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Your data matches 36 different statistics following compositions of up to 3 maps.
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Matching statistic: St000094
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Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000094: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000094: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [[]]
=> 2
[1,2] => [.,[.,.]]
=> [[],[]]
=> 2
[2,1] => [[.,.],.]
=> [[[]]]
=> 3
[1,2,3] => [.,[.,[.,.]]]
=> [[],[],[]]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [[],[[]]]
=> 3
[2,1,3] => [[.,.],[.,.]]
=> [[[]],[]]
=> 3
[2,3,1] => [[.,.],[.,.]]
=> [[[]],[]]
=> 3
[3,1,2] => [[.,[.,.]],.]
=> [[[],[]]]
=> 3
[3,2,1] => [[[.,.],.],.]
=> [[[[]]]]
=> 4
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 2
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[],[],[[]]]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 3
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[],[[],[]]]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[],[[[]]]]
=> 4
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 3
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 3
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 3
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 3
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 3
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 3
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 4
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 3
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [[[],[],[]]]
=> 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [[[],[[]]]]
=> 4
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 4
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 4
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [[[[],[]]]]
=> 4
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[[]]]]]
=> 5
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[],[],[],[],[]]
=> 2
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[],[],[],[[]]]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [[],[],[[],[]]]
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[],[],[[[]]]]
=> 4
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 3
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 3
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 3
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[],[[[]]],[]]
=> 4
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [[],[[[]]],[]]
=> 4
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 3
Description
The depth of an ordered tree.
Matching statistic: St001039
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 88%●distinct values known / distinct values provided: 70%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 88%●distinct values known / distinct values provided: 70%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> ? = 2 - 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[7,8,6,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 7 - 1
[7,6,8,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 7 - 1
[7,6,5,8,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 7 - 1
[7,6,5,4,8,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 7 - 1
[7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5 - 1
[7,5,6,8,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5 - 1
[7,8,5,3,4,6,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5 - 1
[8,3,4,5,6,7,1,2] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 - 1
[7,6,5,4,3,8,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 7 - 1
[7,5,3,4,6,8,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5 - 1
[7,8,6,5,4,1,2,3] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [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]
=> ? = 6 - 1
[7,6,8,5,4,1,2,3] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [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]
=> ? = 6 - 1
[7,6,5,8,4,1,2,3] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [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]
=> ? = 6 - 1
[7,6,5,4,8,1,2,3] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [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]
=> ? = 6 - 1
[7,8,5,6,3,1,2,4] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5 - 1
[7,5,6,8,3,1,2,4] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5 - 1
[7,8,5,3,1,2,4,6] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5 - 1
[8,3,4,5,6,1,2,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 - 1
[8,3,4,5,1,2,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 - 1
[8,3,4,1,2,5,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 - 1
[8,3,1,2,4,5,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 - 1
[8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [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]
=> ? = 3 - 1
[7,6,5,4,3,1,2,8] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 7 - 1
[7,6,5,4,1,2,3,8] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [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]
=> ? = 6 - 1
[7,5,3,1,2,4,6,8] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5 - 1
[1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [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]
=> ? = 2 - 1
[1,2,3,4,8,7,6,5] => [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5 - 1
[1,2,3,4,5,8,7,6] => [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 4 - 1
[1,2,3,4,5,6,8,7] => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[1,2,3,4,8,6,7,5] => [.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[1,2,3,8,5,6,7,4] => [.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[1,2,3,8,6,5,7,4] => [.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[1,2,3,8,6,7,5,4] => [.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[1,2,8,4,5,6,7,3] => [.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[1,8,2,3,4,5,6,7] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [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]
=> ? = 3 - 1
[1,2,8,3,4,5,6,7] => [.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [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]
=> ? = 3 - 1
[1,2,3,8,4,5,6,7] => [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [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]
=> ? = 3 - 1
[1,2,3,4,8,5,6,7] => [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [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]
=> ? = 3 - 1
[1,2,3,4,5,8,6,7] => [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3 - 1
[3,8,1,6,7,4,5,2] => [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 4 - 1
[9,10,8,7,6,5,4,3,1,2] => [[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
=> ?
=> ?
