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Your data matches 104 different statistics following compositions of up to 3 maps.
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Matching statistic: St000660
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(load all 2 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> 2
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 1
Description
The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St000871
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [.,.]
=> [1] => 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 0 = 1 - 1
[[.,.],[.,.]]
=> [3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => 0 = 1 - 1
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0 = 1 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0 = 1 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0 = 1 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0 = 1 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 0 = 1 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0 = 1 - 1
[[[.,[[[.,[.,.]],.],.]],.],.]
=> [3,2,4,5,1,6,7] => [[[.,.],[.,[.,.]]],[.,[.,.]]]
=> [7,6,4,3,1,2,5] => ? = 2 - 1
[[[[.,[[.,.],[.,.]]],.],.],.]
=> [4,2,3,1,5,6,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1 - 1
[[[[.,[[.,[.,.]],.]],.],.],.]
=> [3,2,4,1,5,6,7] => [[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => ? = 1 - 1
[[[[[.,[[.,.],.]],.],.],.],.]
=> [2,3,1,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 1 - 1
[[[[[.,[[.,[.,.]],.]],.],.],.],.]
=> [3,2,4,1,5,6,7,8] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [8,7,6,5,3,1,2,4] => ? = 1 - 1
Description
The number of very big ascents of a permutation.
A very big ascent of a permutation $\pi$ is an index $i$ such that $\pi_{i+1} - \pi_i > 2$.
For the number of ascents, see [[St000245]] and for the number of big ascents, see [[St000646]]. General $r$-ascents were for example be studied in [1, Section 2].
Matching statistic: St000125
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
St000125: Binary trees ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> 0 = 1 - 1
[.,[.,.]]
=> 0 = 1 - 1
[[.,.],.]
=> 0 = 1 - 1
[.,[.,[.,.]]]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> 0 = 1 - 1
[[.,.],[.,.]]
=> 0 = 1 - 1
[[.,[.,.]],.]
=> 0 = 1 - 1
[[[.,.],.],.]
=> 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> 0 = 1 - 1
[[.,.],[[.,.],.]]
=> 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> 0 = 1 - 1
[[[.,.],.],[.,.]]
=> 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> 0 = 1 - 1
[[.,[[.,.],.]],.]
=> 0 = 1 - 1
[[[.,.],[.,.]],.]
=> 0 = 1 - 1
[[[.,[.,.]],.],.]
=> 0 = 1 - 1
[[[[.,.],.],.],.]
=> 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
=> 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
=> 0 = 1 - 1
[[[.,.],.],[.,[.,.]]]
=> 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
=> 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
=> 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> 0 = 1 - 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 - 1
[[[[[.,[[.,[.,.]],.]],.],.],.],.]
=> ? = 1 - 1
[[[[[[[.,.],[.,.]],.],.],.],.],.]
=> ? = 1 - 1
[[[[[[[.,[.,.]],.],.],.],.],.],.]
=> ? = 1 - 1
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 1 - 1
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1 - 1
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> ? = 1 - 1
[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]
=> ? = 1 - 1
[[[[[[[[.,.],[.,.]],.],.],.],.],.],.]
=> ? = 1 - 1
Description
The number of occurrences of the contiguous pattern {{{[.,[[[.,.],.],.]]}}} in a binary tree.
[[oeis:A005773]] counts binary trees avoiding this pattern.
Matching statistic: St001330
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 50%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[.,[.,.]]
=> [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[.,.],.]
