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Your data matches 90 different statistics following compositions of up to 3 maps.
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Matching statistic: St000013
(load all 10 compositions to match this statistic)
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Mp00051: Ordered trees āto Dyck pathā¶ Dyck paths
St000013: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000013: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[]
=> []
=> 0
[[]]
=> [1,0]
=> 1
[[],[]]
=> [1,0,1,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00049: Ordered trees āto binary tree: left brother = left childā¶ Binary trees
Mp00017: Binary trees āto 312-avoiding permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
St000147: Integer partitions ā¶ ā¤Result quality: 79% āvalues known / values provided: 79%ādistinct values known / distinct values provided: 89%
Mp00017: Binary trees āto 312-avoiding permutationā¶ Permutations
Mp00204: Permutations āLLPSā¶ Integer partitions
St000147: Integer partitions ā¶ ā¤Result quality: 79% āvalues known / values provided: 79%ādistinct values known / distinct values provided: 89%
Values
[]
=> .
=> ? => ?
=> ? = 0
[[]]
=> [.,.]
=> [1] => [1]
=> 1
[[],[]]
=> [[.,.],.]
=> [1,2] => [1,1]
=> 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => [2]
=> 2
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 2
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 2
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3]
=> 3
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,1,1]
=> 2
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,1,1]
=> 2
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,1,1]
=> 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [3,1]
=> 3
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> 2
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,1,1]
=> 2
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1]
=> 3
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,1]
=> 2
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1]
=> 3
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> 3
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1]
=> 3
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4]
=> 4
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [2,1,1,1]
=> 2
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [2,1,1,1]
=> 2
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [2,1,1,1]
=> 2
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,1,1]
=> 3
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [2,1,1,1]
=> 2
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [2,2,1]
=> 2
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [2,1,1,1]
=> 2
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [3,1,1]
=> 3
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,1,1,1]
=> 2
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,1,1]
=> 3
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [3,1,1]
=> 3
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [3,1,1]
=> 3
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [4,1]
=> 4
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,1,1]
=> 2
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,2,1]
=> 2
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [2,2,1]
=> 2
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,2,1]
=> 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,2]
=> 3
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [2,1,1,1]
=> 2
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,1,1]
=> 3
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2]
=> 3
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,1,1,1]
=> 2
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [3,1,1]
=> 3
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,1,1]
=> 3
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,1,1]
=> 3
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1]
=> 4
[[],[],[[[]]],[[[]]]]
=> [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [1,2,5,4,3,8,7,6] => ?
=> ? = 3
[[],[],[[[[]]]],[[]]]
=> [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [1,2,6,5,4,3,8,7] => ?
=> ? = 4
[[],[],[[],[[[]]]],[]]
=> [[[[.,.],.],[[.,.],[.,[.,.]]]],.]
=> [1,2,4,7,6,5,3,8] => ?
=> ? = 4
[[],[],[[[[]]],[]],[]]
=> [[[[.,.],.],[[.,[.,[.,.]]],.]],.]
=> [1,2,6,5,4,7,3,8] => ?
=> ? = 4
[[],[],[[[[]],[]]],[]]
=> [[[[.,.],.],[.,[[.,[.,.]],.]]],.]
=> [1,2,6,5,7,4,3,8] => ?
=> ? = 4
[[],[],[[[],[]],[],[]]]
=> [[[.,.],.],[[[.,[[.,.],.]],.],.]]
=> [1,2,5,6,4,7,8,3] => ?
=> ? = 3
[[],[],[[[[],[]]],[]]]
=> [[[.,.],.],[[.,[.,[[.,.],.]]],.]]
=> [1,2,6,7,5,4,8,3] => ?
=> ? = 4
[[],[],[[[],[[],[]]]]]
=> [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> [1,2,5,7,8,6,4,3] => ?
=> ? = 4
[[],[],[[[[[]],[]]]]]
=> [[[.,.],.],[.,[.,[[.,[.,.]],.]]]]
=> [1,2,7,6,8,5,4,3] => ?
=> ? = 5
[[],[[[]]],[[]],[[]]]
=> [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,4,3,2,6,5,8,7] => ?
=> ? = 3
[[],[[[[]]]],[],[[]]]
=> [[[[.,.],[.,[.,[.,.]]]],.],[.,.]]
=> [1,5,4,3,2,6,8,7] => ?
=> ? = 4
[[],[[],[[[]]]],[],[]]
=> [[[[.,.],[[.,.],[.,[.,.]]]],.],.]
=> [1,3,6,5,4,2,7,8] => ?
=> ? = 4
[[],[[[],[[]]],[]],[]]
=> [[[.,.],[[.,[[.,.],[.,.]]],.]],.]
=> [1,4,6,5,3,7,2,8] => ?
=> ? = 4
[[],[[[],[],[],[]]],[]]
=> [[[.,.],[.,[[[[.,.],.],.],.]]],.]
=> [1,4,5,6,7,3,2,8] => ?
=> ? = 3
[[],[[[],[],[]],[],[]]]
=> [[.,.],[[[.,[[[.,.],.],.]],.],.]]
=> [1,4,5,6,3,7,8,2] => ?
=> ? = 3
[[],[[[],[[]]],[],[]]]
=> [[.,.],[[[.,[[.,.],[.,.]]],.],.]]
=> [1,4,6,5,3,7,8,2] => ?
=> ? = 4
[[],[[[],[],[[]]],[]]]
=> [[.,.],[[.,[[[.,.],.],[.,.]]],.]]
=> [1,4,5,7,6,3,8,2] => ?
=> ? = 4
[[],[[[],[[[]]]],[]]]
=> [[.,.],[[.,[[.,.],[.,[.,.]]]],.]]
=> [1,4,7,6,5,3,8,2] => ?
=> ? = 5
[[],[[[[],[],[]]],[]]]
=> [[.,.],[[.,[.,[[[.,.],.],.]]],.]]
=> [1,5,6,7,4,3,8,2] => ?
=> ? = 4
[[],[[[],[[[]],[]]]]]
=> [[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,4,7,6,8,5,3,2] => ?
=> ? = 5
[[],[[[[]],[],[],[]]]]
=> [[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> [1,5,4,6,7,8,3,2] => ?
=> ? = 4
[[],[[[[],[]],[],[]]]]
=> [[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> [1,5,6,4,7,8,3,2] => ?
=> ? = 4
[[[]],[],[[[]]],[[]]]
=> [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [2,1,3,6,5,4,8,7] => ?
=> ? = 3
[[[]],[],[[[[]]]],[]]
=> [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 4
[[[]],[],[[[],[],[]]]]
=> [[[.,[.,.]],.],[.,[[[.,.],.],.]]]
=> [2,1,3,6,7,8,5,4] => ?
=> ? = 3
[[[]],[[]],[[[]]],[]]
=> [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 3
[[[]],[[[]]],[],[[]]]
=> [[[[.,[.,.]],[.,[.,.]]],.],[.,.]]
=> [2,1,5,4,3,6,8,7] => ?
=> ? = 3
[[[]],[[[[]]]],[],[]]
=> [[[[.,[.,.]],[.,[.,[.,.]]]],.],.]
=> [2,1,6,5,4,3,7,8] => ?
=> ? = 4
[[[[]]],[],[[]],[[]]]
=> [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [3,2,1,4,6,5,8,7] => ?
=> ? = 3
[[[[]]],[[]],[],[[]]]
=> [[[[.,[.,[.,.]]],[.,.]],.],[.,.]]
=> [3,2,1,5,4,6,8,7] => ?
=> ? = 3
[[[],[]],[[[[],[]]]]]
=> [[.,[[.,.],.]],[.,[.,[[.,.],.]]]]
=> [2,3,1,7,8,6,5,4] => ?