=> ? = 9 - 1
[7,3,6,1,2,4,5,8] => [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 4 - 1
[5,8,3,4,1,2,6,7] => [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 4 - 1
[5,8,3,6,1,2,4,7] => [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 4 - 1
[7,5,3,6,1,2,4,8] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5 - 1
[7,5,3,8,1,2,4,6] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5 - 1
[5,8,3,6,7,1,2,4] => [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 4 - 1
[7,5,3,6,8,1,2,4] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> ? = 5 - 1
[3,1,8,2,4,7,5,6] => [[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 4 - 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000147
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => [1]
=> 1 = 2 - 1
[1,2] => [.,[.,.]]
=> [2,1] => [1,1]
=> 1 = 2 - 1
[2,1] => [[.,.],.]
=> [1,2] => [2]
=> 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1]
=> 1 = 2 - 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 2 = 3 - 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [3]
=> 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1]
=> 1 = 2 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1]
=> 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,1,1]
=> 2 = 3 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,1,1]
=> 2 = 3 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,1]
=> 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1]
=> 3 = 4 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,1]
=> 2 = 3 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2]
=> 2 = 3 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,1]
=> 2 = 3 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,1]
=> 2 = 3 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2]
=> 2 = 3 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2]
=> 2 = 3 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,1]
=> 2 = 3 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,1]
=> 2 = 3 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,1]
=> 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,1]
=> 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,1]
=> 2 = 3 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,1]
=> 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,1]
=> 2 = 3 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1]
=> 3 = 4 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1]
=> 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1]
=> 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,1]
=> 3 = 4 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [4]
=> 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 2 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1]
=> 2 = 3 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [2,1,1,1]
=> 2 = 3 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [2,1,1,1]
=> 2 = 3 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,1,1]
=> 2 = 3 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 3 = 4 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [2,1,1,1]
=> 2 = 3 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,2,1]
=> 2 = 3 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [2,1,1,1]
=> 2 = 3 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [2,1,1,1]
=> 2 = 3 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,2,1]
=> 2 = 3 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,2,1]
=> 2 = 3 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [2,1,1,1]
=> 2 = 3 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [2,1,1,1]
=> 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [3,1,1]
=> 3 = 4 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [3,1,1]
=> 3 = 4 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [2,1,1,1]
=> 2 = 3 - 1
[1,2,8,4,5,6,7,3] => [.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [7,6,5,3,4,8,2,1] => ?
=> ? = 4 - 1
[1,2,8,5,4,6,7,3] => [.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [7,6,3,4,5,8,2,1] => ?
=> ? = 5 - 1
[1,2,8,5,6,4,7,3] => [.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [7,6,3,4,5,8,2,1] => ?
=> ? = 5 - 1
[1,2,8,5,6,7,4,3] => [.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [7,6,3,4,5,8,2,1] => ?
=> ? = 5 - 1
[1,8,3,4,5,6,7,2] => [.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> [7,6,5,4,2,3,8,1] => ?
=> ? = 4 - 1
[2,1,6,4,5,3,8,7] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [7,8,5,3,4,6,1,2] => ?
=> ? = 4 - 1
[2,1,6,4,5,8,7,3] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [7,8,5,3,4,6,1,2] => ?
=> ? = 4 - 1
[2,1,8,4,6,5,7,3] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [7,5,6,3,4,8,1,2] => ?
=> ? = 4 - 1
[2,1,8,4,6,7,5,3] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [7,5,6,3,4,8,1,2] => ?
=> ? = 4 - 1
[2,6,4,3,5,1,8,7] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [7,8,5,3,4,6,1,2] => ?
=> ? = 4 - 1
[2,6,4,3,5,8,7,1] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [7,8,5,3,4,6,1,2] => ?
=> ? = 4 - 1
[2,8,4,3,6,5,7,1] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [7,5,6,3,4,8,1,2] => ?
=> ? = 4 - 1
[2,8,4,3,6,7,5,1] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [7,5,6,3,4,8,1,2] => ?
=> ? = 4 - 1
[2,8,4,6,5,3,7,1] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [7,5,6,3,4,8,1,2] => ?
=> ? = 4 - 1
[4,2,3,1,6,5,8,7] => [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> [7,8,5,6,3,1,2,4] => ?
=> ? = 4 - 1
[4,2,3,1,6,8,7,5] => [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> [7,8,5,6,3,1,2,4] => ?
