,7] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(3,6),(4,6),(5,6),(6,2)],7)
=> ? = 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,7,4,5,6,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(6,2)],7)
=> ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,4,3,2,5,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ? = 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [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
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [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
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [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
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [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
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> ? = 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => ([(0,5),(4,3),(5,6),(6,1),(6,2),(6,4)],7)
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
=> ? = 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? = 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => ([(0,5),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,6,4,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => ([(0,6),(5,4),(6,1),(6,2),(6,3),(6,5)],7)
=> ? = 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,6,4,5,7,3] => ([(0,5),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ? = 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> ? = 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,7,5,6,3] => ([(0,6),(4,3),(5,2),(5,4),(6,1),(6,5)],7)
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
=> ? = 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => ([(0,6),(4,5),(5,3),(6,1),(6,2),(6,4)],7)
=> ? = 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> ? = 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => ([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
=> ? = 1
[1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,5,4,6,7,2] => ([(0,4),(0,5),(1,6),(2,6),(5,1),(5,2),(6,3)],7)
=> ? = 1
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,5,3,4,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> ? = 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ? = 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,5,7,2] => ([(0,3),(0,4),(1,6),(2,6),(4,5),(5,1),(5,2)],7)
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,6,4,5,2,7] => ([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => ([(0,4),(0,6),(5,3),(6,1),(6,2),(6,5)],7)
=> ? = 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,6,4,5,7,2] => ([(0,3),(0,5),(1,6),(2,6),(4,2),(5,1),(5,4)],7)
=> ? = 1
[1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ? = 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,7,5,6,2] => ([(0,3),(0,5),(4,2),(5,6),(6,1),(6,4)],7)
=> ? = 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ? = 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,7,4,5,6,2] => ([(0,3),(0,6),(4,5),(5,2),(6,1),(6,4)],7)
=> ? = 1
[1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,6,3,4,5,2,7] => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ? = 1
[1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,7,3,5,6,4,2] => ([(0,3),(0,4),(0,6),(5,2),(6,1),(6,5)],7)
=> ? = 1
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,6,3,4,5,7,2] => ([(0,2),(0,3),(0,5),(1,6),(3,6),(4,1),(5,4)],7)
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
Description
The length of the shortest maximal antichain in a poset.
Matching statistic: St000657
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00187: Skew partitions —conjugate⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 90%
Mp00187: Skew partitions —conjugate⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 90%
Values
[1,0,1,0]
=> [[1,1],[]]
=> [[2],[]]
=> [2] => 2
[1,1,0,0]
=> [[2],[]]
=> [[1,1],[]]
=> [1,1] => 1
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [[3],[]]
=> [3] => 3
[1,0,1,1,0,0]
=> [[2,1],[]]
=> [[2,1],[]]
=> [2,1] => 1
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [[2,2],[1]]
=> [1,2] => 1
[1,1,0,1,0,0]
=> [[3],[]]
=> [[1,1,1],[]]
=> [1,1,1] => 1
[1,1,1,0,0,0]
=> [[2,2],[]]
=> [[2,2],[]]
=> [2,2] => 2
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [[4],[]]
=> [4] => 4
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [[3,1],[]]
=> [3,1] => 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [[3,2],[1]]
=> [2,2] => 2
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [[2,1,1],[]]
=> [2,1,1] => 1
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [[3,2],[]]
=> [3,2] => 2
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [[3,3],[2]]
=> [1,3] => 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [[2,2,1],[1]]
=> [1,2,1] => 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [[2,2,2],[1,1]]
=> [1,1,2] => 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [[1,1,1,1],[]]
=> [1,1,1,1] => 1
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [[2,2,2],[1]]
=> [1,2,2] => 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [[3,3],[1]]
=> [2,3] => 2
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [[2,2,1],[]]
=> [2,2,1] => 1
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [[3,3],[]]
=> [3,3] => 3
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [[2,2,2],[]]
=> [2,2,2] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> [[5],[]]
=> [5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [[4,1],[]]
=> [4,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [[4,2],[1]]
=> [3,2] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [[3,1,1],[]]
=> [3,1,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [[4,2],[]]
=> [4,2] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [[4,3],[2]]
=> [2,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [[3,2,1],[1]]
=> [2,2,1] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [[3,2,2],[1,1]]
=> [2,1,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [[2,1,1,1],[]]
=> [2,1,1,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [[3,2,2],[1]]
=> [2,2,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [[4,3],[1]]
=> [3,3] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [[3,2,1],[]]
=> [3,2,1] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [[4,3],[]]
=> [4,3] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [[3,2,2],[]]
=> [3,2,2] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [[4,4],[3]]
=> [1,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [[3,3,1],[2]]