=> ? = 4
[[[[[]]]],[],[[]],[]]
=> [[[[.,[.,[.,[.,.]]]],.],[.,.]],.]
=> [4,3,2,1,5,7,6,8] => ?
=> ? = 4
[[[[],[]]],[],[[],[]]]
=> [[[.,[.,[[.,.],.]]],.],[[.,.],.]]
=> [3,4,2,1,5,7,8,6] => ?
=> ? = 3
[[[],[],[[[]]]],[],[]]
=> [[[.,[[[.,.],.],[.,[.,.]]]],.],.]
=> [2,3,6,5,4,1,7,8] => ?
=> ? = 4
[[[[],[],[],[]]],[],[]]
=> [[[.,[.,[[[[.,.],.],.],.]]],.],.]
=> [3,4,5,6,2,1,7,8] => ?
=> ? = 3
[[[],[[[]],[]]],[[]]]
=> [[.,[[.,.],[[.,[.,.]],.]]],[.,.]]
=> [2,5,4,6,3,1,8,7] => ?
=> ? = 4
[[[[],[],[],[]]],[[]]]
=> [[.,[.,[[[[.,.],.],.],.]]],[.,.]]
=> [3,4,5,6,2,1,8,7] => ?
=> ? = 3
[[[[],[[]],[]]],[[]]]
=> [[.,[.,[[[.,.],[.,.]],.]]],[.,.]]
=> [3,5,4,6,2,1,8,7] => ?
=> ? = 4
[[[[],[],[]],[],[]],[]]
=> [[.,[[[.,[[[.,.],.],.]],.],.]],.]
=> [3,4,5,2,6,7,1,8] => ?
=> ? = 3
[[[[],[[[[]]]]]],[]]
=> [[.,[.,[[.,.],[.,[.,[.,.]]]]]],.]
=> [3,7,6,5,4,2,1,8] => ?
=> ? = 6
[[[],[],[],[[[],[]]]]]
=> [.,[[[[.,.],.],.],[.,[[.,.],.]]]]
=> [2,3,4,7,8,6,5,1] => ?
=> ? = 4
[[[],[],[[]],[],[[]]]]
=> [.,[[[[[.,.],.],[.,.]],.],[.,.]]]
=> [2,3,5,4,6,8,7,1] => ?
=> ? = 3
[[[],[],[[[]]],[[]]]]
=> [.,[[[[.,.],.],[.,[.,.]]],[.,.]]]
=> [2,3,6,5,4,8,7,1] => ?
=> ? = 4
[[[],[],[[[]],[],[]]]]
=> [.,[[[.,.],.],[[[.,[.,.]],.],.]]]
=> [2,3,6,5,7,8,4,1] => ?
=> ? = 4
[[[],[],[[[],[]],[]]]]
=> [.,[[[.,.],.],[[.,[[.,.],.]],.]]]
=> [2,3,6,7,5,8,4,1] => ?
=> ? = 4
[[[],[],[[[],[[]]]]]]
=> [.,[[[.,.],.],[.,[[.,.],[.,.]]]]]
=> [2,3,6,8,7,5,4,1] => ?
=> ? = 5
[[[],[],[[[[],[]]]]]]
=> [.,[[[.,.],.],[.,[.,[[.,.],.]]]]]
=> [2,3,7,8,6,5,4,1] => ?
=> ? = 5
[[[],[[]],[[[]]],[]]]
=> [.,[[[[.,.],[.,.]],[.,[.,.]]],.]]
=> [2,4,3,7,6,5,8,1] => ?
=> ? = 4
[[[],[[]],[[[]],[]]]]
=> [.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [2,4,3,7,6,8,5,1] => ?
=> ? = 4
Description
The largest part of an integer partition.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00049: Ordered trees āto binary tree: left brother = left childā¶ Binary trees
Mp00017: Binary trees āto 312-avoiding permutationā¶ Permutations
Mp00060: Permutations āRobinson-Schensted tableau shapeā¶ Integer partitions
St000010: Integer partitions ā¶ ā¤Result quality: 77% āvalues known / values provided: 77%ādistinct values known / distinct values provided: 89%
Mp00017: Binary trees āto 312-avoiding permutationā¶ Permutations
Mp00060: Permutations āRobinson-Schensted tableau shapeā¶ Integer partitions
St000010: Integer partitions ā¶ ā¤Result quality: 77% āvalues known / values provided: 77%ādistinct values known / distinct values provided: 89%
Values
[]
=> .
=> ? => ?
=> ? = 0
[[]]
=> [.,.]
=> [1] => [1]
=> 1
[[],[]]
=> [[.,.],.]
=> [1,2] => [2]
=> 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => [1,1]
=> 2
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => [3]
=> 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 2
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 2
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1]
=> 3
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [4]
=> 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1]
=> 2
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1]
=> 2
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1]
=> 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1]
=> 3
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,1]
=> 2
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1]
=> 2
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,1]
=> 3
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1]
=> 2
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1]
=> 3
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,1]
=> 3
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1]
=> 3
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1]
=> 4
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [5]
=> 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,1]
=> 2
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [4,1]
=> 2
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,1]
=> 2
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,1,1]
=> 3
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [4,1]
=> 2
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [3,2]
=> 2
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [4,1]
=> 2
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [3,1,1]
=> 3
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [4,1]
=> 2
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,1,1]
=> 3
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [3,1,1]
=> 3
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [3,1,1]
=> 3
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,1,1,1]
=> 4
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [4,1]
=> 2
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [3,2]
=> 2
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [3,2]
=> 2
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,2]
=> 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,2,1]
=> 3
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [4,1]
=> 2
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,1,1]
=> 3
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2]
=> 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,2,1]
=> 3
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,1]
=> 2
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [3,1,1]
=> 3
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,1,1]
=> 3
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,1,1]
=> 3
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [2,1,1,1]
=> 4
[[],[],[[[]]],[[[]]]]
=> [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [1,2,5,4,3,8,7,6] => ?
=> ? = 3
[[],[],[[[[]]]],[[]]]
=> [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [1,2,6,5,4,3,8,7] => ?
=> ? = 4
[[],[],[[],[[[]]]],[]]
=> [[[[.,.],.],[[.,.],[.,[.,.]]]],.]
=> [1,2,4,7,6,5,3,8] => ?
=> ? = 4
[[],[],[[[[]]],[]],[]]
=> [[[[.,.],.],[[.,[.,[.,.]]],.]],.]
=> [1,2,6,5,4,7,3,8] => ?
=> ? = 4
[[],[],[[[[]],[]]],[]]
=> [[[[.,.],.],[.,[[.,[.,.]],.]]],.]
=> [1,2,6,5,7,4,3,8] => ?
=> ? = 4
[[],[],[[[],[]],[],[]]]
=> [[[.,.],.],[[[.,[[.,.],.]],.],.]]
=> [1,2,5,6,4,7,8,3] => ?
=> ? = 3
[[],[],[[[[],[]]],[]]]
=> [[[.,.],.],[[.,[.,[[.,.],.]]],.]]
=> [1,2,6,7,5,4,8,3] => ?
=> ? = 4
[[],[],[[[],[[],[]]]]]
=> [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> [1,2,5,7,8,6,4,3] => ?
=> ? = 4
[[],[],[[[[[]],[]]]]]
=> [[[.,.],.],[.,[.,[[.,[.,.]],.]]]]
=> [1,2,7,6,8,5,4,3] => ?
=> ? = 5
[[],[[[]]],[[]],[[]]]
=> [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,4,3,2,6,5,8,7] => ?
=> ? = 3
[[],[[[[]]]],[],[[]]]
=> [[[[.,.],[.,[.,[.,.]]]],.],[.,.]]
=> [1,5,4,3,2,6,8,7] => ?
=> ? = 4
[[],[[[]],[]],[[]],[]]
=> [[[[.,.],[[.,[.,.]],.]],[.,.]],.]
