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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000862
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => [1] => 1
[[],[]]
=> [[.,.],.]
=> [1,2] => [1,2] => 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => [1,2] => 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => 2
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => 1
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => 1
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => 2
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => 1
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => 1
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => 2
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => 2
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => 2
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => 1
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => 1
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => 2
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => 1
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => 1
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [1,2,3,5,4] => 2
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => 1
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => 1
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [1,2,4,5,3] => 2
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => 2
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [1,2,5,3,4] => 2
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [1,2,5,3,4] => 2
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => 1
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => 2
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => 1
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => 1
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [1,3,4,5,2] => 2
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => 2
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [1,3,5,2,4] => 2
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => 2
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [1,4,5,2,3] => 2
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [1,4,5,2,3] => 2
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => 2
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,5,2,3,4] => 2
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [1,5,2,3,4] => 2
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [1,5,2,4,3] => 2
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [1,5,2,3,4] => 2
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,5,2,3,4] => 2
Description
The number of parts of the shifted shape of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of parts of the shifted shape.
Matching statistic: St000396
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000396: Binary trees ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [.,.]
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [.,[.,.]]
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [[.,.],.]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 2
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 2
[[],[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ? = 1
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 2
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> ? = 1
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> ? = 1
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> ? = 2
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> ? = 1
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]
=> ? = 1
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [[[.,[[.,[.,[.,[.,.]]]],.]],.],.]
=> ? = 1
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> [[[.,[.,[.,[.,.]]]],[[.,.],.]],.]
=> ? = 2
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> [[[.,[.,[.,[.,.]]]],.],[[.,.],.]]
=> ? = 2
[[],[],[],[[[],[[]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [[.,[[[.,[.,[.,[.,.]]]],.],.]],.]
=> ? = 1
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],[[[.,.],.],.]]
=> ? = 2
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> ? = 1
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> ? = 2
[[],[],[[]],[[],[],[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,[[.,[.,.]],.]]]
=> ? = 2
[[],[],[[]],[[[[]]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> ? = 2
[[],[],[[[]]],[[],[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
=> [[[.,[.,[.,.]]],.],[[.,[.,.]],.]]
=> ? = 2
[[],[],[[],[[]]],[[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
=> [[[.,[[.,[.,[.,.]]],.]],[.,.]],.]
=> ? = 2
[[],[],[[],[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> [[.,[.,[[.,[.,[.,.]]],.]]],[.,.]]
=> ? = 2
[[],[],[[[[[]]]]],[]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
=> ? = 2
[[],[],[[],[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [[[.,[.,[[.,[.,[.,.]]],.]]],.],.]
=> ? = 1
[[],[],[[],[[],[[]]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
=> ? = 1
[[],[],[[],[[[[]]]]]]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [[[[.,[[.,[.,[.,.]]],.]],.],.],.]
=> ? = 1
[[],[],[[[],[],[[]]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [[.,[.,[[[.,[.,[.,.]]],.],.]]],.]
=> ? = 1
[[],[],[[[[[[]]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> ? = 1
[[],[[],[],[],[],[[]]]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[[.,[.,.]],.]]]]],.]
=> ? = 1
[[],[[],[[],[[[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [[[.,[[.,[[.,[.,.]],.]],.]],.],.]
=> ? = 1
[[],[[],[[[[]]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,1,0,0]
=> [[.,[[.,[.,.]],.]],[[[.,.],.],.]]
=> ? = 2
[[],[[],[[[[[]]]]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[[.,[.,.]],.]],.],.],.],.]
=> ? = 1
[[],[[[[]]],[[],[]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [[[.,[.,.]],.],[[.,[[.,.],.]],.]]
=> ? = 2
[[],[[[[[[]]]]],[]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[[.,[.,.]],.],.],.],[[.,.],.]]
=> ? = 2
[[],[[[],[[[[]]]]]]]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[.,[[[.,[.,.]],.],.]],.],.],.]
=> ? = 1
[[],[[[[],[[[]]]]]]]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[[.,[[[[.,[.,.]],.],.],.]],.],.]
=> ? = 1
[[],[[[[[],[[]]]]]]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[.,[[[[[.,[.,.]],.],.],.],.]],.]
=> ? = 1
[[],[[[[[[[]]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> ? = 1
[[[]],[[[[[[]]]]]]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> ? = 2
[[[],[]],[],[],[],[[]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[.,[[.,.],[.,[.,[.,[.,.]]]]]],.]
=> ? = 2
[[[[]]],[],[],[],[[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> ? = 2
[[[],[]],[[[[[]]]]]]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[[.,.],[.,.]]],.],.],.],.]
=> ? = 2
[[[[]]],[[],[],[],[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [[[.,.],.],[.,[.,[[.,[.,.]],.]]]]
=> ? = 2
[[[[]]],[[],[[],[]]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [[[.,.],.],[[.,[[.,[.,.]],.]],.]]
=> ? = 2
[[[[]]],[[[[]]],[]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [[[[[.,.],.],[.,.]],.],[[.,.],.]]
=> ? = 2
[[[[]]],[[[[[]]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> ? = 2
[[[[[]]]],[[[]]],[]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> ? = 2
[[[],[],[]],[[[[]]]]]
=> [1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[[.,.],[.,.]]]],.],.],.]
=> ? = 2
[[[],[[]]],[[],[],[]]]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [[.,[[.,.],.]],[.,[[.,[.,.]],.]]]
=> ? = 2
[[[],[[]]],[[[[]]]]]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[.,[[.,.],.]],[.,.]],.],.],.]
=> ? = 2
[[[[]],[]],[[[[]]]]]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[[.,.],[[.,.],[.,.]]],.],.],.]
=> ? = 2
[[[[],[]]],[[[[]]]]]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[.,[[[.,.],.],[.,.]]],.],.],.]
=> ? = 2
[[[[[]]]],[[],[],[]]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> ? = 2
Description
The register function (or Horton-Strahler number) of a binary tree.
This is different from the dimension of the associated poset for the tree $[[[.,.],[.,.]],[[.,.],[.,.]]]$: its register function is 3, whereas the dimension of the associated poset is 2.
Matching statistic: St000679
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000679: Ordered trees ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000679: Ordered trees ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [[]]
=> [.,.]
=> [[]]
=> 1
[[],[]]
=> [[],[]]
=> [.,[.,.]]
=> [[[]]]
=> 1
[[[]]]
=> [[[]]]
=> [[.,.],.]
=> [[],[]]
=> 1
[[],[],[]]
=> [[],[],[]]
=> [.,[.,[.,.]]]
=> [[[[]]]]
=> 1
[[],[[]]]
=> [[[]],[]]
=> [[.,[.,.]],.]
=> [[[]],[]]
=> 1
[[[]],[]]
=> [[],[[]]]
=> [.,[[.,.],.]]
=> [[[],[]]]
=> 2
[[[],[]]]
=> [[[],[]]]
=> [[.,.],[.,.]]
=> [[],[[]]]
=> 1
[[[[]]]]
=> [[[[]]]]
=> [[[.,.],.],.]