=> [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 1
[[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[[[.,[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[.,[[.,.],[[[.,.],.],.]]]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[.,[[[[.,.],.],.],[.,.]]]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[.,[[[.,.],[.,[.,.]]],.]]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[.,[[[.,.],[[.,.],.]],.]]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[.,[[[[.,.],.],[.,.]],.]]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[.,[[[[.,.],[.,.]],.],.]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[.,[[[[[.,.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 1 + 1
[[[.,.],.],[[.,.],[.,.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => ([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[[.,.],.],[[[.,.],.],.]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 1
[[.,[.,[[.,.],.]]],[.,.]]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[[.,[[[.,.],.],.]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
[[[.,.],[[.,.],.]],[.,.]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[[[.,[[.,.],.]],.],[.,.]]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
[[[[[.,.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 1 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => ([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 1
[[[[.,.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 1 + 1
[[[[.,.],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 1 + 1
[[[[.,[.,.]],.],[.,.]],.]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 1 + 1
[[[[[.,.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 1 + 1
[[[.,[.,[.,[.,.]]]],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 1 + 1
[[[[.,.],[.,[.,.]]],.],.]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 1 + 1
[[[[.,[.,.]],[.,.]],.],.]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[[[[[.,.],.],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 1 + 1
[[[[.,[.,[.,.]]],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 1 + 1
[[[[[.,.],[.,.]],.],.],.]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 1 + 1
[[[[[.,[.,.]],.],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 1 + 1
[[[[[[.,.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[[[[[[.,.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> 2 = 1 + 1
[[[[[[.,.],[.,.]],.],.],.],.]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,8,2,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> 2 = 1 + 1
[[[[[[.,[.,.]],.],.],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,8,1,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> 2 = 1 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001719
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1] => [1] => ([(0,1)],2)
=> 1
[.,[.,.]]
=> [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[.,.],.]
=> [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [4,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [5,1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5,1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 1
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [4,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1
[[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 1
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1
[[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [4,3,6,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [6,4,5,2,3,1] => [5,2,4,3,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 1
[.,[[.,.],[[[.,.],.],.]]]
=> [4,5,6,2,3,1] => [6,2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ? = 2
[.,[[[[.,.],.],.],[.,.]]]
=> [6,2,3,4,5,1] => [2,3,4,5,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2
[.,[[[.,.],[.,[.,.]]],.]]
=> [5,4,2,3,6,1] => [4,2,3,6,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,11),(11,10)],12)
=> ? = 2
[.,[[[.,.],[[.,.],.]],.]]
=> [4,5,2,3,6,1] => [6,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ? = 2
[.,[[[[.,.],.],[.,.]],.]]
=> [5,2,3,4,6,1] => [2,3,4,6,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? = 2
[.,[[[[.,.],[.,.]],.],.]]
=> [4,2,3,5,6,1] => [2,3,6,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,7),(5,10),(5,13),(6,11),(6,12),(8,10),(9,7),(10,9),(11,8),(12,8),(13,9)],14)
=> ? = 2
[.,[[[[.,[.,.]],.],.],.]]
=> [3,2,4,5,6,1] => [2,6,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 2
[.,[[[[[.,.],.],.],.],.]]
=> [2,3,4,5,6,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 2
[[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => [4,3,5,1,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
=> ? = 1
[[[.,.],.],[[.,.],[.,.]]]
=> [6,4,5,1,2,3] => [5,1,4,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 1
[[[.,.],.],[[[.,.],.],.]]
=> [4,5,6,1,2,3] => [6,1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 2
[[.,[.,[[.,.],.]]],[.,.]]
=> [6,3,4,2,1,5] => [4,3,2,6,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 1
[[.,[[[.,.],.],.]],[.,.]]
=> [6,2,3,4,1,5] => [2,3,4,6,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 2
[[[.,.],[[.,.],.]],[.,.]]
=> [6,3,4,1,2,5] => [4,3,1,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 1
[[[.,[[.,.],.]],.],[.,.]]
=> [6,2,3,1,4,5] => [2,3,6,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? = 1
[[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 1
[[.,[.,[[.,[.,.]],.]]],.]
=> [4,3,5,2,1,6] => [5,3,2,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
=> ? = 1
[[.,[.,[[[.,.],.],.]]],.]
=> [3,4,5,2,1,6] => [5,2,4,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
=> ? = 2
[[.,[[.,[.,.]],[.,.]]],.]
=> [5,3,2,4,1,6] => [3,2,4,5,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> ? = 1
[[[[.,.],[.,.]],[.,.]],.]
=> [5,3,1,2,4,6] => [3,1,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001720
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001720: Lattices ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1] => [1] => ([(0,1)],2)
=> 2 = 1 + 1
[.,[.,.]]