=> [1,4,3,5,2,7,6,8] => ?
=> ? = 3
[[],[[],[[[]]]],[],[]]
=> [[[[.,.],[[.,.],[.,[.,.]]]],.],.]
=> [1,3,6,5,4,2,7,8] => ?
=> ? = 4
[[],[[[[],[]]]],[[]]]
=> [[[.,.],[.,[.,[[.,.],.]]]],[.,.]]
=> [1,5,6,4,3,2,8,7] => ?
=> ? = 4
[[],[[[]],[[]],[]],[]]
=> [[[.,.],[[[.,[.,.]],[.,.]],.]],.]
=> [1,4,3,6,5,7,2,8] => ?
=> ? = 3
[[],[[[],[[]]],[]],[]]
=> [[[.,.],[[.,[[.,.],[.,.]]],.]],.]
=> [1,4,6,5,3,7,2,8] => ?
=> ? = 4
[[],[[[],[],[],[]]],[]]
=> [[[.,.],[.,[[[[.,.],.],.],.]]],.]
=> [1,4,5,6,7,3,2,8] => ?
=> ? = 3
[[],[[[],[],[]],[],[]]]
=> [[.,.],[[[.,[[[.,.],.],.]],.],.]]
=> [1,4,5,6,3,7,8,2] => ?
=> ? = 3
[[],[[[],[[]]],[],[]]]
=> [[.,.],[[[.,[[.,.],[.,.]]],.],.]]
=> [1,4,6,5,3,7,8,2] => ?
=> ? = 4
[[],[[[[[]]]],[],[]]]
=> [[.,.],[[[.,[.,[.,[.,.]]]],.],.]]
=> [1,6,5,4,3,7,8,2] => ?
=> ? = 5
[[],[[[],[],[[]]],[]]]
=> [[.,.],[[.,[[[.,.],.],[.,.]]],.]]
=> [1,4,5,7,6,3,8,2] => ?
=> ? = 4
[[],[[[],[[[]]]],[]]]
=> [[.,.],[[.,[[.,.],[.,[.,.]]]],.]]
=> [1,4,7,6,5,3,8,2] => ?
=> ? = 5
[[],[[[[],[],[]]],[]]]
=> [[.,.],[[.,[.,[[[.,.],.],.]]],.]]
=> [1,5,6,7,4,3,8,2] => ?
=> ? = 4
[[],[[[],[],[[],[]]]]]
=> [[.,.],[.,[[[.,.],.],[[.,.],.]]]]
=> [1,4,5,7,8,6,3,2] => ?
=> ? = 4
[[],[[[],[[[]],[]]]]]
=> [[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,4,7,6,8,5,3,2] => ?
=> ? = 5
[[],[[[[]],[],[],[]]]]
=> [[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> [1,5,4,6,7,8,3,2] => ?
=> ? = 4
[[],[[[[],[]],[],[]]]]
=> [[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> [1,5,6,4,7,8,3,2] => ?
=> ? = 4
[[],[[[[[]],[],[]]]]]
=> [[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,6,5,7,8,4,3,2] => ?
=> ? = 5
[[[]],[],[[[]]],[[]]]
=> [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [2,1,3,6,5,4,8,7] => ?
=> ? = 3
[[[]],[],[[[[]]]],[]]
=> [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
=> [2,1,3,7,6,5,4,8] => ?
=> ? = 4
[[[]],[],[[[],[],[]]]]
=> [[[.,[.,.]],.],[.,[[[.,.],.],.]]]
=> [2,1,3,6,7,8,5,4] => ?
=> ? = 3
[[[]],[[]],[[[]]],[]]
=> [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
=> [2,1,4,3,7,6,5,8] => ?
=> ? = 3
[[[]],[[[]]],[],[[]]]
=> [[[[.,[.,.]],[.,[.,.]]],.],[.,.]]
=> [2,1,5,4,3,6,8,7] => ?
=> ? = 3
[[[]],[[[[]]]],[],[]]
=> [[[[.,[.,.]],[.,[.,[.,.]]]],.],.]
=> [2,1,6,5,4,3,7,8] => ?
=> ? = 4
[[[[]]],[],[[]],[[]]]
=> [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
=> [3,2,1,4,6,5,8,7] => ?
=> ? = 3
[[[[]]],[[]],[],[[]]]
=> [[[[.,[.,[.,.]]],[.,.]],.],[.,.]]
=> [3,2,1,5,4,6,8,7] => ?
=> ? = 3
[[[[]]],[[[]]],[],[]]
=> [[[[.,[.,[.,.]]],[.,[.,.]]],.],.]
=> [3,2,1,6,5,4,7,8] => ?
=> ? = 3
[[[],[]],[[[]],[]],[]]
=> [[[.,[[.,.],.]],[[.,[.,.]],.]],.]
=> [2,3,1,6,5,7,4,8] => ?
=> ? = 3
[[[],[]],[[[[],[]]]]]
=> [[.,[[.,.],.]],[.,[.,[[.,.],.]]]]
=> [2,3,1,7,8,6,5,4] => ?
=> ? = 4
[[[[[]]]],[],[],[[]]]
=> [[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> [4,3,2,1,5,6,8,7] => ?
=> ? = 4
[[[[[]]]],[],[[]],[]]
=> [[[[.,[.,[.,[.,.]]]],.],[.,.]],.]
=> [4,3,2,1,5,7,6,8] => ?
=> ? = 4
[[[[],[]]],[],[[],[]]]
=> [[[.,[.,[[.,.],.]]],.],[[.,.],.]]
=> [3,4,2,1,5,7,8,6] => ?
=> ? = 3
[[[[[]]]],[[]],[],[]]
=> [[[[.,[.,[.,[.,.]]]],[.,.]],.],.]
=> [4,3,2,1,6,5,7,8] => ?
=> ? = 4
[[[],[[]],[]],[],[[]]]
=> [[[.,[[[.,.],[.,.]],.]],.],[.,.]]
=> [2,4,3,5,1,6,8,7] => ?
=> ? = 3
[[[],[[]],[]],[[]],[]]
=> [[[.,[[[.,.],[.,.]],.]],[.,.]],.]
=> [2,4,3,5,1,7,6,8] => ?
=> ? = 3
[[[],[],[],[[]]],[],[]]
=> [[[.,[[[[.,.],.],.],[.,.]]],.],.]
=> [2,3,4,6,5,1,7,8] => ?
=> ? = 3
[[[],[],[[[]]]],[],[]]
=> [[[.,[[[.,.],.],[.,[.,.]]]],.],.]
=> [2,3,6,5,4,1,7,8] => ?
=> ? = 4
[[[[],[],[],[]]],[],[]]
=> [[[.,[.,[[[[.,.],.],.],.]]],.],.]
=> [3,4,5,6,2,1,7,8] => ?
=> ? = 3
[[[[[[]]],[]]],[],[]]
=> [[[.,[.,[[.,[.,[.,.]]],.]]],.],.]
=> [5,4,3,6,2,1,7,8] => ?
=> ? = 5
Description
The length of the partition.