=> [[],[],[]]
=> 1
[[],[],[],[]]
=> [[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1
[[],[],[[]]]
=> [[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> 1
[[],[[]],[]]
=> [[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2
[[],[[],[]]]
=> [[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 1
[[],[[[]]]]
=> [[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> 1
[[[]],[],[]]
=> [[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2
[[[]],[[]]]
=> [[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [[[],[]],[]]
=> 2
[[[],[]],[]]
=> [[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[[[[]]],[]]
=> [[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 2
[[[],[],[]]]
=> [[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 1
[[[],[[]]]]
=> [[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 1
[[[[]],[]]]
=> [[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> 2
[[[[],[]]]]
=> [[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> 1
[[[[[]]]]]
=> [[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [[],[],[],[]]
=> 1
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1
[[],[],[],[[]]]
=> [[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [[[[[]]]],[]]
=> 1
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [[[[[]]],[]]]
=> 2
[[],[],[[],[]]]
=> [[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [[[]],[[[]]]]
=> 1
[[],[],[[[]]]]
=> [[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [[[[]]],[],[]]
=> 1
[[],[[]],[],[]]
=> [[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> 2
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [[[[]],[]],[]]
=> 2
[[],[[],[]],[]]
=> [[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> 2
[[],[[[]]],[]]
=> [[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [[[[]],[],[]]]
=> 2
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [[[[]]],[[]]]
=> 1
[[],[[],[[]]]]
=> [[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [[[]],[],[[]]]
=> 1
[[],[[[]],[]]]
=> [[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [[[],[]],[[]]]
=> 2
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [[[]],[[]],[]]
=> 1
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [[[]],[],[],[]]
=> 1
[[[]],[],[],[]]
=> [[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 2
[[[]],[],[[]]]
=> [[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [[[[],[]]],[]]
=> 2
[[[]],[[]],[]]
=> [[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [[[[],[]],[]]]
=> 2
[[[]],[[],[]]]
=> [[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [[[]],[[],[]]]
=> 2
[[[]],[[[]]]]
=> [[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [[[],[]],[],[]]
=> 2
[[[],[]],[],[]]
=> [[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2
[[[[]]],[],[]]
=> [[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [[[[],[],[]]]]
=> 2
[[[],[]],[[]]]
=> [[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [[],[[],[[]]]]
=> 2
[[[[]]],[[]]]
=> [[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [[[],[],[]],[]]
=> 2
[[[],[],[]],[]]
=> [[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> 2
[[[],[[]]],[]]
=> [[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> 2
[[[[]],[]],[]]
=> [[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> 2
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> 2
[[[[[]]]],[]]
=> [[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [[[],[],[],[]]]
=> 2
[[],[],[],[],[],[],[],[]]
=> [[],[],[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [[[[[[[[[]]]]]]]]]
=> ? = 1
[[],[],[],[],[],[],[[]]]
=> [[[]],[],[],[],[],[],[]]
=> [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [[[[[[[[]]]]]]],[]]
=> ? = 1
[[],[],[],[],[],[[]],[]]
=> [[],[[]],[],[],[],[],[]]
=> [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [[[[[[[[]]]]]],[]]]
=> ? = 2
[[],[],[],[],[],[[[]]]]
=> [[[[]]],[],[],[],[],[]]
=> [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [[[[[[[]]]]]],[],[]]
=> ? = 1
[[],[],[],[],[[],[]],[]]
=> [[],[[],[]],[],[],[],[]]
=> [.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [[[[]],[[[[[]]]]]]]
=> ? = 2
[[],[],[],[],[[],[],[]]]
=> [[[],[],[]],[],[],[],[]]
=> [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> [[[[]]],[[[[[]]]]]]
=> ? = 1
[[],[],[],[],[[[]],[]]]
=> [[[],[[]]],[],[],[],[]]
=> [[.,[[.,.],.]],[.,[.,[.,[.,.]]]]]
=> [[[],[]],[[[[[]]]]]]
=> ? = 2
[[],[],[],[],[[[[]]]]]
=> [[[[[]]]],[],[],[],[]]
=> [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [[[[[[]]]]],[],[],[]]
=> ? = 1
[[],[],[],[[],[],[],[]]]
=> [[[],[],[],[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [[[[[]]]],[[[[]]]]]
=> ? = 1
[[],[],[],[[],[],[[]]]]
=> [[[[]],[],[]],[],[],[]]
=> [[[.,[.,[.,.]]],.],[.,[.,[.,.]]]]
=> [[[[]]],[],[[[[]]]]]
=> ? = 1
[[],[],[],[[],[[]],[]]]
=> [[[],[[]],[]],[],[],[]]
=> [[.,[[.,[.,.]],.]],[.,[.,[.,.]]]]
=> [[[[]],[]],[[[[]]]]]
=> ? = 2
[[],[],[],[[[]],[[]]]]
=> [[[[]],[[]]],[],[],[]]
=> [[[.,[[.,.],.]],.],[.,[.,[.,.]]]]
=> [[[],[]],[],[[[[]]]]]
=> ? = 2
[[],[],[],[[[],[]],[]]]
=> [[[],[[],[]]],[],[],[]]
=> [[.,[[.,[.,.]],[.,[.,[.,.]]]]],.]
=> [[[[]],[[[[]]]]],[]]
=> ? = 2
[[],[],[],[[[[]]],[]]]
=> [[[],[[[]]]],[],[],[]]
=> [[.,[[[.,.],.],.]],[.,[.,[.,.]]]]
=> [[[],[],[]],[[[[]]]]]
=> ? = 2
[[],[],[],[[[],[],[]]]]
=> [[[[],[],[]]],[],[],[]]
=> [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
=> [[[[]]],[[[[]]]],[]]
=> ? = 1
[[],[],[],[[[[]],[]]]]
=> [[[[],[[]]]],[],[],[]]
=> [[[.,[[.,.],.]],[.,[.,[.,.]]]],.]
=> [[[],[]],[[[[]]]],[]]
=> ? = 2
[[],[],[],[[[[[]]]]]]
=> [[[[[[]]]]],[],[],[]]
=> [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [[[[[]]]],[],[],[],[]]
=> ? = 1
[[],[],[[]],[],[],[],[]]
=> [[],[],[],[],[[]],[],[]]
=> [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [[[[[[[[]]],[]]]]]]
=> ? = 2
[[],[],[[]],[[],[],[]]]
=> [[[],[],[]],[[]],[],[]]
=> [[.,[.,[.,.]]],[[.,[.,[.,.]]],.]]
=> [[[[]]],[[[[]]],[]]]
=> ? = 2
[[],[],[[]],[[[[]]]]]
=> [[[[[]]]],[[]],[],[]]
=> [[[[.,[[.,[.,[.,.]]],.]],.],.],.]
=> [[[[[]]],[]],[],[],[]]
=> ? = 2
[[],[],[[],[],[],[]],[]]
=> [[],[[],[],[],[]],[],[]]
=> [.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [[[[[[]]]],[[[]]]]]
=> ? = 2
[[],[],[[],[],[[]]],[]]
=> [[],[[[]],[],[]],[],[]]
=> [.,[[[.,[.,[.,.]]],.],[.,[.,.]]]]
=> [[[[[]]],[],[[[]]]]]
=> ? = 2
[[],[],[[[],[]],[]],[]]
=> [[],[[],[[],[]]],[],[]]
=> [.,[[.,[[.,[.,.]],[.,[.,.]]]],.]]
=> [[[[[]],[[[]]]],[]]]
=> ? = 2
[[],[],[[[],[],[]]],[]]
=> [[],[[[],[],[]]],[],[]]
=> [.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> [[[[[]]],[[[]]],[]]]
=> ? = 2
[[],[],[[[[[]]]]],[]]
=> [[],[[[[[]]]]],[],[]]
=> [.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [[[[[]]],[],[],[],[]]]
=> ? = 2
[[],[],[[],[],[],[],[]]]
=> [[[],[],[],[],[]],[],[]]
=> [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [[[[[[]]]]],[[[]]]]
=> ? = 1
[[],[],[[],[[],[]],[]]]
=> [[[],[[],[]],[]],[],[]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],[.,.]]
=> [[[[]],[[[]]]],[[]]]
=> ? = 2
[[],[],[[[],[],[],[]]]]
=> [[[[],[],[],[]]],[],[]]
=> [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [[[[[]]]],[[[]]],[]]
=> ? = 1
[[],[],[[[],[],[[]]]]]
=> [[[[[]],[],[]]],[],[]]
=> [[[[.,[.,[.,.]]],.],[.,[.,.]]],.]