=> ? = 4 - 1
[4,2,3,1,8,6,7,5] => [[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> [7,5,6,8,3,1,2,4] => ?
=> ? = 4 - 1
[4,2,3,6,5,8,7,1] => [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> [7,8,5,6,3,1,2,4] => ?
=> ? = 4 - 1
[6,2,4,3,5,1,8,7] => [[[.,.],[[.,.],[.,.]]],[[.,.],.]]
=> [7,8,5,3,4,1,2,6] => ?
=> ? = 4 - 1
[8,2,4,3,6,5,7,1] => [[[.,.],[[.,.],[[.,.],[.,.]]]],.]
=> [7,5,6,3,4,1,2,8] => ?
=> ? = 4 - 1
[8,2,4,3,6,7,5,1] => [[[.,.],[[.,.],[[.,.],[.,.]]]],.]
=> [7,5,6,3,4,1,2,8] => ?
=> ? = 4 - 1
[8,2,6,4,5,3,7,1] => [[[.,.],[[[.,.],[.,.]],[.,.]]],.]
=> [7,5,3,4,6,1,2,8] => ?
=> ? = 5 - 1
[8,2,6,4,5,7,3,1] => [[[.,.],[[[.,.],[.,.]],[.,.]]],.]
=> [7,5,3,4,6,1,2,8] => ?
=> ? = 5 - 1
[3,8,1,6,7,4,5,2] => [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,2,1,3] => ?
=> ? = 4 - 1
[8,4,6,2,7,3,5,1] => [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [7,5,6,3,1,2,4,8] => ?
=> ? = 5 - 1
[7,3,6,1,2,4,5,8] => [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,2,1,3,7] => ?
=> ? = 4 - 1
[4,2,6,8,1,3,5,7] => [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> [7,8,5,6,3,1,2,4] => ?
=> ? = 4 - 1
[6,2,4,8,1,3,5,7] => [[[.,.],[[.,.],[.,.]]],[[.,.],.]]
=> [7,8,5,3,4,1,2,6] => ?
=> ? = 4 - 1
[5,8,3,4,1,2,6,7] => [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [7,6,8,4,2,1,3,5] => ?
=> ? = 4 - 1
[5,8,3,6,1,2,4,7] => [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [7,6,8,4,2,1,3,5] => ?
=> ? = 4 - 1
[8,4,2,6,1,3,5,7] => [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [7,5,6,3,1,2,4,8] => ?
=> ? = 5 - 1
[8,6,2,4,1,3,5,7] => [[[[.,.],[[.,.],[.,.]]],[.,.]],.]
=> [7,5,3,4,1,2,6,8] => ?
=> ? = 5 - 1
[5,8,3,6,7,1,2,4] => [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [7,6,8,4,2,1,3,5] => ?
=> ? = 4 - 1
[8,6,2,4,7,1,3,5] => [[[[.,.],[[.,.],[.,.]]],[.,.]],.]
=> [7,5,3,4,1,2,6,8] => ?
=> ? = 5 - 1
[8,4,6,2,5,1,3,7] => [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [7,5,6,3,1,2,4,8] => ?
=> ? = 5 - 1
[8,6,7,2,4,1,3,5] => [[[[.,.],[[.,.],[.,.]]],[.,.]],.]
=> [7,5,3,4,1,2,6,8] => ?
=> ? = 5 - 1
[3,1,8,2,6,4,5,7] => [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,2,1,3] => ?
=> ? = 4 - 1
[3,1,8,2,6,7,4,5] => [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,2,1,3] => ?
=> ? = 4 - 1
[5,1,4,2,8,3,6,7] => [[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [7,6,8,3,2,4,1,5] => ?
=> ? = 4 - 1
[3,1,8,6,2,4,5,7] => [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,2,1,3] => ?
=> ? = 4 - 1
[3,1,8,6,2,7,4,5] => [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,2,1,3] => ?
=> ? = 4 - 1
[5,3,1,2,8,4,6,7] => [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [7,6,8,4,2,1,3,5] => ?
=> ? = 4 - 1
[7,3,1,2,6,4,8,5] => [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,2,1,3,7] => ?
=> ? = 4 - 1
[7,3,1,6,2,4,8,5] => [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,2,1,3,7] => ?