Matching statistic: St000011
Mp00051: Ordered trees āto Dyck pathā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
St000011: Dyck paths ā¶ ā¤Result quality: 66% āvalues known / values provided: 66%ādistinct values known / distinct values provided: 100%
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
Mp00099: Dyck paths ābounce pathā¶ Dyck paths
St000011: Dyck paths ā¶ ā¤Result quality: 66% āvalues known / values provided: 66%ādistinct values known / distinct values provided: 100%
Values
[]
=> []
=> []
=> []
=> 0
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 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
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[],[[]],[[[[]]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[],[[[[]]]],[[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[],[[],[[[]]]],[]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[],[[[[]]],[]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[],[[[],[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[[],[],[[[],[[]]]],[]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[],[[[[]],[]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[],[[[[],[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[],[[],[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[],[],[[[]],[],[],[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[],[],[[[],[]],[],[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[],[],[[[[]]],[],[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4
[[],[],[[[],[],[]],[]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[[],[],[[[],[[]]],[]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4
[[],[],[[[[],[]]],[]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4
[[],[],[[[[[]]]],[]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[[],[],[[[],[],[[]]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
[[],[],[[[],[[],[]]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 4
[[],[],[[[[],[],[]]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[[],[],[[[[[]],[]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 5
[[],[[]],[],[[[[]]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[[]],[[[[]]]],[]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[[]],[[[[[]]]]]]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[[],[[],[]],[[[[]]]]]
=> [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4
[[],[[[]]],[[[[]]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4
[[],[[[[]]]],[],[[]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[[[[]]]],[[]],[]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[[[],[]]],[[[]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[[],[[],[[[]]]],[],[]]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[[[[]]],[]],[],[]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[[[],[],[]]],[],[]]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[[],[[[],[[]]]],[],[]]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[[[[]],[]]],[],[]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[[[[],[]]]],[],[]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[],[[[[],[]]]],[[]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4
[[],[[[[[]]]]],[[]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[[],[[[]],[[]],[]],[]]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[],[[[],[]],[],[]],[]]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 3
[[],[[[],[[]]],[]],[]]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4
[[],[[[],[],[],[]]],[]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[[],[[[],[],[]],[],[]]]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[[],[[[],[[]]],[],[]]]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 4
[[],[[[[],[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4
[[],[[[[[]],[]]],[]]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 5
[[],[[[],[],[],[],[]]]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 3
[[],[[[],[],[[],[]]]]]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 4
[[],[[[],[],[[[]]]]]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 5
[[],[[[],[[[]],[]]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 5
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000097
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00049: Ordered trees āto binary tree: left brother = left childā¶ Binary trees
Mp00017: Binary trees āto 312-avoiding permutationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000097: Graphs ā¶ ā¤Result quality: 66% āvalues known / values provided: 66%ādistinct values known / distinct values provided: 89%
Mp00017: Binary trees āto 312-avoiding permutationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000097: Graphs ā¶ ā¤Result quality: 66% āvalues known / values provided: 66%ādistinct values known / distinct values provided: 89%
Values
[]
=> .
=> ? => ?
=> ? = 0
[[]]
=> [.,.]
=> [1] => ([],1)
=> 1
[[],[]]
=> [[.,.],.]
=> [1,2] => ([],2)
=> 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 2
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 2
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 2
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => ([],5)
=> 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 2
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 2
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 2
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 2
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 2
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[]],[[]],[[],[]]]
=> [[[.,[.,.]],[.,.]],[[.,.],.]]
=> [2,1,4,3,6,7,5] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? = 2
[[[]],[[]],[[[]]]]
=> [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[],[]],[[]]]
=> [[[.,[.,.]],[[.,.],.]],[.,.]]
=> [2,1,4,5,3,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? = 2
[[[]],[[[]]],[[]]]
=> [[[.,[.,.]],[.,[.,.]]],[.,.]]
=> [2,1,5,4,3,7,6] => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[],[],[],[]]]
=> [[.,[.,.]],[[[[.,.],.],.],.]]
=> [2,1,4,5,6,7,3] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
[[[]],[[],[],[[]]]]
=> [[.,[.,.]],[[[.,.],.],[.,.]]]
=> [2,1,4,5,7,6,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[],[[]],[]]]
=> [[.,[.,.]],[[[.,.],[.,.]],.]]
=> [2,1,4,6,5,7,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[],[[],[]]]]
=> [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [2,1,4,6,7,5,3] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[],[[[]]]]]
=> [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,7,6,5,3] => ([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[]],[[[]],[],[]]]
=> [[.,[.,.]],[[[.,[.,.]],.],.]]
=> [2,1,5,4,6,7,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[[]],[[]]]]
=> [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [2,1,5,4,7,6,3] => ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[[],[]],[]]]
=> [[.,[.,.]],[[.,[[.,.],.]],.]]
=> [2,1,5,6,4,7,3] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[[[]]],[]]]
=> [[.,[.,.]],[[.,[.,[.,.]]],.]]
=> [2,1,6,5,4,7,3] => ([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[]],[[[],[],[]]]]
=> [[.,[.,.]],[.,[[[.,.],.],.]]]
=> [2,1,5,6,7,4,3] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[[],[[]]]]]
=> [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [2,1,5,7,6,4,3] => ([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[]],[[[[]],[]]]]
=> [[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [2,1,6,5,7,4,3] => ([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[]],[[[[],[]]]]]
=> [[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [2,1,6,7,5,4,3] => ([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[]],[[[[[]]]]]]
=> [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[[],[]],[[]],[[]]]
=> [[[.,[[.,.],.]],[.,.]],[.,.]]
=> [2,3,1,5,4,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? = 2
[[[[]]],[[]],[[]]]
=> [[[.,[.,[.,.]]],[.,.]],[.,.]]
=> [3,2,1,5,4,7,6] => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[]],[[],[],[]]]
=> [[.,[[.,.],.]],[[[.,.],.],.]]
=> [2,3,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 2
[[[],[]],[[],[[]]]]
=> [[.,[[.,.],.]],[[.,.],[.,.]]]
=> [2,3,1,5,7,6,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[]],[[[]],[]]]
=> [[.,[[.,.],.]],[[.,[.,.]],.]]
=> [2,3,1,6,5,7,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[]],[[[],[]]]]
=> [[.,[[.,.],.]],[.,[[.,.],.]]]
=> [2,3,1,6,7,5,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ? = 3
[[[],[]],[[[[]]]]]
=> [[.,[[.,.],.]],[.,[.,[.,.]]]]
=> [2,3,1,7,6,5,4] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[[]]],[[],[],[]]]
=> [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [3,2,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ? = 3
[[[[]]],[[],[[]]]]
=> [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [3,2,1,5,7,6,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[]]],[[[]],[]]]
=> [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [3,2,1,6,5,7,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[]]],[[[],[]]]]
=> [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [3,2,1,6,7,5,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[]]],[[[[]]]]]
=> [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[],[],[]],[[],[]]]
=> [[.,[[[.,.],.],.]],[[.,.],.]]
=> [2,3,4,1,6,7,5] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 2
[[[],[],[]],[[[]]]]
=> [[.,[[[.,.],.],.]],[.,[.,.]]]
=> [2,3,4,1,7,6,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ? = 3
[[[],[[]]],[[],[]]]
=> [[.,[[.,.],[.,.]]],[[.,.],.]]
=> [2,4,3,1,6,7,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[[]]],[[[]]]]
=> [[.,[[.,.],[.,.]]],[.,[.,.]]]
=> [2,4,3,1,7,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[]],[]],[[],[]]]
=> [[.,[[.,[.,.]],.]],[[.,.],.]]
=> [3,2,4,1,6,7,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[]],[]],[[[]]]]
=> [[.,[[.,[.,.]],.]],[.,[.,.]]]
=> [3,2,4,1,7,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[],[]]],[[],[]]]
=> [[.,[.,[[.,.],.]]],[[.,.],.]]
=> [3,4,2,1,6,7,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ? = 3
[[[[],[]]],[[[]]]]
=> [[.,[.,[[.,.],.]]],[.,[.,.]]]
=> [3,4,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[[]]]],[[],[]]]
=> [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [4,3,2,1,6,7,5] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[[[]]]],[[[]]]]
=> [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[],[],[],[]],[[]]]
=> [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
[[[],[],[[]]],[[]]]
=> [[.,[[[.,.],.],[.,.]]],[.,.]]
=> [2,3,5,4,1,7,6] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[[]],[]],[[]]]
=> [[.,[[[.,.],[.,.]],.]],[.,.]]