=> [[[[]]],[],[[[]]],[]]
=> ? = 1
[[],[],[[[[],[],[]]]]]
=> [[[[[],[],[]]]],[],[]]
=> [[[[.,[.,[.,.]]],[.,[.,.]]],.],.]
=> [[[[]]],[[[]]],[],[]]
=> ? = 1
[[],[[],[],[],[],[],[]]]
=> [[[],[],[],[],[],[]],[]]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [[[[[[[]]]]]],[[]]]
=> ? = 1
[[],[[],[],[],[],[[]]]]
=> [[[[]],[],[],[],[]],[]]
=> [[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [[[[[[]]]]],[],[[]]]
=> ? = 1
[[],[[],[[],[],[],[]]]]
=> [[[[],[],[],[]],[]],[]]
=> [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [[[[[]]]],[[]],[[]]]
=> ? = 1
[[],[[],[[],[[]],[]]]]
=> [[[[],[[]],[]],[]],[]]
=> [[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [[[[]],[]],[[]],[[]]]
=> ? = 2
[[],[[],[[[],[]],[]]]]
=> [[[[],[[],[]]],[]],[]]
=> [[[.,[[.,[.,.]],[.,.]]],.],[.,.]]
=> [[[[]],[[]]],[],[[]]]
=> ? = 2
[[],[[],[[[[]]],[]]]]
=> [[[[],[[[]]]],[]],[]]
=> [[[.,[[[.,.],.],.]],[.,.]],[.,.]]
=> [[[],[],[]],[[]],[[]]]
=> ? = 2
[[],[[[]],[],[],[],[]]]
=> [[[],[],[],[],[[]]],[]]
=> [[.,[.,[.,[.,[[.,.],.]]]]],[.,.]]
=> [[[[[[],[]]]]],[[]]]
=> ? = 2
[[],[[[],[],[],[]],[]]]
=> [[[],[[],[],[],[]]],[]]
=> [[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> [[[[[[]]]],[[]]],[]]
=> ? = 2
[[],[[[],[[],[]]],[]]]
=> [[[],[[[],[]],[]]],[]]
=> [[.,[[[.,[.,.]],[.,.]],[.,.]]],.]
=> [[[[]],[[]],[[]]],[]]
=> ? = 2
[[],[[[[],[]],[]],[]]]
=> [[[],[[],[[],[]]]],[]]
=> [[.,[[.,[[.,[.,.]],[.,.]]],.]],.]
=> [[[[[]],[[]]],[]],[]]
=> ? = 2
[[],[[[[],[],[]]],[]]]
=> [[[],[[[],[],[]]]],[]]
=> [[.,[[[.,[.,[.,.]]],[.,.]],.]],.]
=> [[[[[]]],[[]],[]],[]]
=> ? = 2
[[],[[[[[[]]]]],[]]]
=> [[[],[[[[[]]]]]],[]]
=> [[.,[[[[[.,.],.],.],.],.]],[.,.]]
=> [[[],[],[],[],[]],[[]]]
=> ? = 2
[[],[[[],[],[],[],[]]]]
=> [[[[],[],[],[],[]]],[]]
=> [[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [[[[[[]]]]],[[]],[]]
=> ? = 1
[[],[[[[],[],[],[]]]]]
=> [[[[[],[],[],[]]]],[]]
=> [[[[.,[.,[.,[.,.]]]],[.,.]],.],.]
=> [[[[[]]]],[[]],[],[]]
=> ? = 1
[[[]],[[],[],[],[],[]]]
=> [[[],[],[],[],[]],[[]]]
=> [[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [[[[[[]]]]],[[],[]]]
=> ? = 2
[[[]],[[[],[],[],[]]]]
=> [[[[],[],[],[]]],[[]]]
=> [[[.,[.,[.,[.,.]]]],[[.,.],.]],.]
=> [[[[[]]]],[[],[]],[]]
=> ? = 2
[[[[]]],[],[],[],[[]]]
=> [[[]],[],[],[],[[[]]]]
=> [[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> [[[[[[],[],[]]]]],[]]
=> ? = 2
[[[],[]],[[],[],[],[]]]
=> [[[],[],[],[]],[[],[]]]
=> [[.,[.,[.,[.,.]]]],[[.,.],[.,.]]]
=> [[[[[]]]],[[],[[]]]]
=> ? = 2
[[[],[]],[[[],[]],[]]]
=> [[[],[[],[]]],[[],[]]]
=> [[.,[[.,[.,.]],[[.,.],[.,.]]]],.]
=> [[[[]],[[],[[]]]],[]]
=> ? = 2
[[[[]]],[[],[],[],[]]]
=> [[[],[],[],[]],[[[]]]]
=> [[.,[.,[.,[.,.]]]],[[[.,.],.],.]]
=> [[[[[]]]],[[],[],[]]]
=> ? = 2
Description
The pruning number of an ordered tree.
A hanging branch of an ordered tree is a proper factor of the form $[^r]^r$ for some $r\geq 1$. A hanging branch is a maximal hanging branch if it is not a proper factor of another hanging branch.
A pruning of an ordered tree is the act of deleting all its maximal hanging branches. The pruning order of an ordered tree is the number of prunings required to reduce it to $[]$.
Matching statistic: St001741
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001741: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => [1] => 1
[[],[]]
=> [[.,.],.]
=> [1,2] => [1,2] => 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => [1,2] => 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => 2
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => 1
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => 1
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => 2
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => 1
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => 1
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => 2
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => 2
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => 2
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => 1
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => 1
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => 2
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => 1
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => 1
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [1,2,3,5,4] => 2
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => 1
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => 1
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [1,2,4,5,3] => 2
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => 2
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [1,2,5,3,4] => 2
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [1,2,5,3,4] => 2
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => 1
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => 1
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => 2
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => 1
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => 1
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [1,3,4,5,2] => 2
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => 2
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [1,3,5,2,4] => 2
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => 2
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [1,4,5,2,3] => 2
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [1,4,5,2,3] => 2
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => 2
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,5,2,3,4] => 2
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [1,5,2,3,4] => 2
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [1,5,2,4,3] => 2
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [1,5,2,3,4] => 2
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,5,2,3,4] => 2
[[[],[]],[[]],[],[]]
=> [[[[.,[[.,.],.]],[.,.]],.],.]
=> [5,2,3,1,4,6,7] => [1,4,6,7,2,3,5] => ? = 2
[[[[]]],[[]],[],[]]
=> [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [5,3,2,1,4,6,7] => [1,4,6,7,2,3,5] => ? = 2
[[[],[]],[[],[]],[]]
=> [[[.,[[.,.],.]],[[.,.],.]],.]
=> [5,6,2,3,1,4,7] => [1,4,7,2,3,5,6] => ? = 2
[[[],[]],[[[]]],[]]
=> [[[.,[[.,.],.]],[.,[.,.]]],.]
=> [6,5,2,3,1,4,7] => [1,4,7,2,3,5,6] => ? = 2
[[[[]]],[[],[]],[]]
=> [[[.,[.,[.,.]]],[[.,.],.]],.]
=> [5,6,3,2,1,4,7] => [1,4,7,2,3,5,6] => ? = 2
[[[[]]],[[[]]],[]]
=> [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [6,5,3,2,1,4,7] => [1,4,7,2,3,5,6] => ? = 2
[[[],[],[]],[],[],[]]
=> [[[[.,[[[.,.],.],.]],.],.],.]
=> [2,3,4,1,5,6,7] => [1,5,6,7,2,3,4] => ? = 2
[[[],[[]]],[],[],[]]
=> [[[[.,[[.,.],[.,.]]],.],.],.]
=> [4,2,3,1,5,6,7] => [1,5,6,7,2,3,4] => ? = 2
[[[[],[]]],[],[],[]]
=> [[[[.,[.,[[.,.],.]]],.],.],.]