=> ? = 4 - 1
[2,6,4,1,3,8,5,7] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [7,8,5,3,4,6,1,2] => ?
=> ? = 4 - 1
[2,8,4,1,6,3,5,7] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [7,5,6,3,4,8,1,2] => ?
=> ? = 4 - 1
[2,8,4,1,6,7,3,5] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [7,5,6,3,4,8,1,2] => ?
=> ? = 4 - 1
[5,3,4,1,8,2,6,7] => [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [7,6,8,4,2,1,3,5] => ?
=> ? = 4 - 1
[7,3,8,1,6,2,4,5] => [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,2,1,3,7] => ?
=> ? = 4 - 1
[2,8,4,6,1,3,5,7] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [7,5,6,3,4,8,1,2] => ?
=> ? = 4 - 1
Description
The largest part of an integer partition.
Matching statistic: St000013
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 90%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 90%
Values
[1] => [.,.]
=> [1,0]
=> 1 = 2 - 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1 = 2 - 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[8,6,7,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 6 - 1
[8,6,4,5,7,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 6 - 1
[8,6,4,2,3,5,7,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 6 - 1
[8,6,7,4,2,1,3,5] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 6 - 1
[8,6,4,2,3,5,1,7] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 6 - 1
[8,6,4,2,3,1,5,7] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 6 - 1
[8,6,4,2,1,3,5,7] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 6 - 1
[2,1,4,3,8,6,7,5] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 4 - 1
[2,1,4,8,6,5,7,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 4 - 1
[2,1,4,8,6,7,5,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 4 - 1
[2,1,6,4,5,3,8,7] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 4 - 1
[2,1,6,4,5,8,7,3] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 4 - 1
[2,1,8,4,6,5,7,3] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 4 - 1
[2,1,8,4,6,7,5,3] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 4 - 1
[2,4,3,8,6,5,7,1] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 4 - 1
[2,6,4,3,5,1,8,7] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 4 - 1
[2,6,4,3,5,8,7,1] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 4 - 1
[2,8,4,3,6,5,7,1] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 4 - 1
[2,8,4,3,6,7,5,1] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 4 - 1
[2,8,4,6,5,3,7,1] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 4 - 1
[4,2,3,1,6,5,8,7] => [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[4,2,3,1,6,8,7,5] => [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[4,2,3,1,8,6,7,5] => [[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 4 - 1
[4,2,3,6,5,8,7,1] => [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[6,2,4,3,5,1,8,7] => [[[.,.],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 4 - 1
[8,2,6,4,5,3,7,1] => [[[.,.],[[[.,.],[.,.]],[.,.]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 5 - 1
[8,2,6,4,5,7,3,1] => [[[.,.],[[[.,.],[.,.]],[.,.]]],.]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 5 - 1
[6,4,5,2,3,1,8,7] => [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 5 - 1
[2,1,8,6,7,4,5,3] => [[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 5 - 1
[8,4,6,2,7,3,5,1] => [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 5 - 1
[6,4,8,2,7,1,5,3] => [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 5 - 1
[9,10,8,7,6,5,4,3,1,2] => [[[[[[[[.,[.,.]],.],.],.],.],.],.],[.,.]]
=> ?