=> [2,4,3,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[[],[]]],[[]]]
=> [[.,[[.,.],[[.,.],.]]],[.,.]]
=> [2,4,5,3,1,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[[[]]]],[[]]]
=> [[.,[[.,.],[.,[.,.]]]],[.,.]]
=> [2,5,4,3,1,7,6] => ([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[[]],[],[]],[[]]]
=> [[.,[[[.,[.,.]],.],.]],[.,.]]
=> [3,2,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[]],[[]]],[[]]]
=> [[.,[[.,[.,.]],[.,.]]],[.,.]]
=> [3,2,5,4,1,7,6] => ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3
[[[[],[]],[]],[[]]]
=> [[.,[[.,[[.,.],.]],.]],[.,.]]
=> [3,4,2,5,1,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[[]]],[]],[[]]]
=> [[.,[[.,[.,[.,.]]],.]],[.,.]]
=> [4,3,2,5,1,7,6] => ([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St001039
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00051: Ordered trees āto Dyck pathā¶ Dyck paths
Mp00121: Dyck paths āCori-Le Borgne involutionā¶ Dyck paths
Mp00229: Dyck paths āDelest-Viennotā¶ Dyck paths
St001039: Dyck paths ā¶ ā¤Result quality: 62% āvalues known / values provided: 62%ādistinct values known / distinct values provided: 89%
Mp00121: Dyck paths āCori-Le Borgne involutionā¶ Dyck paths
Mp00229: Dyck paths āDelest-Viennotā¶ Dyck paths
St001039: Dyck paths ā¶ ā¤Result quality: 62% āvalues known / values provided: 62%ādistinct values known / distinct values provided: 89%
Values
[]
=> []
=> []
=> []
=> ? = 0
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[[],[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2
[[],[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 3
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 2
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> ? = 3
[[],[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 3
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 3
[[],[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 3
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 4
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 3
[[],[],[],[[],[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 3
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 3
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 4
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 4
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 5
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[[],[],[[]],[[],[],[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 2
[[],[],[[[]]],[],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 3
[[],[],[[[[]]]],[[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 4
[[],[],[[],[[[]]]],[]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 4
[[],[],[[[[]]],[]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 4
[[],[],[[[],[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3
[[],[],[[[],[[]]]],[]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4
[[],[],[[[[]],[]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 4
[[],[],[[[[],[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4
[[],[],[[[[[]]]]],[]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 5
[[],[],[[],[],[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[[],[],[[],[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 3
[[],[],[[[]],[],[],[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 3
[[],[],[[[],[]],[],[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 3
[[],[],[[[[]]],[],[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> ? = 4
[[],[],[[[],[],[]],[]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 3
[[],[],[[[],[[]]],[]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 4
[[],[],[[[[],[]]],[]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 4
[[],[],[[[[[]]]],[]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> ? = 5
[[],[],[[[],[],[[]]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 4
[[],[],[[[],[[],[]]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 4
[[],[],[[[[],[],[]]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 4
[[],[],[[[[[]],[]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 5
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000442
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00051: Ordered trees āto Dyck pathā¶ Dyck paths
Mp00121: Dyck paths āCori-Le Borgne involutionā¶ Dyck paths
St000442: Dyck paths ā¶ ā¤Result quality: 62% āvalues known / values provided: 62%ādistinct values known / distinct values provided: 78%
Mp00121: Dyck paths āCori-Le Borgne involutionā¶ Dyck paths
St000442: Dyck paths ā¶ ā¤Result quality: 62% āvalues known / values provided: 62%ādistinct values known / distinct values provided: 78%
Values
[]
=> []
=> []
=> ? = 0 - 1
[[]]
=> [1,0]
=> [1,0]
=> ? = 1 - 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 3 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3 = 4 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[],[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 3 - 1
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 4 - 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 5 - 1
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[],[],[[]],[[[[]]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[],[],[[[]]],[],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[[],[],[[[]]],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
[[],[],[[[[]]]],[[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 4 - 1
[[],[],[[],[[[]]]],[]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 4 - 1
[[],[],[[[[]]],[]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 4 - 1
[[],[],[[[],[[]]]],[]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 4 - 1
[[],[],[[[[]],[]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 4 - 1
[[],[],[[[[[]]]]],[]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> ? = 5 - 1
[[],[],[[],[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
[[],[],[[[]],[],[],[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[],[],[[[[]]],[],[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 4 - 1
[[],[],[[[],[[]]],[]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 4 - 1
[[],[],[[[[[]]]],[]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> ? = 5 - 1
[[],[],[[[],[],[[]]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[],[],[[[[[]],[]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[[],[],[[[[[[]]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 6 - 1
[[],[[]],[],[],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2 - 1
[[],[[]],[],[[[[]]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[],[[]],[[]],[[[]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 3 - 1
[[],[[]],[[[]]],[[]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[[]],[[[[]]]],[]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 4 - 1
[[],[[]],[[[[[]]]]]]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> ? = 5 - 1
[[],[[[]]],[],[],[],[]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[[],[[],[]],[],[],[[]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 2 - 1
[[],[[[]]],[],[[],[]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[],[[[]]],[],[[[]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 3 - 1
[[],[[[]]],[[]],[[]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[[[]]],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 3 - 1
[[],[[],[]],[[[[]]]]]
=> [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[],[[[]]],[[[[]]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[[],[[[[]]]],[],[[]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 4 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000093
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00050: Ordered trees āto binary tree: right brother = right childā¶ Binary trees
Mp00014: Binary trees āto 132-avoiding permutationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000093: Graphs ā¶ ā¤Result quality: 54% āvalues known / values provided: 54%ādistinct values known / distinct values provided: 78%
Mp00014: Binary trees āto 132-avoiding permutationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000093: Graphs ā¶ ā¤Result quality: 54% āvalues known / values provided: 54%ādistinct values known / distinct values provided: 78%
Values
[]
=> .
=> ? => ?
=> ? = 0
[[]]
=> [.,.]
=> [1] => ([],1)
=> 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 1
[[[]]]
=> [[.,.],.]
=> [1,2] => ([],2)
=> 2
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 3
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 3
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 4
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [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
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[[[]],[[]],[[],[]]]
=> [[.,.],[[.,.],[[.,[.,.]],.]]]
=> [6,5,7,3,4,1,2] => ([(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),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[[[]],[[]],[[[]]]]
=> [[.,.],[[.,.],[[[.,.],.],.]]]
=> [5,6,7,3,4,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
[[[]],[[],[]],[[]]]
=> [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [6,7,4,3,5,1,2] => ([(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),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[[[]],[[[]]],[[]]]
=> [[.,.],[[[.,.],.],[[.,.],.]]]
=> [6,7,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
[[[]],[[],[],[],[]]]
=> [[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [6,5,4,3,7,1,2] => ([(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)],7)
=> ? = 2
[[[]],[[],[],[[]]]]
=> [[.,.],[[.,[.,[[.,.],.]]],.]]
=> [5,6,4,3,7,1,2] => ([(0,5),(0,6),(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)],7)
=> ? = 3
[[[]],[[],[[]],[]]]
=> [[.,.],[[.,[[.,.],[.,.]]],.]]
=> [6,4,5,3,7,1,2] => ([(0,5),(0,6),(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)],7)
=> ? = 3
[[[]],[[],[[],[]]]]
=> [[.,.],[[.,[[.,[.,.]],.]],.]]
=> [5,4,6,3,7,1,2] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
[[[]],[[[]],[],[]]]
=> [[.,.],[[[.,.],[.,[.,.]]],.]]
=> [6,5,3,4,7,1,2] => ([(0,5),(0,6),(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)],7)
=> ? = 3
[[[]],[[[]],[[]]]]
=> [[.,.],[[[.,.],[[.,.],.]],.]]