=> [3,4,2,1,5,6,7] => [1,5,6,7,2,3,4] => ? = 2
[[[[[]]]],[],[],[]]
=> [[[[.,[.,[.,[.,.]]]],.],.],.]
=> [4,3,2,1,5,6,7] => [1,5,6,7,2,3,4] => ? = 2
[[[],[],[]],[[]],[]]
=> [[[.,[[[.,.],.],.]],[.,.]],.]
=> [6,2,3,4,1,5,7] => [1,5,7,2,3,4,6] => ? = 2
[[[],[[]]],[[]],[]]
=> [[[.,[[.,.],[.,.]]],[.,.]],.]
=> [6,4,2,3,1,5,7] => [1,5,7,2,3,4,6] => ? = 2
[[[[],[]]],[[]],[]]
=> [[[.,[.,[[.,.],.]]],[.,.]],.]
=> [6,3,4,2,1,5,7] => [1,5,7,2,3,4,6] => ? = 2
[[[[[]]]],[[]],[]]
=> [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [6,4,3,2,1,5,7] => [1,5,7,2,3,4,6] => ? = 2
[[[],[],[],[]],[],[]]
=> [[[.,[[[[.,.],.],.],.]],.],.]
=> [2,3,4,5,1,6,7] => [1,6,7,2,3,4,5] => ? = 2
[[[],[],[[]]],[],[]]
=> [[[.,[[[.,.],.],[.,.]]],.],.]
=> [5,2,3,4,1,6,7] => [1,6,7,2,3,4,5] => ? = 2
[[[],[[],[]]],[],[]]
=> [[[.,[[.,.],[[.,.],.]]],.],.]
=> [4,5,2,3,1,6,7] => [1,6,7,2,3,4,5] => ? = 2
[[[],[[[]]]],[],[]]
=> [[[.,[[.,.],[.,[.,.]]]],.],.]
=> [5,4,2,3,1,6,7] => [1,6,7,2,3,4,5] => ? = 2
[[[[],[],[]]],[],[]]
=> [[[.,[.,[[[.,.],.],.]]],.],.]
=> [3,4,5,2,1,6,7] => [1,6,7,2,3,4,5] => ? = 2
[[[[],[[]]]],[],[]]
=> [[[.,[.,[[.,.],[.,.]]]],.],.]
=> [5,3,4,2,1,6,7] => [1,6,7,2,3,4,5] => ? = 2
[[[[[],[]]]],[],[]]
=> [[[.,[.,[.,[[.,.],.]]]],.],.]
=> [4,5,3,2,1,6,7] => [1,6,7,2,3,4,5] => ? = 2
[[[[[[]]]]],[],[]]
=> [[[.,[.,[.,[.,[.,.]]]]],.],.]
=> [5,4,3,2,1,6,7] => [1,6,7,2,3,4,5] => ? = 2
[[[],[],[],[],[]],[]]
=> [[.,[[[[[.,.],.],.],.],.]],.]
=> [2,3,4,5,6,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[],[],[[]]],[]]
=> [[.,[[[[.,.],.],.],[.,.]]],.]
=> [6,2,3,4,5,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[],[[],[]]],[]]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> [5,6,2,3,4,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[],[[[]]]],[]]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> [6,5,2,3,4,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[[],[],[]]],[]]
=> [[.,[[.,.],[[[.,.],.],.]]],.]
=> [4,5,6,2,3,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[[],[[]]]],[]]
=> [[.,[[.,.],[[.,.],[.,.]]]],.]
=> [6,4,5,2,3,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[[[],[]]]],[]]
=> [[.,[[.,.],[.,[[.,.],.]]]],.]
=> [5,6,4,2,3,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[],[[[[]]]]],[]]
=> [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [6,5,4,2,3,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[[],[],[],[]]],[]]
=> [[.,[.,[[[[.,.],.],.],.]]],.]
=> [3,4,5,6,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[[],[],[[]]]],[]]
=> [[.,[.,[[[.,.],.],[.,.]]]],.]
=> [6,3,4,5,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[[],[[],[]]]],[]]
=> [[.,[.,[[.,.],[[.,.],.]]]],.]
=> [5,6,3,4,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[[],[[[]]]]],[]]
=> [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [6,5,3,4,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[[[],[],[]]]],[]]
=> [[.,[.,[.,[[[.,.],.],.]]]],.]
=> [4,5,6,3,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[[[],[[]]]]],[]]
=> [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [6,4,5,3,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[[[[],[]]]]],[]]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> [5,6,4,3,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[[[[[[]]]]]],[]]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => [1,7,2,3,4,5,6] => ? = 2
[[],[],[],[],[],[],[],[]]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? = 1
[[],[],[],[],[],[],[[]]]
=> [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7,8] => ? = 1
[[],[],[],[],[],[[]],[]]
=> [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [7,1,2,3,4,5,6,8] => [1,2,3,4,5,6,8,7] => ? = 2
[[],[],[],[],[],[[],[]]]
=> [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [7,8,1,2,3,4,5,6] => [1,2,3,4,5,6,7,8] => ? = 1
[[],[],[],[],[],[[[]]]]
=> [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [8,7,1,2,3,4,5,6] => [1,2,3,4,5,6,7,8] => ? = 1
[[],[],[],[],[[],[]],[]]
=> [[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [6,7,1,2,3,4,5,8] => [1,2,3,4,5,8,6,7] => ? = 2
[[],[],[],[],[[],[],[]]]
=> [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [6,7,8,1,2,3,4,5] => [1,2,3,4,5,6,7,8] => ? = 1
[[],[],[],[],[[],[[]]]]
=> [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> [8,6,7,1,2,3,4,5] => [1,2,3,4,5,6,7,8] => ? = 1
[[],[],[],[],[[[]],[]]]
=> [[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [7,6,8,1,2,3,4,5] => [1,2,3,4,5,6,8,7] => ? = 2
[[],[],[],[],[[[],[]]]]
=> [[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> [7,8,6,1,2,3,4,5] => [1,2,3,4,5,6,7,8] => ? = 1
[[],[],[],[],[[[[]]]]]
=> [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [8,7,6,1,2,3,4,5] => [1,2,3,4,5,6,7,8] => ? = 1
[[],[],[],[[],[],[],[]]]
=> [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [5,6,7,8,1,2,3,4] => [1,2,3,4,5,6,7,8] => ? = 1
Description
The largest integer such that all patterns of this size are contained in the permutation.
Matching statistic: St000455
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 45%●distinct values known / distinct values provided: 33%
Values
[[]]
=> [.,.]
=> ([],1)
=> ([],1)
=> ? = 1 - 2
[[],[]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 2
[[[]]]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ? = 1 - 2
[[],[],[]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 2
[[],[[]]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 2
[[[]],[]]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 0 = 2 - 2
[[[],[]]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 2
[[[[]]]]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 1 - 2
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0 = 2 - 2
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0 = 2 - 2
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 1 - 2
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 2 - 2
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[[],[],[[]]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[[],[[]],[]]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[[],[[[]]]]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[[[]],[],[]]]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[[]],[[]]]]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[[[]]],[]]]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[[[[],[],[]]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[[[],[[]]]]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[[[[]],[]]]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0 = 2 - 2
[[[[[],[]]]]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[[[[[]]]]]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 1 - 2
[[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[],[],[],[[]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[],[],[[]],[]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 0 = 2 - 2
[[],[],[],[[],[]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[],[],[[[]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[],[[]],[],[]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[],[[]],[[]]]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[],[[],[]],[]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[],[[[]]],[]]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[],[[],[],[]]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[],[[],[[]]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[],[[[]],[]]]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 0 = 2 - 2
[[],[],[[[],[]]]]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[],[[[[]]]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[[]],[],[],[]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[]],[],[[]]]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[]],[[]],[]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[],[[]],[[],[]]]
=> [.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[]],[[[]]]]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[],[]],[],[]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 2 - 2
[[],[[[]]],[],[]]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 2 - 2
[[],[[],[]],[[]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 2 - 2
[[],[[[]]],[[]]]
=> [.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 2 - 2
[[],[[],[],[]],[]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[],[[]]],[]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[[]],[]],[]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[[],[[[],[]]],[]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[[[]]]],[]]
=> [.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 2 - 2
[[],[[],[],[],[]]]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[[],[],[[]]]]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[[],[[]],[]]]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 0 = 2 - 2
[[],[[],[[],[]]]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[[],[[[]]]]]
=> [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[[[],[],[]]]]
=> [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[[[],[[]]]]]
=> [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
[[],[[[[],[]]]]]
=> [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ? = 1 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000397
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000397: Ordered trees ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Values
[[]]
=> [.,.]