=> ? = 9 - 1
[4,2,6,8,1,3,5,7] => [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[6,2,4,8,1,3,5,7] => [[[.,.],[[.,.],[.,.]]],[[.,.],.]]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 4 - 1
[6,4,2,8,1,3,5,7] => [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 5 - 1
[8,4,2,6,1,3,5,7] => [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 5 - 1
[8,6,2,4,1,3,5,7] => [[[[.,.],[[.,.],[.,.]]],[.,.]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 5 - 1
[8,6,2,4,7,1,3,5] => [[[[.,.],[[.,.],[.,.]]],[.,.]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 5 - 1
[6,4,8,2,5,1,3,7] => [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 5 - 1
[6,4,8,2,7,1,3,5] => [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 5 - 1
[8,4,6,2,5,1,3,7] => [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 5 - 1
[8,6,7,2,4,1,3,5] => [[[[.,.],[[.,.],[.,.]]],[.,.]],.]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 5 - 1
[6,8,4,7,2,5,1,3] => [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 5 - 1
[8,6,4,7,2,5,1,3] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 6 - 1
[2,4,1,8,6,3,5,7] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 4 - 1
[2,4,1,8,6,7,3,5] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 4 - 1
[2,6,4,1,3,8,5,7] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 4 - 1
[2,8,4,1,6,3,5,7] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 4 - 1
[2,8,4,1,6,7,3,5] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 4 - 1
[2,8,4,6,1,3,5,7] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 4 - 1
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: St000442
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 81%●distinct values known / distinct values provided: 70%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 81%●distinct values known / distinct values provided: 70%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> ? = 2 - 2
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 2 - 2
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1 = 3 - 2
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 3 - 2
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2 = 4 - 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 4 - 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 4 - 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 4 - 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 4 - 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 4 - 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 3 - 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 4 - 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 3 - 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 4 - 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 4 - 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 3 - 2
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 4 - 2
[7,8,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 8 - 2
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 8 - 2
[7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 8 - 2
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 8 - 2
[7,8,6,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 7 - 2
[7,6,8,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 7 - 2
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 7 - 2
[8,3,4,5,6,7,2,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
[7,6,5,4,3,8,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 8 - 2
[7,6,5,3,4,8,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 7 - 2
[7,8,6,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 7 - 2
[7,6,8,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 7 - 2
[7,6,5,8,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 7 - 2
[8,6,7,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 6 - 2
[8,6,4,5,7,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 6 - 2
[7,6,5,4,8,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 7 - 2
[7,8,6,5,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 7 - 2
[7,6,8,5,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 7 - 2
[7,6,5,8,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 7 - 2
[8,3,4,5,6,2,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
[8,6,4,2,3,5,7,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 6 - 2
[8,3,4,5,2,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
[8,3,4,2,5,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
[8,3,2,4,5,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
[8,2,3,4,5,6,7,1] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 - 2
[7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 8 - 2
[7,6,5,3,4,2,8,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 7 - 2
[7,6,5,4,2,3,8,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 7 - 2
[7,6,5,3,2,4,8,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 7 - 2
[7,8,6,5,4,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 7 - 2
[7,6,8,5,4,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 7 - 2
[7,6,5,8,4,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 7 - 2
[7,6,5,4,8,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 7 - 2
[7,8,6,5,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 7 - 2
[7,6,8,5,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 7 - 2
[7,6,5,8,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> ? = 7 - 2
[8,6,7,4,2,1,3,5] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 6 - 2
[8,3,4,5,6,2,1,7] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
[8,6,4,2,3,5,1,7] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 6 - 2
[8,3,4,2,5,6,1,7] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
[8,3,2,4,5,6,1,7] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
[8,2,3,4,5,6,1,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 - 2
[8,6,4,2,3,1,5,7] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 6 - 2
[8,6,4,2,1,3,5,7] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 6 - 2
[8,3,4,5,2,1,6,7] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
[8,2,3,4,5,1,6,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 - 2
[8,3,4,2,1,5,6,7] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
[8,2,3,4,1,5,6,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 - 2
[8,3,2,1,4,5,6,7] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000786
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 71%●distinct values known / distinct values provided: 70%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 71%●distinct values known / distinct values provided: 70%
Values
[1] => [.,.]
=> [1] => ([],1)
=> 1 = 2 - 1
[1,2] => [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => [[.,.],.]