=> [1,3,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [[3,3,2],[2,1]]
=> [1,2,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [[2,2,1,1],[1]]
=> [1,2,1,1] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [[3,3,2],[2]]
=> [1,3,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [[3,3,3],[2,2]]
=> [1,1,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [[2,2,2,1],[1,1]]
=> [1,1,2,1] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [[2,2,2,2],[1,1]]
=> [1,1,2,2] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [[3,3,3],[2,1]]
=> [1,2,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [[2,2,2,1],[1]]
=> [1,2,2,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [[3,3,3],[2]]
=> [1,3,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [[2,2,2,2],[1]]
=> [1,2,2,2] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [[4,4],[2]]
=> [2,4] => 2
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> [[4,3,3],[]]
=> [4,3,3] => ? = 3
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> [[3,3,3,3],[2]]
=> [1,3,3,3] => ? = 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,2],[]]
=> [[5,5],[]]
=> [5,5] => ? = 5
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [[3,3,2,2],[]]
=> [[4,4,2],[]]
=> [4,4,2] => ? = 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [[3,3,3,2],[1]]
=> [[4,4,3],[1]]
=> [3,4,3] => ? = 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [[4,4,2],[]]
=> [[3,3,2,2],[]]
=> [3,3,2,2] => ? = 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2],[]]
=> [[4,4,3],[]]
=> [4,4,3] => ? = 3
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [[4,4,4],[2]]
=> [2,4,4] => ? = 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> [[3,3,3,2],[1]]
=> [2,3,3,2] => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [[4,4,4],[2]]
=> [[3,3,3,3],[1,1]]
=> [2,2,3,3] => ? = 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [[5,5],[]]
=> [[2,2,2,2,2],[]]
=> [2,2,2,2,2] => ? = 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4],[1]]
=> [[3,3,3,3],[1]]
=> [2,3,3,3] => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> [[4,4,4],[1,1]]
=> [3,3,4] => ? = 3
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [[4,3,3],[]]
=> [[3,3,3,1],[]]
=> [3,3,3,1] => ? = 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [[4,4,4],[1]]
=> [3,4,4] => ? = 3
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [[4,4,3],[]]
=> [[3,3,3,2],[]]
=> [3,3,3,2] => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3],[]]
=> [[4,4,4],[]]
=> [4,4,4] => ? = 4
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> [[3,3,3,3],[]]
=> [3,3,3,3] => ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1,1],[]]
=> [[6,4],[]]
=> [6,4] => ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1,1],[]]
=> [[5,3,2],[]]
=> [5,3,2] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1,1],[]]
=> [[4,2,2,2],[]]
=> [4,2,2,2] => ? = 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4,1],[1,1]]
=> [[4,3,3,3],[2]]
=> [2,3,3,3] => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [[3,2,2,2,1],[]]
=> [[5,4,1],[]]
=> [5,4,1] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [[4,3,2,1],[]]
=> [[4,3,2,1],[]]
=> [4,3,2,1] => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [[4,4,2,1],[]]
=> [[4,3,2,2],[]]
=> [4,3,2,2] => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2,1],[]]
=> [[5,4,3],[]]
=> [5,4,3] => ? = 3
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3,1],[1]]
=> [[4,3,3,2],[1]]
=> [3,3,3,2] => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [[5,4,1],[]]
=> [[3,2,2,2,1],[]]
=> [3,2,2,2,1] => ? = 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [[4,4,4,1],[2]]
=> [[4,3,3,3],[1,1]]
=> [3,2,3,3] => ? = 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [[5,5,1],[]]
=> [[3,2,2,2,2],[]]
=> [3,2,2,2,2] => ? = 2
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4,1],[1]]
=> [[4,3,3,3],[1]]
=> [3,3,3,3] => ? = 3
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[4,3,3,1],[]]
=> [[4,3,3,1],[]]
=> [4,3,3,1] => ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3,1],[1]]
=> [[5,4,4],[1]]
=> [4,4,4] => ? = 4
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [[4,4,3,1],[]]
=> [[4,3,3,2],[]]
=> [4,3,3,2] => ? = 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[4,4,4,2],[1,1,1]]
=> [[4,4,3,3],[3]]
=> [1,4,3,3] => ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[6,6],[4]]
=> [[2,2,2,2,2,2],[1,1,1,1]]
=> ? => ? = 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[6,5],[3]]
=> [[2,2,2,2,2,1],[1,1,1]]
=> ? => ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[5,5,5],[3,3]]
=> [[3,3,3,3,3],[2,2,2]]
=> ? => ? = 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[6,6],[3]]
=> [[2,2,2,2,2,2],[1,1,1]]
=> ? => ? = 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[6,4],[2]]
=> [[2,2,2,2,1,1],[1,1]]
=> ? => ? = 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [[5,5,4],[3,2]]
=> [[3,3,3,3,2],[2,2,1]]
=> ? => ? = 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [[5,4,4],[2,2]]
=> [[3,3,3,3,1],[2,2]]
=> ? => ? = 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[4,4,4,4],[2,2,2]]
=> [[4,4,4,4],[3,3]]
=> ? => ? = 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [[5,5,4],[2,2]]
=> [[3,3,3,3,2],[2,2]]
=> ? => ? = 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [[6,5],[2]]
=> [[2,2,2,2,2,1],[1,1]]
=> ? => ? = 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [[5,5,5],[3,2]]
=> [[3,3,3,3,3],[2,2,1]]
=> ? => ? = 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [[6,6],[2]]
=> [[2,2,2,2,2,2],[1,1]]
=> ? => ? = 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[5,5,5],[2,2]]
=> [[3,3,3,3,3],[2,2]]
=> ? => ? = 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[6,3],[1]]
=> [[2,2,2,1,1,1],[1]]
=> ? => ? = 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [[5,5,3],[3,1]]
=> [[3,3,3,2,2],[2,1,1]]
=> ? => ? = 1
Description
The smallest part of an integer composition.
Matching statistic: St001107
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 61%●distinct values known / distinct values provided: 60%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 61%●distinct values known / distinct values provided: 60%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7 - 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 3 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 4 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 1 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
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