=> [[],[]]
=> 2 = 1 + 1
[[],[]]
=> [.,[.,.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[[]]]
=> [[.,.],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 2 = 1 + 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> 2 = 1 + 1
[[[]],[]]
=> [[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 3 = 2 + 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> 2 = 1 + 1
[[[[]]]]
=> [[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 2 = 1 + 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 2 = 1 + 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> 2 = 1 + 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> 3 = 2 + 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> 2 = 1 + 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> 2 = 1 + 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> 3 = 2 + 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> 3 = 2 + 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> 3 = 2 + 1
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> 3 = 2 + 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> 2 = 1 + 1
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> 2 = 1 + 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> 3 = 2 + 1
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> 2 = 1 + 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> 2 = 1 + 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> 2 = 1 + 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [[],[[],[[],[[[],[]],[]]]]]
=> 2 = 1 + 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> 3 = 2 + 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [[],[[],[[[],[[],[]]],[]]]]
=> 2 = 1 + 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [[],[[],[[[[],[]],[]],[]]]]
=> 2 = 1 + 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> 3 = 2 + 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> 3 = 2 + 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> 3 = 2 + 1
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> 3 = 2 + 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [[],[[[],[[],[[],[]]]],[]]]
=> 2 = 1 + 1
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [[],[[[],[[[],[]],[]]],[]]]
=> 2 = 1 + 1
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> 3 = 2 + 1
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [[],[[[[],[[],[]]],[]],[]]]
=> 2 = 1 + 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [[],[[[[[],[]],[]],[]],[]]]
=> 2 = 1 + 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> 3 = 2 + 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> 3 = 2 + 1
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> 3 = 2 + 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> 3 = 2 + 1
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> 3 = 2 + 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> 3 = 2 + 1
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> 3 = 2 + 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> 3 = 2 + 1
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [[[[],[]],[]],[[[],[]],[]]]
=> 3 = 2 + 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> 3 = 2 + 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> 3 = 2 + 1
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> 3 = 2 + 1
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> 3 = 2 + 1
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> 3 = 2 + 1
[[],[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[],[[],[[],[[],[[],[[],[[],[]]]]]]]]
=> ? = 1 + 1
[[],[],[],[],[],[[]]]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [[],[[],[[],[[],[[],[[[],[]],[]]]]]]]
=> ? = 1 + 1
[[],[],[],[],[[]],[]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [[],[[],[[],[[],[[[],[]],[[],[]]]]]]]
=> ? = 2 + 1
[[],[],[],[],[[],[]]]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [[],[[],[[],[[],[[[],[[],[]]],[]]]]]]
=> ? = 1 + 1
[[],[],[],[],[[[]]]]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [[],[[],[[],[[],[[[[],[]],[]],[]]]]]]
=> ? = 1 + 1
[[],[],[],[[]],[],[]]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [[],[[],[[],[[[],[]],[[],[[],[]]]]]]]
=> ? = 2 + 1
[[],[],[],[[]],[[]]]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [[],[[],[[],[[[],[]],[[[],[]],[]]]]]]
=> ? = 2 + 1
[[],[],[],[[],[]],[]]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [[],[[],[[],[[[],[[],[]]],[[],[]]]]]]
=> ? = 2 + 1
[[],[],[],[[[]]],[]]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [[],[[],[[],[[[[],[]],[]],[[],[]]]]]]
=> ? = 2 + 1
[[],[],[],[[],[],[]]]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [[],[[],[[],[[[],[[],[[],[]]]],[]]]]]
=> ? = 1 + 1
[[],[],[],[[],[[]]]]
=> [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [[],[[],[[],[[[],[[[],[]],[]]],[]]]]]
=> ? = 1 + 1
[[],[],[],[[[]],[]]]
=> [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [[],[[],[[],[[[[],[]],[[],[]]],[]]]]]
=> ? = 2 + 1
[[],[],[],[[[],[]]]]
=> [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [[],[[],[[],[[[[],[[],[]]],[]],[]]]]]
=> ? = 1 + 1
[[],[],[],[[[[]]]]]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [[],[[],[[],[[[[[],[]],[]],[]],[]]]]]
=> ? = 1 + 1
[[],[],[[]],[],[],[]]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [[],[[],[[[],[]],[[],[[],[[],[]]]]]]]
=> ? = 2 + 1
[[],[],[[]],[],[[]]]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [[],[[],[[[],[]],[[],[[[],[]],[]]]]]]
=> ? = 2 + 1
[[],[],[[]],[[]],[]]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [[],[[],[[[],[]],[[[],[]],[[],[]]]]]]
=> ? = 2 + 1
[[],[],[[]],[[],[]]]
=> [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [[],[[],[[[],[]],[[[],[[],[]]],[]]]]]
=> ? = 2 + 1
[[],[],[[]],[[[]]]]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [[],[[],[[[],[]],[[[[],[]],[]],[]]]]]
=> ? = 2 + 1
[[],[],[[],[]],[],[]]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[],[[],[[[],[[],[]]],[[],[[],[]]]]]]
=> ? = 2 + 1
[[],[],[[[]]],[],[]]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[],[[],[[[[],[]],[]],[[],[[],[]]]]]]
=> ? = 2 + 1
[[],[],[[],[]],[[]]]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[],[[],[[[],[[],[]]],[[[],[]],[]]]]]
=> ? = 2 + 1
[[],[],[[[]]],[[]]]
=> [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[],[[],[[[[],[]],[]],[[[],[]],[]]]]]
=> ? = 2 + 1
[[],[],[[],[],[]],[]]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [[],[[],[[[],[[],[[],[]]]],[[],[]]]]]
=> ? = 2 + 1
[[],[],[[],[[]]],[]]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [[],[[],[[[],[[[],[]],[]]],[[],[]]]]]
=> ? = 2 + 1
[[],[],[[[]],[]],[]]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [[],[[],[[[[],[]],[[],[]]],[[],[]]]]]
=> ? = 2 + 1
[[],[],[[[],[]]],[]]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [[],[[],[[[[],[[],[]]],[]],[[],[]]]]]
=> ? = 2 + 1
[[],[],[[[[]]]],[]]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [[],[[],[[[[[],[]],[]],[]],[[],[]]]]]
=> ? = 2 + 1
[[],[],[[],[],[],[]]]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [[],[[],[[[],[[],[[],[[],[]]]]],[]]]]
=> ? = 1 + 1
[[],[],[[],[],[[]]]]
=> [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [[],[[],[[[],[[],[[[],[]],[]]]],[]]]]
=> ? = 1 + 1
[[],[],[[],[[]],[]]]
=> [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [[],[[],[[[],[[[],[]],[[],[]]]],[]]]]
=> ? = 2 + 1
[[],[],[[],[[],[]]]]
=> [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [[],[[],[[[],[[[],[[],[]]],[]]],[]]]]
=> ? = 1 + 1
[[],[],[[],[[[]]]]]
=> [.,[.,[[.,[[[.,.],.],.]],.]]]