=> [5,6,3,4,7,1,2] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
[[[]],[[[],[]],[]]]
=> [[.,.],[[[.,[.,.]],[.,.]],.]]
=> [6,4,3,5,7,1,2] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
[[[]],[[[],[],[]]]]
=> [[.,.],[[[.,[.,[.,.]]],.],.]]
=> [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)
=> ? = 3
[[[]],[[[],[[]]]]]
=> [[.,.],[[[.,[[.,.],.]],.],.]]
=> [4,5,3,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
[[[]],[[[[]],[]]]]
=> [[.,.],[[[[.,.],[.,.]],.],.]]
=> [5,3,4,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
[[[]],[[[[],[]]]]]
=> [[.,.],[[[[.,[.,.]],.],.],.]]
=> [4,3,5,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
[[[],[]],[[]],[[]]]
=> [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [6,7,4,5,2,1,3] => ([(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),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2
[[[[]]],[[]],[[]]]
=> [[[.,.],.],[[.,.],[[.,.],.]]]
=> [6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
[[[],[]],[[],[],[]]]
=> [[.,[.,.]],[[.,[.,[.,.]]],.]]
=> [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
[[[],[]],[[],[[]]]]
=> [[.,[.,.]],[[.,[[.,.],.]],.]]
=> [5,6,4,7,2,1,3] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[]],[[[]],[]]]
=> [[.,[.,.]],[[[.,.],[.,.]],.]]
=> [6,4,5,7,2,1,3] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[]],[[[],[]]]]
=> [[.,[.,.]],[[[.,[.,.]],.],.]]
=> [5,4,6,7,2,1,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ? = 3
[[[],[]],[[[[]]]]]
=> [[.,[.,.]],[[[[.,.],.],.],.]]
=> [4,5,6,7,2,1,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ? = 4
[[[[]]],[[],[],[]]]
=> [[[.,.],.],[[.,[.,[.,.]]],.]]
=> [6,5,4,7,1,2,3] => ([(0,1),(0,2),(0,3),(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)
=> ? = 3
[[[[]]],[[],[[]]]]
=> [[[.,.],.],[[.,[[.,.],.]],.]]
=> [5,6,4,7,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3
[[[[]]],[[[]],[]]]
=> [[[.,.],.],[[[.,.],[.,.]],.]]
=> [6,4,5,7,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3
[[[[]]],[[[],[]]]]
=> [[[.,.],.],[[[.,[.,.]],.],.]]
=> [5,4,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 3
[[[[]]],[[[[]]]]]
=> [[[.,.],.],[[[[.,.],.],.],.]]
=> [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 4
[[[],[],[]],[[],[]]]
=> [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [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
[[[],[],[]],[[[]]]]
=> [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [5,6,7,3,2,1,4] => ([(0,1),(0,2),(0,3),(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)
=> ? = 3
[[[],[[]]],[[],[]]]
=> [[.,[[.,.],.]],[[.,[.,.]],.]]
=> [6,5,7,2,3,1,4] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[[]]],[[[]]]]
=> [[.,[[.,.],.]],[[[.,.],.],.]]
=> [5,6,7,2,3,1,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3
[[[[]],[]],[[],[]]]
=> [[[.,.],[.,.]],[[.,[.,.]],.]]
=> [6,5,7,3,1,2,4] => ([(0,3),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[]],[]],[[[]]]]
=> [[[.,.],[.,.]],[[[.,.],.],.]]
=> [5,6,7,3,1,2,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3
[[[[],[]]],[[],[]]]
=> [[[.,[.,.]],.],[[.,[.,.]],.]]
=> [6,5,7,2,1,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ? = 3
[[[[],[]]],[[[]]]]
=> [[[.,[.,.]],.],[[[.,.],.],.]]
=> [5,6,7,2,1,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 3
[[[[[]]]],[[],[]]]
=> [[[[.,.],.],.],[[.,[.,.]],.]]
=> [6,5,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ? = 4
[[[[[]]]],[[[]]]]
=> [[[[.,.],.],.],[[[.,.],.],.]]
=> [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 4
[[[],[],[],[]],[[]]]
=> [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [6,7,4,3,2,1,5] => ([(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)],7)
=> ? = 2
[[[],[],[[]]],[[]]]
=> [[.,[.,[[.,.],.]]],[[.,.],.]]
=> [6,7,3,4,2,1,5] => ([(0,5),(0,6),(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)],7)
=> ? = 3
[[[],[[]],[]],[[]]]
=> [[.,[[.,.],[.,.]]],[[.,.],.]]
=> [6,7,4,2,3,1,5] => ([(0,5),(0,6),(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)],7)
=> ? = 3
[[[],[[],[]]],[[]]]
=> [[.,[[.,[.,.]],.]],[[.,.],.]]
=> [6,7,3,2,4,1,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
[[[[]],[],[]],[[]]]
=> [[[.,.],[.,[.,.]]],[[.,.],.]]
=> [6,7,4,3,1,2,5] => ([(0,5),(0,6),(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)],7)
=> ? = 3
[[[[]],[[]]],[[]]]
=> [[[.,.],[[.,.],.]],[[.,.],.]]
=> [6,7,3,4,1,2,5] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
[[[[],[]],[]],[[]]]
=> [[[.,[.,.]],[.,.]],[[.,.],.]]
=> [6,7,4,2,1,3,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
[[[[],[],[]]],[[]]]
=> [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [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)
=> ? = 3
[[[[],[[]]]],[[]]]
=> [[[.,[[.,.],.]],.],[[.,.],.]]
=> [6,7,2,3,1,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
[[[[[]],[]]],[[]]]
=> [[[[.,.],[.,.]],.],[[.,.],.]]
=> [6,7,3,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
[[[[[],[]]]],[[]]]
=> [[[[.,[.,.]],.],.],[[.,.],.]]
=> [6,7,2,1,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 4
[[],[],[],[[[]]],[[]]]
=> [.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [7,8,4,5,6,3,2,1] => ?
=> ? = 3
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000786
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00050: Ordered trees āto binary tree: right brother = right childā¶ Binary trees
Mp00014: Binary trees āto 132-avoiding permutationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000786: Graphs ā¶ ā¤Result quality: 54% āvalues known / values provided: 54%ādistinct values known / distinct values provided: 78%
Mp00014: Binary trees āto 132-avoiding permutationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000786: Graphs ā¶ ā¤Result quality: 54% āvalues known / values provided: 54%ādistinct values known / distinct values provided: 78%
Values
[]
=> .
=> ? => ?
=> ? = 0
[[]]
=> [.,.]
=> [1] => ([],1)
=> 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 1
[[[]]]
=> [[.,.],.]
=> [1,2] => ([],2)
=> 2
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 3
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 3
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 4
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [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
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[[[]],[[[],[],[]]]]
=> [[.,.],[[[.,[.,[.,.]]],.],.]]
=> [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)
=> ? = 3
[[[[],[],[]]],[[]]]
=> [[[.,[.,[.,.]]],.],[[.,.],.]]
=> [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)
=> ? = 3
[[],[],[],[],[[]],[[]]]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [7,8,5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(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
[[],[],[],[[]],[],[[]]]
=> [.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [7,8,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(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
[[],[],[],[[[]]],[[]]]
=> [.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [7,8,4,5,6,3,2,1] => ?
=> ? = 3
[[],[],[[]],[],[],[[]]]
=> [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(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
[[],[],[[]],[],[[],[]]]
=> [.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,3,4,2,1] => ([(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),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[],[],[[]],[[]],[[]]]
=> [.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> [7,8,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[],[],[[]],[[],[],[]]]
=> [.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> [7,6,5,8,3,4,2,1] => ([(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),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[],[],[[]],[[[[]]]]]
=> [.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> [5,6,7,8,3,4,2,1] => ([(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),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[[],[],[[[]]],[[[]]]]
=> [.,[.,[[[.,.],.],[[[.,.],.],.]]]]