=> [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[.,.],.]
=> [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 1 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 2 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 2 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [4,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [5,1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5,1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 1 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1 + 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 1 + 1
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 1
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1 + 1
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 + 1
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2 = 1 + 1
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [4,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1 + 1
[[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
=> 2 = 1 + 1
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1 + 1
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1 + 1
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 1 + 1
[[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 1 + 1
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 1 + 1
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 1 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 1 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [4,3,6,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
=> ? = 1 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [6,4,5,2,3,1] => [5,2,4,3,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 1 + 1
[.,[[.,.],[[[.,.],.],.]]]
=> [4,5,6,2,3,1] => [6,2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
=> ? = 2 + 1
[.,[[[[.,.],.],.],[.,.]]]
=> [6,2,3,4,5,1] => [2,3,4,5,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 2 + 1
[.,[[[.,.],[.,[.,.]]],.]]
=> [5,4,2,3,6,1] => [4,2,3,6,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,11),(11,10)],12)
=> ? = 2 + 1
[.,[[[.,.],[[.,.],.]],.]]
=> [4,5,2,3,6,1] => [6,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
=> ? = 2 + 1
[.,[[[[.,.],.],[.,.]],.]]
=> [5,2,3,4,6,1] => [2,3,4,6,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
=> ? = 2 + 1
[.,[[[[.,.],[.,.]],.],.]]
=> [4,2,3,5,6,1] => [2,3,6,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,7),(5,10),(5,13),(6,11),(6,12),(8,10),(9,7),(10,9),(11,8),(12,8),(13,9)],14)
=> ? = 2 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> [3,2,4,5,6,1] => [2,6,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
=> ? = 2 + 1
[.,[[[[[.,.],.],.],.],.]]
=> [2,3,4,5,6,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,15),(2,14),(3,19),(3,21),(4,20),(4,21),(5,14),(5,19),(6,15),(6,20),(8,10),(9,11),(10,12),(11,13),(12,7),(13,7),(14,8),(15,9),(16,10),(16,18),(17,11),(17,18),(18,12),(18,13),(19,8),(19,16),(20,9),(20,17),(21,16),(21,17)],22)
=> ? = 2 + 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [6,5,4,3,1,2] => [4,3,5,1,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(8,9)],10)
=> ? = 1 + 1
[[[.,.],.],[[.,.],[.,.]]]
=> [6,4,5,1,2,3] => [5,1,4,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 1 + 1
[[[.,.],.],[[[.,.],.],.]]
=> [4,5,6,1,2,3] => [6,1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 2 + 1
[[.,[.,[[.,.],.]]],[.,.]]
=> [6,3,4,2,1,5] => [4,3,2,6,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 1 + 1
[[.,[[[.,.],.],.]],[.,.]]
=> [6,2,3,4,1,5] => [2,3,4,6,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 2 + 1
[[[.,.],[[.,.],.]],[.,.]]
=> [6,3,4,1,2,5] => [4,3,1,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> 2 = 1 + 1
[[[.,[[.,.],.]],.],[.,.]]
=> [6,2,3,1,4,5] => [2,3,6,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
=> ? = 1 + 1
[[[[[.,.],.],.],.],[.,.]]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
=> ? = 1 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> [4,3,5,2,1,6] => [5,3,2,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
=> ? = 1 + 1
[[.,[.,[[[.,.],.],.]]],.]
=> [3,4,5,2,1,6] => [5,2,4,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
=> ? = 2 + 1
[[.,[[.,[.,.]],[.,.]]],.]
=> [5,3,2,4,1,6] => [3,2,4,5,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> ? = 1 + 1
[[[[.,.],[.,.]],[.,.]],.]