=> [[],[[],[[[],[[[[],[]],[]],[]]],[]]]]
=> ? = 1 + 1
[[],[],[[[]],[],[]]]
=> [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [[],[[],[[[[],[]],[[],[[],[]]]],[]]]]
=> ? = 2 + 1
[[],[],[[[]],[[]]]]
=> [.,[.,[[[.,.],[[.,.],.]],.]]]
=> [[],[[],[[[[],[]],[[[],[]],[]]],[]]]]
=> ? = 2 + 1
[[],[],[[[],[]],[]]]
=> [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [[],[[],[[[[],[[],[]]],[[],[]]],[]]]]
=> ? = 2 + 1
[[],[],[[[[]]],[]]]
=> [.,[.,[[[[.,.],.],[.,.]],.]]]
=> [[],[[],[[[[[],[]],[]],[[],[]]],[]]]]
=> ? = 2 + 1
[[],[],[[[],[],[]]]]
=> [.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [[],[[],[[[[],[[],[[],[]]]],[]],[]]]]
=> ? = 1 + 1
[[],[],[[[],[[]]]]]
=> [.,[.,[[[.,[[.,.],.]],.],.]]]
=> [[],[[],[[[[],[[[],[]],[]]],[]],[]]]]
=> ? = 1 + 1
[[],[],[[[[]],[]]]]
=> [.,[.,[[[[.,.],[.,.]],.],.]]]
=> [[],[[],[[[[[],[]],[[],[]]],[]],[]]]]
=> ? = 2 + 1
[[],[],[[[[],[]]]]]
=> [.,[.,[[[[.,[.,.]],.],.],.]]]
=> [[],[[],[[[[[],[[],[]]],[]],[]],[]]]]
=> ? = 1 + 1
[[],[],[[[[[]]]]]]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> [[],[[],[[[[[[],[]],[]],[]],[]],[]]]]
=> ? = 1 + 1
[[],[[]],[],[],[],[]]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [[],[[[],[]],[[],[[],[[],[[],[]]]]]]]
=> ? = 2 + 1
[[],[[]],[],[],[[]]]
=> [.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [[],[[[],[]],[[],[[],[[[],[]],[]]]]]]
=> ? = 2 + 1
[[],[[]],[],[[]],[]]
=> [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [[],[[[],[]],[[],[[[],[]],[[],[]]]]]]
=> ? = 2 + 1
[[],[[]],[],[[],[]]]
=> [.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [[],[[[],[]],[[],[[[],[[],[]]],[]]]]]
=> ? = 2 + 1
[[],[[]],[],[[[]]]]
=> [.,[[.,.],[.,[[[.,.],.],.]]]]
=> [[],[[[],[]],[[],[[[[],[]],[]],[]]]]]
=> ? = 2 + 1
[[],[[]],[[]],[],[]]
=> [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [[],[[[],[]],[[[],[]],[[],[[],[]]]]]]
=> ? = 2 + 1
[[],[[]],[[]],[[]]]
=> [.,[[.,.],[[.,.],[[.,.],.]]]]
=> [[],[[[],[]],[[[],[]],[[[],[]],[]]]]]
=> ? = 2 + 1
[[],[[]],[[],[]],[]]
=> [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [[],[[[],[]],[[[],[[],[]]],[[],[]]]]]
=> ? = 2 + 1
Description
The Strahler number of a rooted tree.
Matching statistic: St001235
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Values
[[]]
=> [1,0]
=> [1] => [1] => 1
[[],[]]
=> [1,0,1,0]
=> [2,1] => [2] => 1
[[[]]]
=> [1,1,0,0]
=> [1,2] => [2] => 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,3,1] => [3] => 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [2,1,3] => [3] => 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,2] => 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [3,1,2] => [3] => 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => [3] => 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4] => 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4] => 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2] => 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4] => 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4] => 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3] => 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3] => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2] => 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,2] => 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4] => 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4] => 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3] => 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4] => 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4] => 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5] => 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5] => 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2] => 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5] => 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5] => 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,3] => 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,3] => 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2] => 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2] => 2
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5] => 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5] => 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,3] => 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5] => 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5] => 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,4] => 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4] => 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,2,2] => 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4] => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,4] => 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,3] => 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [2,3] => 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,3] => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [2,3] => 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2] => 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2] => 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,2,2] => 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,2] => 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [3,2] => 2
[[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => [7] => ? = 1
[[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => [7] => ? = 1
[[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => [5,2] => ? = 2
[[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => [7] => ? = 1
[[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => [7] => ? = 1
[[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => [4,3] => ? = 2
[[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => [4,3] => ? = 2
[[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5] => [5,2] => ? = 2
[[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,5,7,6] => [5,2] => ? = 2
[[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,6,7,1,5] => [7] => ? = 1
[[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5,7] => [7] => ? = 1
[[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,1,7,5,6] => [4,3] => ? = 2
[[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => [7] => ? = 1
[[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => [7] => ? = 1
[[],[],[[]],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => [3,4] => ? = 2
[[],[],[[]],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => [3,4] => ? = 2
[[],[],[[]],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => [3,2,2] => ? = 2
[[],[],[[]],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,7,4,6] => [3,4] => ? = 2
[[],[],[[]],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => [3,4] => ? = 2
[[],[],[[],[]],[],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,7,4] => [4,3] => ? = 2
[[],[],[[[]]],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => [4,3] => ? = 2
[[],[],[[],[]],[[]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,5,1,6,4,7] => [4,3] => ? = 2
[[],[],[[[]]],[[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => [4,3] => ? = 2
[[],[],[[],[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4] => [5,2] => ? = 2
[[],[],[[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,7,6] => [5,2] => ? = 2
[[],[],[[[]],[]],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,4,7,5] => [3,2,2] => ? = 2
[[],[],[[[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,6,1,4,7,5] => [5,2] => ? = 2
[[],[],[[[[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [5,2] => ? = 2
[[],[],[[],[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => [7] => ? = 1
[[],[],[[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => [7] => ? = 1
[[],[],[[],[[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,7,4,6] => [4,3] => ? = 2
[[],[],[[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => [7] => ? = 1
[[],[],[[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => [7] => ? = 1
[[],[],[[[]],[],[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => [3,4] => ? = 2
[[],[],[[[]],[[]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => [3,4] => ? = 2
[[],[],[[[],[]],[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => [4,3] => ? = 2
[[],[],[[[[]]],[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => [4,3] => ? = 2
[[],[],[[[],[],[]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => [7] => ? = 1
[[],[],[[[],[[]]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => [7] => ? = 1
[[],[],[[[[]],[]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => [3,4] => ? = 2
[[],[],[[[[],[]]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [7] => ? = 1
[[],[],[[[[[]]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [7] => ? = 1
[[],[[]],[],[],[],[]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => [2,5] => ? = 2
[[],[[]],[],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => [2,5] => ? = 2
[[],[[]],[],[[]],[]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [2,3,2] => ? = 2
[[],[[]],[],[[],[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,4,5,7,3,6] => [2,5] => ? = 2
[[],[[]],[],[[[]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => [2,5] => ? = 2
[[],[[]],[[]],[],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [2,2,3] => ? = 2
[[],[[]],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => [2,2,3] => ? = 2
[[],[[]],[[],[]],[]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [2,3,2] => ? = 2
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001174
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 67%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 67%
Values
[[]]
=> [.,.]
=> [1] => [1] => ? = 1 - 1
[[],[]]
=> [[.,.],.]