=> [1,2] => ([],2)
=> 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 3 = 4 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [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 = 2 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [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 = 3 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [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 = 3 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [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 = 3 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [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 = 3 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [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 = 3 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [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 = 3 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[8,3,4,5,6,7,2,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => ([(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)
=> ? = 5 - 1
[8,6,7,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [7,5,3,1,2,4,6,8] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[8,6,4,5,7,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [7,5,3,1,2,4,6,8] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[8,3,4,5,6,2,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => ([(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)
=> ? = 5 - 1
[8,6,4,2,3,5,7,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [7,5,3,1,2,4,6,8] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[8,3,4,5,2,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => ([(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)
=> ? = 5 - 1
[8,3,4,2,5,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => ([(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)
=> ? = 5 - 1
[8,3,2,4,5,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => ([(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)
=> ? = 5 - 1
[8,2,3,4,5,6,7,1] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [7,6,5,4,3,1,2,8] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[8,3,4,5,6,7,1,2] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,2,1,3,8] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[8,6,7,4,2,1,3,5] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [7,5,3,1,2,4,6,8] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[8,3,4,5,6,2,1,7] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => ([(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)
=> ? = 5 - 1
[8,6,4,2,3,5,1,7] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [7,5,3,1,2,4,6,8] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[8,3,4,2,5,6,1,7] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => ([(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)
=> ? = 5 - 1
[8,3,2,4,5,6,1,7] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => ([(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)
=> ? = 5 - 1
[8,2,3,4,5,6,1,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [7,6,5,4,3,1,2,8] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[8,3,4,5,6,1,2,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,2,1,3,8] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[8,6,4,2,3,1,5,7] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [7,5,3,1,2,4,6,8] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[8,6,4,2,1,3,5,7] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [7,5,3,1,2,4,6,8] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6 - 1
[8,3,4,5,2,1,6,7] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => ([(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)
=> ? = 5 - 1
[8,2,3,4,5,1,6,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [7,6,5,4,3,1,2,8] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[8,3,4,5,1,2,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,2,1,3,8] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[8,3,4,2,1,5,6,7] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => ([(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)
=> ? = 5 - 1
[8,2,3,4,1,5,6,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [7,6,5,4,3,1,2,8] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[8,3,4,1,2,5,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,2,1,3,8] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[8,3,2,1,4,5,6,7] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => ([(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)
=> ? = 5 - 1
[8,2,3,1,4,5,6,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [7,6,5,4,3,1,2,8] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[8,3,1,2,4,5,6,7] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,2,1,3,8] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[8,2,1,3,4,5,6,7] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [7,6,5,4,3,1,2,8] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 1
[8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[2,4,6,8,7,5,3,1] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[2,4,6,5,8,7,3,1] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[2,4,3,6,8,7,5,1] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[2,4,6,5,3,8,7,1] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[2,4,3,6,5,8,7,1] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[2,1,4,6,8,7,5,3] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[2,1,4,6,5,8,7,3] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[2,4,3,1,6,8,7,5] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[2,1,4,3,6,8,7,5] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[2,4,6,5,3,1,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[2,4,3,6,5,1,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[2,1,4,6,5,3,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[2,4,3,1,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
[2,1,4,3,8,6,7,5] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [7,5,6,8,3,4,1,2] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4 - 1
[2,1,4,8,6,5,7,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [7,5,6,8,3,4,1,2] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4 - 1
[2,1,4,8,6,7,5,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [7,5,6,8,3,4,1,2] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4 - 1
[2,1,6,4,5,3,8,7] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [7,8,5,3,4,6,1,2] => ?
=> ? = 4 - 1
[2,1,6,4,5,8,7,3] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [7,8,5,3,4,6,1,2] => ?
=> ? = 4 - 1
[2,1,8,4,6,5,7,3] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [7,5,6,3,4,8,1,2] => ?
=> ? = 4 - 1
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: St000093
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 70%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 70%
Values
[1] => [.,.]
=> [1] => ([],1)
=> 1 = 2 - 1
[1,2] => [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => [[.,.],.]