=> [5,3,1,2,4,6] => [3,1,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9)
=> 2 = 1 + 1
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St000181
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1,0]
=> [1,0]
=> ([],1)
=> 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 2
[[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1
[[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 1
[[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1
[[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 1
[[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1
[[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1
[[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1
[[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 2
[.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 2
[.,[[[.,.],[.,[.,.]]],.]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[.,[[[.,.],[[.,.],.]],.]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2
[.,[[[[.,.],.],[.,.]],.]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[.,[[[[.,.],[.,.]],.],.]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[.,[[[[.,[.,.]],.],.],.]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 2
[.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1
[[[.,.],.],[[.,.],[.,.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1
[[[.,.],.],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 2
[[.,[.,[[.,.],.]]],[.,.]]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 1
[[.,[[[.,.],.],.]],[.,.]]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 2
[[[.,.],[[.,.],.]],[.,.]]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 1
[[[.,[[.,.],.]],.],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 1
[[[[[.,.],.],.],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 1
[[.,[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 1
[[.,[.,[[[.,.],.],.]]],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 2
[[.,[[.,[.,.]],[.,.]]],.]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 1
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St001490
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00019: Binary trees —right rotate⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [.,.]
=> [1,0]
=> [[1],[]]
=> 1
[.,[.,.]]
=> [[.,.],.]
=> [1,1,0,0]
=> [[2],[]]
=> 1
[[.,.],.]
=> [.,[.,.]]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1
[.,[.,[.,.]]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 1
[.,[[.,.],.]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 1
[[.,.],[.,.]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1
[[[.,.],.],.]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1
[.,[.,[.,[.,.]]]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 1
[.,[.,[[.,.],.]]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 1
[.,[[.,[.,.]],.]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? = 1
[.,[[[.,.],.],.]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 2
[[.,.],[.,[.,.]]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[[.,.],[[.,.],.]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1
[[.,[.,.]],[.,.]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1
[[.,[.,[.,.]]],.]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1
[[.,[[.,.],.]],.]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1
[[[.,.],[.,.]],.]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1
[[[.,[.,.]],.],.]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 1
[[[[.,.],.],.],.]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ? = 1
[.,[.,[[.,.],[.,.]]]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? = 1
[.,[.,[[.,[.,.]],.]]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? = 1
[.,[.,[[[.,.],.],.]]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? = 2
[.,[[.,.],[.,[.,.]]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? = 1
[.,[[.,.],[[.,.],.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ? = 1
[.,[[.,[.,.]],[.,.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? = 1
[.,[[[.,.],.],[.,.]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? = 2
[.,[[.,[.,[.,.]]],.]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? = 1
[.,[[.,[[.,.],.]],.]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? = 1
[.,[[[.,.],[.,.]],.]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? = 2
[.,[[[.,[.,.]],.],.]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? = 2
[.,[[[[.,.],.],.],.]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 2
[[.,.],[.,[.,[.,.]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 1
[[.,.],[[[.,.],.],.]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 2
[[.,[.,.]],[.,[.,.]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 1
[[[.,.],.],[[.,.],.]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 1
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 1
[[[[.,.],.],.],[.,.]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? = 1
[[.,[.,[.,[.,.]]]],.]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 1
[[.,[.,[[.,.],.]]],.]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? = 1
[[.,[[.,.],[.,.]]],.]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? = 1
[[.,[[.,[.,.]],.]],.]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? = 1
[[.,[[[.,.],.],.]],.]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 2
[[[.,.],[.,[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1
[[[.,.],[[.,.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? = 1
[[[.,[.,.]],[.,.]],.]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 1
[[[[.,.],.],[.,.]],.]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? = 1
[[[.,[.,[.,.]]],.],.]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 1
[[[.,[[.,.],.]],.],.]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? = 1
[[[[.,.],[.,.]],.],.]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? = 1
[[[[.,[.,.]],.],.],.]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? = 1
[[[[[.,.],.],.],.],.]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? = 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> ? = 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[5,5],[3]]
=> ? = 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [[[.,.],[[.,.],[.,.]]],.]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[4,3,2],[1]]
=> ? = 1
[.,[[.,.],[[[.,.],.],.]]]
=> [[[.,.],[[[.,.],.],.]],.]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [[4,4,2],[1]]
=> ? = 2
[.,[[[[.,.],.],.],[.,.]]]
=> [[[[[.,.],.],.],[.,.]],.]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [[4,3,3],[]]
=> ? = 2
[.,[[[.,.],[.,[.,.]]],.]]