=> [6,7,8,3,4,5,2,1] => ([(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),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[[],[],[[[[]]]],[[]]]
=> [.,[.,[[[[.,.],.],.],[[.,.],.]]]]
=> [7,8,3,4,5,6,2,1] => ([(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),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[[],[],[[],[[[]]]],[]]
=> [.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> [8,4,5,6,3,7,2,1] => ?
=> ? = 4
[[],[],[[[[]]],[]],[]]
=> [.,[.,[[[[.,.],.],[.,.]],[.,.]]]]
=> [8,6,3,4,5,7,2,1] => ?
=> ? = 4
[[],[],[[[],[[]]]],[]]
=> [.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> [8,4,5,3,6,7,2,1] => ?
=> ? = 4
[[],[],[[[],[[]]],[]]]
=> [.,[.,[[[.,[[.,.],.]],[.,.]],.]]]
=> [7,4,5,3,6,8,2,1] => ?
=> ? = 4
[[],[],[[[[],[]]],[]]]
=> [.,[.,[[[[.,[.,.]],.],[.,.]],.]]]
=> [7,4,3,5,6,8,2,1] => ?
=> ? = 4
[[],[],[[[],[],[[]]]]]
=> [.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> [5,6,4,3,7,8,2,1] => ?
=> ? = 4
[[],[],[[[],[[],[]]]]]
=> [.,[.,[[[.,[[.,[.,.]],.]],.],.]]]
=> [5,4,6,3,7,8,2,1] => ?
=> ? = 4
[[],[[]],[],[],[],[[]]]
=> [.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(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
[[],[[]],[],[],[[],[]]]
=> [.,[[.,.],[.,[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,4,2,3,1] => ([(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),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[],[[]],[],[[[[]]]]]
=> [.,[[.,.],[.,[[[[.,.],.],.],.]]]]
=> [5,6,7,8,4,2,3,1] => ([(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),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[[],[[]],[[]],[[]],[]]
=> [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> [8,6,7,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[],[[]],[[]],[[[]]]]
=> [.,[[.,.],[[.,.],[[[.,.],.],.]]]]
=> [6,7,8,4,5,2,3,1] => ?
=> ? = 3
[[],[[]],[[[]]],[[]]]
=> [.,[[.,.],[[[.,.],.],[[.,.],.]]]]
=> [7,8,4,5,6,2,3,1] => ([(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),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[],[[]],[[[[]]]],[]]
=> [.,[[.,.],[[[[.,.],.],.],[.,.]]]]
=> [8,4,5,6,7,2,3,1] => ([(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),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[[],[[]],[[[[[]]]]]]
=> [.,[[.,.],[[[[[.,.],.],.],.],.]]]
=> [4,5,6,7,8,2,3,1] => ([(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),(5,7),(6,7)],8)
=> ? = 5
[[],[[],[]],[],[],[[]]]
=> [.,[[.,[.,.]],[.,[.,[[.,.],.]]]]]
=> [7,8,6,5,3,2,4,1] => ([(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),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[[],[[],[]],[],[[],[]]]
=> [.,[[.,[.,.]],[.,[[.,[.,.]],.]]]]
=> [7,6,8,5,3,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(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
[[],[[[]]],[],[[],[]]]
=> [.,[[[.,.],.],[.,[[.,[.,.]],.]]]]
=> [7,6,8,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(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)
=> ? = 3
[[],[[[]]],[],[[[]]]]
=> [.,[[[.,.],.],[.,[[[.,.],.],.]]]]
=> [6,7,8,5,2,3,4,1] => ([(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),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[[],[[[]]],[[]],[[]]]
=> [.,[[[.,.],.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,2,3,4,1] => ([(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),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[],[[[]]],[[[]]],[]]
=> [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [8,5,6,7,2,3,4,1] => ([(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),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[[],[[],[]],[[[[]]]]]
=> [.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [5,6,7,8,3,2,4,1] => ([(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),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[[],[[[]]],[[[[]]]]]
=> [.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [5,6,7,8,2,3,4,1] => ([(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),(4,7),(5,7),(6,7)],8)
=> ? = 4
[[],[[[[]]]],[],[[]]]
=> [.,[[[[.,.],.],.],[.,[[.,.],.]]]]
=> [7,8,6,2,3,4,5,1] => ([(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),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[[],[[[]],[]],[[]],[]]
=> [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [8,6,7,4,2,3,5,1] => ?
=> ? = 3
[[],[[[[]]]],[[]],[]]
=> [.,[[[[.,.],.],.],[[.,.],[.,.]]]]
=> [8,6,7,2,3,4,5,1] => ([(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),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4
[[],[[[],[]]],[[[]]]]
=> [.,[[[.,[.,.]],.],[[[.,.],.],.]]]
=> [6,7,8,3,2,4,5,1] => ([(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),(4,7),(5,7),(6,7)],8)
=> ? = 3
[[],[[],[[[]]]],[],[]]
=> [.,[[.,[[[.,.],.],.]],[.,[.,.]]]]
=> [8,7,3,4,5,2,6,1] => ?
=> ? = 4
[[],[[[[]]],[]],[],[]]
=> [.,[[[[.,.],.],[.,.]],[.,[.,.]]]]
=> [8,7,5,2,3,4,6,1] => ?
=> ? = 4
[[],[[[],[[]]]],[],[]]
=> [.,[[[.,[[.,.],.]],.],[.,[.,.]]]]
=> [8,7,3,4,2,5,6,1] => ?
=> ? = 4
[[],[[[[]],[]]],[],[]]
=> [.,[[[[.,.],[.,.]],.],[.,[.,.]]]]
=> [8,7,4,2,3,5,6,1] => ?
=> ? = 4
[[],[[[[],[]]]],[],[]]
=> [.,[[[[.,[.,.]],.],.],[.,[.,.]]]]
=> [8,7,3,2,4,5,6,1] => ?
=> ? = 4
[[],[[[[],[]]]],[[]]]
=> [.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [7,8,3,2,4,5,6,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 4
[[],[[[[[]]]]],[[]]]
=> [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [7,8,2,3,4,5,6,1] => ([(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),(5,7),(6,7)],8)
=> ? = 5
[[],[[[]],[[]],[]],[]]
=> [.,[[[.,.],[[.,.],[.,.]]],[.,.]]]
=> [8,6,4,5,2,3,7,1] => ?
=> ? = 3
[[],[[[],[[]]],[]],[]]
=> [.,[[[.,[[.,.],.]],[.,.]],[.,.]]]
=> [8,6,3,4,2,5,7,1] => ?
=> ? = 4
[[],[[[],[]],[[],[]]]]
=> [.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [6,5,7,3,2,4,8,1] => ([(0,7),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
Matching statistic: St000098
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00049: Ordered trees āto binary tree: left brother = left childā¶ Binary trees
Mp00017: Binary trees āto 312-avoiding permutationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000098: Graphs ā¶ ā¤Result quality: 53% āvalues known / values provided: 53%ādistinct values known / distinct values provided: 89%
Mp00017: Binary trees āto 312-avoiding permutationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000098: Graphs ā¶ ā¤Result quality: 53% āvalues known / values provided: 53%ādistinct values known / distinct values provided: 89%
Values
[]
=> .
=> ? => ?
=> ? = 0
[[]]
=> [.,.]
=> [1] => ([],1)
=> 1
[[],[]]
=> [[.,.],.]
=> [1,2] => ([],2)
=> 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 2
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 2
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 2
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 2
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => ([],5)
=> 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 2
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 2
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 2
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 2
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 2
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 2
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 3
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 2
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[]],[[]],[[],[]]]
=> [[[.,[.,.]],[.,.]],[[.,.],.]]