=> [1,2] => ([],2)
=> 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 3 = 4 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [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 = 2 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [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 = 3 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [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 = 3 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [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 = 3 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [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 = 3 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [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 = 3 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [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 = 3 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,1,4,3,7,6,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,1,4,7,3,6,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,1,4,7,6,3,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,1,4,7,6,5,3] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,1,5,4,7,3,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,1,5,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,1,5,7,4,3,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,1,5,7,4,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,1,5,7,6,4,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,1,3,7,6,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,1,7,3,6,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,1,7,6,3,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,1,7,6,5,3] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,3,1,7,6,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,3,7,1,6,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,3,7,6,1,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,3,7,6,5,1] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,7,1,3,6,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,7,1,6,3,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,7,1,6,5,3] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,7,3,1,6,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,7,3,6,1,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,7,3,6,5,1] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,7,6,1,3,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,7,6,1,5,3] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,7,6,3,1,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,7,6,3,5,1] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,7,6,5,1,3] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,4,7,6,5,3,1] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,1,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,1,4,7,3,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,1,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,1,7,4,3,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,1,7,4,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,1,7,6,4,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,4,1,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,4,1,7,3,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,4,1,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,4,3,1,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,4,3,7,1,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,4,3,7,6,1] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,4,7,1,3,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,4,7,1,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,4,7,3,1,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,4,7,3,6,1] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,4,7,6,1,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,4,7,6,3,1] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,7,1,4,3,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
[2,5,7,1,4,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [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)
=> ? = 4 - 1
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: St001203
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 70%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 70%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[7,8,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[7,8,6,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,8,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[8,3,4,5,6,7,2,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[7,6,5,4,3,8,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[7,6,5,3,4,8,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,8,6,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,8,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,5,8,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[8,6,7,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 6 - 1
[8,6,4,5,7,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 6 - 1
[7,6,5,4,8,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,8,6,5,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,8,5,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,5,8,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[8,3,4,5,6,2,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[8,6,4,2,3,5,7,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 6 - 1
[8,3,4,5,2,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[8,3,4,2,5,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[8,3,2,4,5,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 5 - 1
[8,2,3,4,5,6,7,1] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 4 - 1
[7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[7,6,5,3,4,2,8,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,5,4,2,3,8,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,5,3,2,4,8,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,8,6,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,8,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,5,8,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,5,4,8,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 5 - 1
[7,5,6,8,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 5 - 1
[7,8,5,3,4,6,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 5 - 1
[8,3,4,5,6,7,1,2] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 4 - 1
[7,6,5,4,3,8,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,5,3,4,6,8,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 5 - 1
[7,8,6,5,4,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,8,5,4,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,5,8,4,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,5,4,8,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,8,6,5,4,1,2,3] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[7,6,8,5,4,1,2,3] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[7,6,5,8,4,1,2,3] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[7,6,5,4,8,1,2,3] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[7,8,6,5,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,8,5,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,6,5,8,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
Description
We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
In the list $L$ delete the first entry $c_0$ and substract from all other entries $n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra.
Example:
[5,6,6,6,6] goes into [2,2,2,2,1].
Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra.
The statistic return the global dimension of the CNakayama algebra divided by 2.
Matching statistic: St000451
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 90%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 90%
Values
[1] => [.,.]
=> [1,0]
=> [1] => 1 = 2 - 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,2] => 1 = 2 - 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [2,1] => 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 2 - 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 3 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 2 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 3 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3 = 4 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 3 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 3 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 3 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 3 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 3 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2 = 3 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 2 = 3 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 3 = 4 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 3 = 4 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = 2 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2 = 3 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2 = 3 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2 = 3 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2 = 3 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 3 = 4 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2 = 3 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 3 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2 = 3 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2 = 3 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 3 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 3 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2 = 3 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 3 = 4 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 3 = 4 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2 = 3 - 1
