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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St000052
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(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 0
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000204
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 67% ●values known / values provided: 70%●distinct values known / distinct values provided: 67%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 67% ●values known / values provided: 70%●distinct values known / distinct values provided: 67%
Values
[1] => [.,.]
=> [1] => [.,.]
=> 0
[1,2] => [.,[.,.]]
=> [2,1] => [[.,.],.]
=> 0
[2,1] => [[.,.],.]
=> [1,2] => [.,[.,.]]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [[.,[.,.]],.]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [[.,.],[.,.]]
=> 0
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => [[.,.],[.,.]]
=> 0
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 0
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 0
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 0
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 0
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> 1
[7,8,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 0
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 0
[7,8,5,6,4,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [8,6,1,2,3,4,5,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 0
[7,5,6,8,4,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [8,6,1,2,3,4,5,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,8,6,4,5,3,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [8,5,1,2,3,4,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,8,4,5,3,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [8,5,1,2,3,4,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 0
[7,5,6,4,8,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [8,6,1,2,3,4,5,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,4,5,8,3,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [8,5,1,2,3,4,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,5,4,6,8,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [8,6,1,2,3,4,5,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,8,6,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [8,4,1,2,3,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,8,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [8,4,1,2,3,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [8,4,1,2,3,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,8,6,4,3,5,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [8,5,1,2,3,4,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,8,4,3,5,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [8,5,1,2,3,4,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[8,7,6,3,4,5,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [5,4,1,2,3,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,8,5,4,3,6,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [8,6,1,2,3,4,5,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[8,3,4,5,6,7,2,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 0
[7,6,5,4,3,8,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 0
[7,5,6,4,3,8,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [8,6,1,2,3,4,5,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,4,5,3,8,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [8,5,1,2,3,4,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,5,4,6,3,8,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [8,6,1,2,3,4,5,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,5,3,4,8,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [8,4,1,2,3,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,4,3,5,8,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [8,5,1,2,3,4,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,5,4,3,6,8,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [8,6,1,2,3,4,5,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[3,4,5,6,7,8,2,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [8,7,6,5,4,1,2,3] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0
[7,8,6,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [8,3,1,2,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,8,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [8,3,1,2,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,5,8,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [8,3,1,2,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,5,4,8,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [8,3,1,2,4,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[4,5,6,7,8,2,3,1] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [8,7,6,5,3,1,2,4] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0
[7,8,6,5,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [8,4,1,2,3,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,8,5,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [8,4,1,2,3,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,5,8,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [8,4,1,2,3,5,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[8,7,6,5,2,3,4,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [4,3,1,2,5,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[5,6,7,8,2,3,4,1] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [8,7,6,4,3,1,2,5] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0
[7,8,6,4,3,2,5,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [8,5,1,2,3,4,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,6,8,4,3,2,5,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [8,5,1,2,3,4,6,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[8,7,6,3,4,2,5,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [5,4,1,2,3,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[8,7,6,3,2,4,5,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [5,4,1,2,3,6,7,8] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[7,8,5,4,3,2,6,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [8,6,1,2,3,4,5,7] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0
[8,7,2,3,4,5,6,1] => [[[[.,.],[.,[.,[.,[.,.]]]]],.],.]
=> [6,5,4,3,1,2,7,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 0
[7,8,2,3,4,5,6,1] => [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
=> [8,6,5,4,3,1,2,7] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0
[8,3,4,5,6,2,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 0
[8,3,4,5,2,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 0
[8,3,4,2,5,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 0
[8,3,2,4,5,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 0
[8,2,3,4,5,6,7,1] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [7,6,5,4,3,1,2,8] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0
[7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 0
Description
The number of internal nodes of a binary tree.