=> [1,2] => [2,1] => 0 = 1 - 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => [1,2] => 0 = 1 - 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 0 = 1 - 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => 0 = 1 - 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => 1 = 2 - 1
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 0 = 1 - 1
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 0 = 1 - 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => 0 = 1 - 1
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => 1 = 2 - 1
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 0 = 1 - 1
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => 0 = 1 - 1
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => 1 = 2 - 1
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => 1 = 2 - 1
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => 1 = 2 - 1
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => 1 = 2 - 1
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => 0 = 1 - 1
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 0 = 1 - 1
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => 1 = 2 - 1
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => 0 = 1 - 1
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [5,4,3,2,1] => 0 = 1 - 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => 0 = 1 - 1
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [5,3,2,1,4] => 1 = 2 - 1
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => 0 = 1 - 1
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => 0 = 1 - 1
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [5,4,2,1,3] => 1 = 2 - 1
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => 1 = 2 - 1
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [5,2,1,4,3] => 1 = 2 - 1
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [5,2,1,3,4] => 1 = 2 - 1
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => 0 = 1 - 1
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => 0 = 1 - 1
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => 1 = 2 - 1
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => 0 = 1 - 1
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => 0 = 1 - 1
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [5,4,3,1,2] => 1 = 2 - 1
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => 1 = 2 - 1
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [5,3,1,2,4] => 1 = 2 - 1
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => 1 = 2 - 1
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => 1 = 2 - 1
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [5,4,1,3,2] => 1 = 2 - 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [5,4,1,2,3] => 1 = 2 - 1
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => 1 = 2 - 1
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => 1 = 2 - 1
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [5,1,4,3,2] => 1 = 2 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [5,1,3,2,4] => 1 = 2 - 1
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [5,1,4,2,3] => 1 = 2 - 1
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [5,1,2,4,3] => 1 = 2 - 1
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => 1 = 2 - 1
[[[],[],[],[]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => 0 = 1 - 1
[[],[],[],[],[],[],[]]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1 - 1
[[],[],[],[],[],[[]]]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => ? = 1 - 1
[[],[],[],[],[[]],[]]
=> [[[[[[.,.],.],.],.],[.,.]],.]
=> [6,1,2,3,4,5,7] => [7,5,4,3,2,1,6] => ? = 2 - 1
[[],[],[],[],[[],[]]]
=> [[[[[.,.],.],.],.],[[.,.],.]]
=> [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => ? = 1 - 1
[[],[],[],[],[[[]]]]
=> [[[[[.,.],.],.],.],[.,[.,.]]]
=> [7,6,1,2,3,4,5] => [5,4,3,2,1,6,7] => ? = 1 - 1
[[],[],[],[[]],[],[]]
=> [[[[[[.,.],.],.],[.,.]],.],.]
=> [5,1,2,3,4,6,7] => [7,6,4,3,2,1,5] => ? = 2 - 1
[[],[],[],[[]],[[]]]
=> [[[[[.,.],.],.],[.,.]],[.,.]]
=> [7,5,1,2,3,4,6] => [6,4,3,2,1,5,7] => ? = 2 - 1
[[],[],[],[[],[]],[]]
=> [[[[[.,.],.],.],[[.,.],.]],.]
=> [5,6,1,2,3,4,7] => [7,4,3,2,1,6,5] => ? = 2 - 1
[[],[],[],[[[]]],[]]
=> [[[[[.,.],.],.],[.,[.,.]]],.]
=> [6,5,1,2,3,4,7] => [7,4,3,2,1,5,6] => ? = 2 - 1
[[],[],[],[[],[],[]]]
=> [[[[.,.],.],.],[[[.,.],.],.]]
=> [5,6,7,1,2,3,4] => [4,3,2,1,7,6,5] => ? = 1 - 1
[[],[],[],[[],[[]]]]
=> [[[[.,.],.],.],[[.,.],[.,.]]]
=> [7,5,6,1,2,3,4] => [4,3,2,1,6,5,7] => ? = 1 - 1
[[],[],[],[[[]],[]]]
=> [[[[.,.],.],.],[[.,[.,.]],.]]
=> [6,5,7,1,2,3,4] => [4,3,2,1,7,5,6] => ? = 2 - 1
[[],[],[],[[[],[]]]]
=> [[[[.,.],.],.],[.,[[.,.],.]]]
=> [6,7,5,1,2,3,4] => [4,3,2,1,5,7,6] => ? = 1 - 1
[[],[],[],[[[[]]]]]
=> [[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => ? = 1 - 1
[[],[],[[]],[],[],[]]
=> [[[[[[.,.],.],[.,.]],.],.],.]
=> [4,1,2,3,5,6,7] => [7,6,5,3,2,1,4] => ? = 2 - 1
[[],[],[[]],[],[[]]]
=> [[[[[.,.],.],[.,.]],.],[.,.]]
=> [7,4,1,2,3,5,6] => [6,5,3,2,1,4,7] => ? = 2 - 1
[[],[],[[]],[[]],[]]
=> [[[[[.,.],.],[.,.]],[.,.]],.]
=> [6,4,1,2,3,5,7] => [7,5,3,2,1,4,6] => ? = 2 - 1
[[],[],[[]],[[],[]]]
=> [[[[.,.],.],[.,.]],[[.,.],.]]
=> [6,7,4,1,2,3,5] => [5,3,2,1,4,7,6] => ? = 2 - 1
[[],[],[[]],[[[]]]]
=> [[[[.,.],.],[.,.]],[.,[.,.]]]
=> [7,6,4,1,2,3,5] => [5,3,2,1,4,6,7] => ? = 2 - 1
[[],[],[[],[]],[],[]]
=> [[[[[.,.],.],[[.,.],.]],.],.]
=> [4,5,1,2,3,6,7] => [7,6,3,2,1,5,4] => ? = 2 - 1
[[],[],[[[]]],[],[]]
=> [[[[[.,.],.],[.,[.,.]]],.],.]
=> [5,4,1,2,3,6,7] => [7,6,3,2,1,4,5] => ? = 2 - 1
[[],[],[[],[]],[[]]]
=> [[[[.,.],.],[[.,.],.]],[.,.]]
=> [7,4,5,1,2,3,6] => [6,3,2,1,5,4,7] => ? = 2 - 1
[[],[],[[[]]],[[]]]
=> [[[[.,.],.],[.,[.,.]]],[.,.]]
=> [7,5,4,1,2,3,6] => [6,3,2,1,4,5,7] => ? = 2 - 1
[[],[],[[],[],[]],[]]
=> [[[[.,.],.],[[[.,.],.],.]],.]
=> [4,5,6,1,2,3,7] => [7,3,2,1,6,5,4] => ? = 2 - 1
[[],[],[[],[[]]],[]]
=> [[[[.,.],.],[[.,.],[.,.]]],.]
=> [6,4,5,1,2,3,7] => [7,3,2,1,5,4,6] => ? = 2 - 1
[[],[],[[[]],[]],[]]
=> [[[[.,.],.],[[.,[.,.]],.]],.]
=> [5,4,6,1,2,3,7] => [7,3,2,1,6,4,5] => ? = 2 - 1
[[],[],[[[],[]]],[]]
=> [[[[.,.],.],[.,[[.,.],.]]],.]
=> [5,6,4,1,2,3,7] => [7,3,2,1,4,6,5] => ? = 2 - 1
[[],[],[[[[]]]],[]]
=> [[[[.,.],.],[.,[.,[.,.]]]],.]
=> [6,5,4,1,2,3,7] => [7,3,2,1,4,5,6] => ? = 2 - 1
[[],[],[[],[],[],[]]]
=> [[[.,.],.],[[[[.,.],.],.],.]]
=> [4,5,6,7,1,2,3] => [3,2,1,7,6,5,4] => ? = 1 - 1
[[],[],[[],[],[[]]]]
=> [[[.,.],.],[[[.,.],.],[.,.]]]
=> [7,4,5,6,1,2,3] => [3,2,1,6,5,4,7] => ? = 1 - 1
[[],[],[[],[[]],[]]]
=> [[[.,.],.],[[[.,.],[.,.]],.]]