[1,5,4,3,2,6,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 5 - 1
[1,5,4,3,6,2,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 5 - 1
[1,5,4,3,6,7,2] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 5 - 1
[1,5,4,6,3,2,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 5 - 1
[1,5,4,6,3,7,2] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 5 - 1
[1,5,4,6,7,3,2] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 5 - 1
[1,5,6,4,3,2,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 5 - 1
[1,5,6,4,3,7,2] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 5 - 1
[1,5,6,4,7,3,2] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 5 - 1
[1,5,6,7,4,3,2] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 5 - 1
[1,6,5,4,3,2,7] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => ? = 6 - 1
[1,6,5,4,3,7,2] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => ? = 6 - 1
[1,6,5,4,7,3,2] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => ? = 6 - 1
[1,6,5,7,4,3,2] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => ? = 6 - 1
[1,6,7,5,4,3,2] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => ? = 6 - 1
[1,7,4,3,2,5,6] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,3,6,7,4] => ? = 5 - 1
[1,7,4,3,5,2,6] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,3,6,7,4] => ? = 5 - 1
[1,7,4,3,5,6,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,3,6,7,4] => ? = 5 - 1
[1,7,4,5,3,2,6] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,3,6,7,4] => ? = 5 - 1
[1,7,4,5,3,6,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,3,6,7,4] => ? = 5 - 1
[1,7,4,5,6,3,2] => [.,[[[[.,.],.],[.,[.,.]]],.]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,2,3,6,7,4] => ? = 5 - 1
[1,7,5,4,2,3,6] => [.,[[[[.,[.,.]],.],[.,.]],.]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,2,3,7,4] => ? = 5 - 1
[1,7,5,4,2,6,3] => [.,[[[[.,[.,.]],.],[.,.]],.]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,2,3,7,4] => ? = 5 - 1
[1,7,5,4,6,2,3] => [.,[[[[.,[.,.]],.],[.,.]],.]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,2,3,7,4] => ? = 5 - 1
[1,7,5,6,4,2,3] => [.,[[[[.,[.,.]],.],[.,.]],.]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,2,3,7,4] => ? = 5 - 1
[2,1,3,4,5,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 3 - 1
[2,1,3,4,6,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 3 - 1
[2,1,3,4,6,7,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 3 - 1
[2,1,3,5,4,6,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 3 - 1
[2,1,3,5,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 1
[2,1,3,5,6,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 3 - 1
[2,1,3,5,6,7,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 3 - 1
[2,1,3,5,7,4,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 1
[2,1,3,5,7,6,4] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 1
[2,1,3,6,5,4,7] => [[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,4,5,7] => ? = 4 - 1
[2,1,3,6,5,7,4] => [[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,4,5,7] => ? = 4 - 1
[2,1,3,6,7,5,4] => [[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,4,5,7] => ? = 4 - 1
[2,1,4,3,5,6,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 3 - 1
[2,1,4,3,5,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 3 - 1
[2,1,4,3,6,5,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3 - 1
[2,1,4,3,6,7,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3 - 1
[2,1,4,3,7,6,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 4 - 1
[2,1,4,5,3,6,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 3 - 1
[2,1,4,5,3,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 3 - 1
[2,1,4,5,6,3,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 3 - 1
[2,1,4,5,6,7,3] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 3 - 1
[2,1,4,5,7,3,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 3 - 1
[2,1,4,5,7,6,3] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 3 - 1
[2,1,4,6,3,5,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3 - 1
[2,1,4,6,3,7,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3 - 1
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St001337
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 70%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 70%
Values
[1] => [.,.]
=> [1] => ([],1)
=> 1 = 2 - 1
[1,2] => [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[2,1] => [[.,.],.]
=> [1,2] => ([],2)
=> 2 = 3 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 2 = 3 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 3 = 4 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 3 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> 3 = 4 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 3 = 4 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 4 = 5 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [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 = 2 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [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 = 3 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [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 = 3 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [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 = 3 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [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 = 3 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [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 = 3 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [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 = 3 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [7,5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [7,5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,3,5,6,4,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [7,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,3,5,7,4,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,3,5,7,6,4] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,3,7,4,5,6] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [7,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,5,3,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [7,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,5,3,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,5,6,3,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [7,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,5,6,7,3] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [7,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,5,7,3,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,5,7,6,3] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,6,3,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,6,3,7,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,6,5,3,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,6,5,7,3] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,6,7,3,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,4,6,7,5,3] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,2,7,3,4,5,6] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,2,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,2,4,5,7,6] => [.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,2,4,6,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,2,4,6,7,5] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,2,5,4,6,7] => [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,2,5,4,7,6] => [.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,2,5,6,4,7] => [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,2,5,6,7,4] => [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,2,5,7,4,6] => [.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,2,5,7,6,4] => [.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,4,2,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,4,2,5,7,6] => [.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,4,2,6,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,4,2,6,7,5] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,4,5,2,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,4,5,2,7,6] => [.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,4,5,6,2,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,4,5,6,7,2] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [7,6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,4,5,7,2,6] => [.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,4,5,7,6,2] => [.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
[1,3,4,6,2,5,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 1
Description
The upper domination number of a graph.
This is the maximum cardinality of a minimal dominating set of $G$.
The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001338The upper irredundance number of a graph. St000141The maximum drop size of a permutation. St000306The bounce count of a Dyck path. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St000662The staircase size of the code of a permutation. St001717The largest size of an interval in a poset. St000245The number of ascents of a permutation. 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. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000314The number of left-to-right-maxima of a permutation. St000080The rank of the poset. St001330The hat guessing number of a graph. St001651The Frankl number of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001589The nesting number of a perfect matching. St001875The number of simple modules with projective dimension at most 1. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000454The largest eigenvalue of a graph if it is integral. St001624The breadth of a lattice.
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