=> [[[[.,.],[.,[.,.]]],.],.]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> ? = 2
[.,[[[.,.],[[.,.],.]],.]]
=> [[[[.,.],[[.,.],.]],.],.]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3],[1]]
=> ? = 2
[.,[[[[.,.],.],[.,.]],.]]
=> [[[[[.,.],.],[.,.]],.],.]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> ? = 2
[.,[[[[.,.],[.,.]],.],.]]
=> [[[[[.,.],[.,.]],.],.],.]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [[4,4,3],[]]
=> ? = 2
[.,[[[[.,[.,.]],.],.],.]]
=> [[[[[.,[.,.]],.],.],.],.]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3],[]]
=> ? = 2
[.,[[[[[.,.],.],.],.],.]]
=> [[[[[[.,.],.],.],.],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> ? = 2
[[.,.],[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> ? = 1
[[[.,.],.],[[.,.],[.,.]]]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> ? = 1
[[[.,.],.],[[[.,.],.],.]]
=> [[.,.],[.,[[[.,.],.],.]]]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> ? = 2
[[.,[.,[[.,.],.]]],[.,.]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ? = 1
[[.,[[[.,.],.],.]],[.,.]]
=> [.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> ? = 2
[[[.,.],[[.,.],.]],[.,.]]
=> [[.,.],[[[.,.],.],[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ? = 1
[[[.,[[.,.],.]],.],[.,.]]
=> [[.,[[.,.],.]],[.,[.,.]]]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2,1]]
=> ? = 1
[[[[[.,.],.],.],.],[.,.]]
=> [[[[.,.],.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> ? = 1
[[.,[.,[[.,[.,.]],.]]],.]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> ? = 1
[[.,[.,[[[.,.],.],.]]],.]
=> [.,[[.,[[[.,.],.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> ? = 2
[[.,[[.,[.,.]],[.,.]]],.]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> ? = 1
Description
The number of connected components of a skew partition.
Matching statistic: St001882
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => [3,1,2] => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0 = 1 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 0 = 1 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0 = 1 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0 = 1 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,3,2,4] => [1,5,3,2,4] => 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 1 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 1 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 1 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0 = 1 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 - 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1 - 1
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 2 - 1
[[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0 = 1 - 1
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0 = 1 - 1
[[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 1 - 1
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0 = 1 - 1
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 1 - 1
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1 - 1
[[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0 = 1 - 1
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 1 - 1
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 1 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [6,1,2,3,5,4] => [6,1,2,3,5,4] => ? = 1 - 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [2,4,6,5,3,1] => [6,2,1,4,3,5] => [6,2,1,4,3,5] => ? = 1 - 1
[.,[[.,.],[[[.,.],.],.]]]
=> [2,4,5,6,3,1] => [6,2,1,4,5,3] => [6,2,1,4,5,3] => ? = 2 - 1
[.,[[[[.,.],.],.],[.,.]]]
=> [2,3,4,6,5,1] => [6,2,3,4,1,5] => [6,2,3,4,1,5] => ? = 2 - 1
[.,[[[.,.],[.,[.,.]]],.]]
=> [2,5,4,3,6,1] => [6,2,5,3,4,1] => [6,2,5,3,4,1] => ? = 2 - 1
[.,[[[.,.],[[.,.],.]],.]]
=> [2,4,5,3,6,1] => [6,2,5,4,3,1] => [6,2,5,4,3,1] => ? = 2 - 1
[.,[[[[.,.],.],[.,.]],.]]
=> [2,3,5,4,6,1] => [6,2,3,5,4,1] => [6,2,3,5,4,1] => ? = 2 - 1
[.,[[[[.,.],[.,.]],.],.]]
=> [2,4,3,5,6,1] => [6,2,4,3,5,1] => [6,2,4,3,5,1] => ? = 2 - 1
[.,[[[[.,[.,.]],.],.],.]]
=> [3,2,4,5,6,1] => [6,3,2,4,5,1] => [6,3,2,4,5,1] => ? = 2 - 1
[.,[[[[[.,.],.],.],.],.]]