=> [2,1,4,3,6,7,5] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? = 2
[[[]],[[]],[[[]]]]
=> [[[.,[.,.]],[.,.]],[.,[.,.]]]
=> [2,1,4,3,7,6,5] => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[],[]],[[]]]
=> [[[.,[.,.]],[[.,.],.]],[.,.]]
=> [2,1,4,5,3,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? = 2
[[[]],[[[]]],[[]]]
=> [[[.,[.,.]],[.,[.,.]]],[.,.]]
=> [2,1,5,4,3,7,6] => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[],[],[],[]]]
=> [[.,[.,.]],[[[[.,.],.],.],.]]
=> [2,1,4,5,6,7,3] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
[[[]],[[],[],[[]]]]
=> [[.,[.,.]],[[[.,.],.],[.,.]]]
=> [2,1,4,5,7,6,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[],[[]],[]]]
=> [[.,[.,.]],[[[.,.],[.,.]],.]]
=> [2,1,4,6,5,7,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[],[[],[]]]]
=> [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [2,1,4,6,7,5,3] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[],[[[]]]]]
=> [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,7,6,5,3] => ([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[]],[[[]],[],[]]]
=> [[.,[.,.]],[[[.,[.,.]],.],.]]
=> [2,1,5,4,6,7,3] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[[]],[[]]]]
=> [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [2,1,5,4,7,6,3] => ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[[],[]],[]]]
=> [[.,[.,.]],[[.,[[.,.],.]],.]]
=> [2,1,5,6,4,7,3] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[[[]]],[]]]
=> [[.,[.,.]],[[.,[.,[.,.]]],.]]
=> [2,1,6,5,4,7,3] => ([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[]],[[[],[],[]]]]
=> [[.,[.,.]],[.,[[[.,.],.],.]]]
=> [2,1,5,6,7,4,3] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[]],[[[],[[]]]]]
=> [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [2,1,5,7,6,4,3] => ([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[]],[[[[]],[]]]]
=> [[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [2,1,6,5,7,4,3] => ([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[]],[[[[],[]]]]]
=> [[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [2,1,6,7,5,4,3] => ([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[]],[[[[[]]]]]]
=> [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[[[],[]],[[]],[[]]]
=> [[[.,[[.,.],.]],[.,.]],[.,.]]
=> [2,3,1,5,4,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? = 2
[[[[]]],[[]],[[]]]
=> [[[.,[.,[.,.]]],[.,.]],[.,.]]
=> [3,2,1,5,4,7,6] => ([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[]],[[],[],[]]]
=> [[.,[[.,.],.]],[[[.,.],.],.]]
=> [2,3,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 2
[[[],[]],[[],[[]]]]
=> [[.,[[.,.],.]],[[.,.],[.,.]]]
=> [2,3,1,5,7,6,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[]],[[[]],[]]]
=> [[.,[[.,.],.]],[[.,[.,.]],.]]
=> [2,3,1,6,5,7,4] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[]],[[[],[]]]]
=> [[.,[[.,.],.]],[.,[[.,.],.]]]
=> [2,3,1,6,7,5,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ? = 3
[[[],[]],[[[[]]]]]
=> [[.,[[.,.],.]],[.,[.,[.,.]]]]
=> [2,3,1,7,6,5,4] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[[]]],[[],[],[]]]
=> [[.,[.,[.,.]]],[[[.,.],.],.]]
=> [3,2,1,5,6,7,4] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ? = 3
[[[[]]],[[],[[]]]]
=> [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [3,2,1,5,7,6,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[]]],[[[]],[]]]
=> [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [3,2,1,6,5,7,4] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[]]],[[[],[]]]]
=> [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [3,2,1,6,7,5,4] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[]]],[[[[]]]]]
=> [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[],[],[]],[[],[]]]
=> [[.,[[[.,.],.],.]],[[.,.],.]]
=> [2,3,4,1,6,7,5] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 2
[[[],[],[]],[[[]]]]
=> [[.,[[[.,.],.],.]],[.,[.,.]]]
=> [2,3,4,1,7,6,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
=> ? = 3
[[[],[[]]],[[],[]]]
=> [[.,[[.,.],[.,.]]],[[.,.],.]]
=> [2,4,3,1,6,7,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[[]]],[[[]]]]
=> [[.,[[.,.],[.,.]]],[.,[.,.]]]
=> [2,4,3,1,7,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[]],[]],[[],[]]]
=> [[.,[[.,[.,.]],.]],[[.,.],.]]
=> [3,2,4,1,6,7,5] => ([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[]],[]],[[[]]]]
=> [[.,[[.,[.,.]],.]],[.,[.,.]]]
=> [3,2,4,1,7,6,5] => ([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[],[]]],[[],[]]]
=> [[.,[.,[[.,.],.]]],[[.,.],.]]
=> [3,4,2,1,6,7,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
=> ? = 3
[[[[],[]]],[[[]]]]
=> [[.,[.,[[.,.],.]]],[.,[.,.]]]
=> [3,4,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[[]]]],[[],[]]]
=> [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [4,3,2,1,6,7,5] => ([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[[[]]]],[[[]]]]
=> [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[],[],[],[]],[[]]]
=> [[.,[[[[.,.],.],.],.]],[.,.]]
=> [2,3,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 2
[[[],[],[[]]],[[]]]
=> [[.,[[[.,.],.],[.,.]]],[.,.]]
=> [2,3,5,4,1,7,6] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[[]],[]],[[]]]
=> [[.,[[[.,.],[.,.]],.]],[.,.]]
=> [2,4,3,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[[],[]]],[[]]]
=> [[.,[[.,.],[[.,.],.]]],[.,.]]
=> [2,4,5,3,1,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[],[[[]]]],[[]]]
=> [[.,[[.,.],[.,[.,.]]]],[.,.]]
=> [2,5,4,3,1,7,6] => ([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[[[[]],[],[]],[[]]]
=> [[.,[[[.,[.,.]],.],.]],[.,.]]
=> [3,2,4,5,1,7,6] => ([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[]],[[]]],[[]]]
=> [[.,[[.,[.,.]],[.,.]]],[.,.]]
=> [3,2,5,4,1,7,6] => ([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 3
[[[[],[]],[]],[[]]]
=> [[.,[[.,[[.,.],.]],.]],[.,.]]
=> [3,4,2,5,1,7,6] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[[[[[]]],[]],[[]]]
=> [[.,[[.,[.,[.,.]]],.]],[.,.]]
=> [4,3,2,5,1,7,6] => ([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
The following 80 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000306The bounce count of a Dyck path. St000730The maximal arc length of a set partition. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001494The Alon-Tarsi number of a graph. St000053The number of valleys of the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St000527The width of the poset. St001717The largest size of an interval in a poset. St000662The staircase size of the code of a permutation. St000141The maximum drop size of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St000245The number of ascents of a permutation. St000470The number of runs in a permutation. St000907The number of maximal antichains of minimal length in a poset. St000308The height of the tree associated to a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000734The last entry in the first row of a standard tableau. St000676The number of odd rises of a Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St000542The number of left-to-right-minima of a permutation. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000808The number of up steps of the associated bargraph. St000166The depth minus 1 of an ordered tree. St000094The depth of an ordered tree. St000062The length of the longest increasing subsequence of the permutation. St000015The number of peaks of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000822The Hadwiger number of the graph. St000877The depth of the binary word interpreted as a path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000021The number of descents of a permutation. St000080The rank of the poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000451The length of the longest pattern of the form k 1 2. St001330The hat guessing number of a graph. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001674The number of vertices of the largest induced star graph in the graph. St001589The nesting number of a perfect matching. St001323The independence gap of a graph. St001590The crossing number of a perfect matching. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001621The number of atoms of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000299The number of nonisomorphic vertex-induced subtrees. St000983The length of the longest alternating subword. St001820The size of the image of the pop stack sorting operator.
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