=> [6,4,5,7,1,2,3] => [3,2,1,7,5,4,6] => ? = 2 - 1
[[],[],[[],[[],[]]]]
=> [[[.,.],.],[[.,.],[[.,.],.]]]
=> [6,7,4,5,1,2,3] => [3,2,1,5,4,7,6] => ? = 1 - 1
[[],[],[[],[[[]]]]]
=> [[[.,.],.],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,1,2,3] => [3,2,1,5,4,6,7] => ? = 1 - 1
[[],[],[[[]],[],[]]]
=> [[[.,.],.],[[[.,[.,.]],.],.]]
=> [5,4,6,7,1,2,3] => [3,2,1,7,6,4,5] => ? = 2 - 1
[[],[],[[[]],[[]]]]
=> [[[.,.],.],[[.,[.,.]],[.,.]]]
=> [7,5,4,6,1,2,3] => [3,2,1,6,4,5,7] => ? = 2 - 1
[[],[],[[[],[]],[]]]
=> [[[.,.],.],[[.,[[.,.],.]],.]]
=> [5,6,4,7,1,2,3] => [3,2,1,7,4,6,5] => ? = 2 - 1
[[],[],[[[[]]],[]]]
=> [[[.,.],.],[[.,[.,[.,.]]],.]]
=> [6,5,4,7,1,2,3] => [3,2,1,7,4,5,6] => ? = 2 - 1
[[],[],[[[],[],[]]]]
=> [[[.,.],.],[.,[[[.,.],.],.]]]
=> [5,6,7,4,1,2,3] => [3,2,1,4,7,6,5] => ? = 1 - 1
[[],[],[[[],[[]]]]]
=> [[[.,.],.],[.,[[.,.],[.,.]]]]
=> [7,5,6,4,1,2,3] => [3,2,1,4,6,5,7] => ? = 1 - 1
[[],[],[[[[]],[]]]]
=> [[[.,.],.],[.,[[.,[.,.]],.]]]
=> [6,5,7,4,1,2,3] => [3,2,1,4,7,5,6] => ? = 2 - 1
[[],[],[[[[],[]]]]]
=> [[[.,.],.],[.,[.,[[.,.],.]]]]
=> [6,7,5,4,1,2,3] => [3,2,1,4,5,7,6] => ? = 1 - 1
[[],[],[[[[[]]]]]]
=> [[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => [3,2,1,4,5,6,7] => ? = 1 - 1
[[],[[]],[],[],[],[]]
=> [[[[[[.,.],[.,.]],.],.],.],.]
=> [3,1,2,4,5,6,7] => [7,6,5,4,2,1,3] => ? = 2 - 1
[[],[[]],[],[],[[]]]
=> [[[[[.,.],[.,.]],.],.],[.,.]]
=> [7,3,1,2,4,5,6] => [6,5,4,2,1,3,7] => ? = 2 - 1
[[],[[]],[],[[]],[]]
=> [[[[[.,.],[.,.]],.],[.,.]],.]
=> [6,3,1,2,4,5,7] => [7,5,4,2,1,3,6] => ? = 2 - 1
[[],[[]],[],[[],[]]]
=> [[[[.,.],[.,.]],.],[[.,.],.]]
=> [6,7,3,1,2,4,5] => [5,4,2,1,3,7,6] => ? = 2 - 1
[[],[[]],[],[[[]]]]
=> [[[[.,.],[.,.]],.],[.,[.,.]]]
=> [7,6,3,1,2,4,5] => [5,4,2,1,3,6,7] => ? = 2 - 1
[[],[[]],[[]],[],[]]
=> [[[[[.,.],[.,.]],[.,.]],.],.]
=> [5,3,1,2,4,6,7] => [7,6,4,2,1,3,5] => ? = 2 - 1
[[],[[]],[[]],[[]]]
=> [[[[.,.],[.,.]],[.,.]],[.,.]]
=> [7,5,3,1,2,4,6] => [6,4,2,1,3,5,7] => ? = 2 - 1
Description
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001621
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
[[]]
=> [.,.]
=> ([],1)
=> ([(0,1)],2)
=> 1
[[],[]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[[]]]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[[]]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[]],[]]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[],[]]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[[]]]]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[],[],[[]]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[],[[]],[]]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[],[[[]]]]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[[]],[],[]]]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[[[]],[[]]]]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[[[[]]],[]]]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[[[],[],[]]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[[],[[]]]]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[[[]],[]]]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[[[[[],[]]]]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[[[[]]]]]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[],[],[[]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[],[[]],[]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 2
[[],[],[],[[],[]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[],[[[]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[[]],[],[]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[],[[]],[[]]]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[],[[],[]],[]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[],[[[]]],[]]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[],[[],[],[]]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[[],[[]]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[[[]],[]]]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 2
[[],[],[[[],[]]]]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[[[[]]]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[[]],[],[],[]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[]],[],[[]]]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[]],[[]],[]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 2
[[],[[]],[[],[]]]
=> [.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[]],[[[]]]]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[],[]],[],[]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 2
[[],[[[]]],[],[]]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 2
[[],[[],[]],[[]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 2
[[],[[[]]],[[]]]
=> [.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 2
[[],[[],[],[]],[]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[],[[]]],[]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[[]],[]],[]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 2
[[],[[[],[]]],[]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[[[]]]],[]]
=> [.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[],[[]],[]]]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 2
[[],[[[]],[],[]]]
=> [.,[[[.,.],[.,[.,.]]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[[[]],[[]]]]
=> [.,[[[.,.],[[.,.],.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[[[],[]],[]]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[[[[]]],[]]]
=> [.,[[[[.,.],.],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[[[[]],[]]]]
=> [.,[[[[.,.],[.,.]],.],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 2
[[[]],[],[],[],[]]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 2
[[[]],[],[],[[]]]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 2
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001624
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Mp00013: Binary trees —to poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
[[]]
=> [.,.]
=> ([],1)
=> ([(0,1)],2)
=> 1
[[],[]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[[]]]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[[]]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[]],[]]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[[[],[]]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[[]]]]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? = 2
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[],[],[[]]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[],[[]],[]]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[],[[[]]]]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[[]],[],[]]]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[[[]],[[]]]]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[[[[]]],[]]]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[[[],[],[]]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[[],[[]]]]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[[[]],[]]]]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[[[[[],[]]]]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[[[[[]]]]]]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[],[],[[]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[],[[]],[]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 2
[[],[],[],[[],[]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[],[[[]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[[]],[],[]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[],[[]],[[]]]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[],[[],[]],[]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[],[[[]]],[]]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[],[[],[],[]]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[[],[[]]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[[[]],[]]]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 2
[[],[],[[[],[]]]]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[],[[[[]]]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[],[[]],[],[],[]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[]],[],[[]]]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[]],[[]],[]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 2
[[],[[]],[[],[]]]
=> [.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[]],[[[]]]]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[],[]],[],[]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 2
[[],[[[]]],[],[]]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 2
[[],[[],[]],[[]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 2
[[],[[[]]],[[]]]
=> [.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 2
[[],[[],[],[]],[]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[],[[]]],[]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[[]],[]],[]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 2
[[],[[[],[]]],[]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[[[]]]],[]]
=> [.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 2
[[],[[],[[]],[]]]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 2
[[],[[[]],[],[]]]
=> [.,[[[.,.],[.,[.,.]]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[[[]],[[]]]]
=> [.,[[[.,.],[[.,.],.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[[[],[]],[]]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[[[[]]],[]]]
=> [.,[[[[.,.],.],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 2
[[],[[[[]],[]]]]
=> [.,[[[[.,.],[.,.]],.],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 2
[[[]],[],[],[],[]]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 2
[[[]],[],[],[[]]]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 2
Description
The breadth of a lattice.
The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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