=> [2,3,4,5,6,1] => [6,2,3,4,5,1] => [6,2,3,4,5,1] => ? = 2 - 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,6,5,4,3,2] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => ? = 1 - 1
[[[.,.],.],[[.,.],[.,.]]]
=> [1,2,4,6,5,3] => [1,2,6,4,3,5] => [1,2,6,4,3,5] => ? = 1 - 1
[[[.,.],.],[[[.,.],.],.]]
=> [1,2,4,5,6,3] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? = 2 - 1
[[.,[.,[[.,.],.]]],[.,.]]
=> [3,4,2,1,6,5] => [4,1,3,2,6,5] => [4,1,3,2,6,5] => ? = 1 - 1
[[.,[[[.,.],.],.]],[.,.]]
=> [2,3,4,1,6,5] => [4,2,3,1,6,5] => [4,2,3,1,6,5] => ? = 2 - 1
[[[.,.],[[.,.],.]],[.,.]]
=> [1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => ? = 1 - 1
[[[.,[[.,.],.]],.],[.,.]]
=> [2,3,1,4,6,5] => [3,2,1,4,6,5] => [3,2,1,4,6,5] => ? = 1 - 1
[[[[[.,.],.],.],.],[.,.]]
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 1 - 1
[[.,[.,[[.,[.,.]],.]]],.]
=> [4,3,5,2,1,6] => [5,1,4,3,2,6] => [5,1,4,3,2,6] => ? = 1 - 1
[[.,[.,[[[.,.],.],.]]],.]
=> [3,4,5,2,1,6] => [5,1,3,4,2,6] => [5,1,3,4,2,6] => ? = 2 - 1
[[.,[[.,[.,.]],[.,.]]],.]
=> [3,2,5,4,1,6] => [5,3,2,1,4,6] => [5,3,2,1,4,6] => ? = 1 - 1
Description
The number of occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
Matching statistic: St001890
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [1,0]
=> [1,0]
=> ([],1)
=> ? = 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 2
[[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 1
[[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 1
[[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1
[[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 1
[[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1
[[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1
[[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 1
[[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1
[.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 2
[.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 2
[.,[[[.,.],[.,[.,.]]],.]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 2
[.,[[[.,.],[[.,.],.]],.]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 2
[.,[[[[.,.],.],[.,.]],.]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[.,[[[[.,.],[.,.]],.],.]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 2
[.,[[[[.,[.,.]],.],.],.]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 2
[.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2
[[.,.],[.,[.,[.,[.,.]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1
[[[.,.],.],[[.,.],[.,.]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1
[[[.,.],.],[[[.,.],.],.]]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 2
[[.,[.,[[.,.],.]]],[.,.]]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 1
[[.,[[[.,.],.],.]],[.,.]]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 2
[[[.,.],[[.,.],.]],[.,.]]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 1
[[[.,[[.,.],.]],.],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 1
[[[[[.,.],.],.],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 1
[[.,[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 1
[[.,[.,[[[.,.],.],.]]],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? = 2
Description
The maximum magnitude of the Möbius function of a poset.
The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
The following 94 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001811The Castelnuovo-Mumford regularity of a permutation. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001845The number of join irreducibles minus the rank of a lattice. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001964The interval resolution global dimension of a poset. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000456The monochromatic index of a connected graph. St000535The rank-width of a graph. St000544The cop number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000948The chromatic discriminant of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001363The Euler characteristic of a graph according to Knill. St001395The number of strictly unfriendly partitions of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001826The maximal number of leaves on a vertex of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000268The number of strongly connected orientations of a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000637The length of the longest cycle in a graph. St001029The size of the core of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001073The number of nowhere zero 3-flows of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001316The domatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001367The smallest number which does not occur as degree of a vertex in a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001797The number of overfull subgraphs of a graph. St000523The number of 2-protected nodes of a rooted tree. St000700The protection number of an ordered tree. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000441The number of successions of a permutation. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000661The number of rises of length 3 of a Dyck path. St000665The number of rafts of a permutation. St000674The number of hills of a Dyck path. St000842The breadth of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000439The position of the first down step of a Dyck path. St000981The length of the longest zigzag subpath.
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