That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St001167
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 70%●distinct values known / distinct values provided: 67%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 70%●distinct values known / distinct values provided: 67%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[7,8,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[7,8,5,6,4,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[7,5,6,8,4,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,6,4,5,3,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,6,8,4,5,3,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[7,5,6,4,8,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[7,6,4,5,8,3,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,5,4,6,8,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,6,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
[7,6,8,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
[7,8,6,4,3,5,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,6,8,4,3,5,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[8,7,6,3,4,5,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[7,8,5,4,3,6,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[8,3,4,5,6,7,2,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 0
[7,6,5,4,3,8,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[7,5,6,4,3,8,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[7,6,4,5,3,8,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,5,4,6,3,8,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[7,6,5,3,4,8,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
[7,6,4,3,5,8,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,5,4,3,6,8,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[3,4,5,6,7,8,2,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 0
[7,8,6,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[7,6,8,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[7,6,5,8,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[7,6,5,4,8,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[4,5,6,7,8,2,3,1] => [[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 0
[7,8,6,5,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
[7,6,8,5,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
[7,6,5,8,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
[8,7,6,5,2,3,4,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[5,6,7,8,2,3,4,1] => [[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 0
[7,8,6,4,3,2,5,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[7,6,8,4,3,2,5,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[8,7,6,3,4,2,5,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[8,7,6,3,2,4,5,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[7,8,5,4,3,2,6,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,2,3,4,5,6,1] => [[[[.,.],[.,[.,[.,[.,.]]]]],.],.]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
[7,8,2,3,4,5,6,1] => [[[.,.],[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 0
[8,3,4,5,6,2,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 0
[8,3,4,5,2,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 0
[8,3,4,2,5,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 0
[8,3,2,4,5,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> ? = 0
[8,2,3,4,5,6,7,1] => [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 0
[7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001323
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001323: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 70%●distinct values known / distinct values provided: 67%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001323: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 70%●distinct values known / distinct values provided: 67%
Values
[1] => [.,.]
=> [1] => ([],1)
=> 0
[1,2] => [.,[.,.]]
=> [2,1] => ([(0,1)],2)
=> 0
[2,1] => [[.,.],.]
=> [1,2] => ([],2)
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 0
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> 0
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 0
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> 0
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> 0
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,3,5,6,4,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,3,6,4,5,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,3,6,4,7,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,3,7,4,5,6] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,4,5,3,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,4,5,6,3,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,4,5,6,7,3] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,7,3,4,5,6] => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,3,2,4,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,3,4,2,5,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,3,4,5,2,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,3,4,5,6,2,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,3,4,5,6,7,2] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [2,7,6,5,4,3,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,7,2,3,4,5,6] => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,1,3,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[2,3,1,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[2,3,4,1,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[2,3,4,5,1,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[2,3,4,5,6,1,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[2,3,4,5,6,7,1] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[7,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[7,8,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 0
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 0
[7,8,5,6,4,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 0
[7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 0
[7,5,6,8,4,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 0
[7,8,6,4,5,3,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 0
[7,6,8,4,5,3,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 0
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 0
[7,5,6,4,8,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 0
[7,6,4,5,8,3,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 0
[7,5,4,6,8,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 0
[7,8,6,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 0
[7,6,8,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 0
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [1,2,4,3,5,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 0
[7,8,6,4,3,5,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 0
[7,6,8,4,3,5,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,2,3,5,4,6,8,7] => ([(4,7),(5,6)],8)
=> ? = 0
[8,7,6,3,4,5,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [1,2,5,4,3,6,7,8] => ([(5,6),(5,7),(6,7)],8)
=> ? = 0
[7,8,5,4,3,6,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,2,3,4,6,5,8,7] => ([(4,7),(5,6)],8)
=> ? = 0
[8,3,4,5,6,7,2,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [1,2,7,6,5,4,3,8] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[7,6,5,4,3,8,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? = 0
Description
The independence gap of a graph.
This is the difference between the independence number [[St000093]] and the minimal size of a maximally independent set of a graph.
In particular, this statistic is $0$ for well covered graphs
Matching statistic: St000373
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 56%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 56%
Values
[1] => [.,.]
=> [1] => [1] => 0
[1,2] => [.,[.,.]]
=> [2,1] => [2,1] => 0
[2,1] => [[.,.],.]
=> [1,2] => [1,2] => 0
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => 0
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 0
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 0
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 0
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 0
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => 0
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,3,2] => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,1,3,4,2] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => 1
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => [7,1,2,3,4,6,5] => ? = 1
[1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => [7,1,2,3,5,4,6] => ? = 1
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => [7,1,2,3,5,4,6] => ? = 1
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => [7,1,2,3,6,5,4] => ? = 1
[1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => [7,1,2,3,5,6,4] => ? = 2
[1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [7,1,2,4,3,5,6] => ? = 1
[1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [7,1,2,4,3,6,5] => ? = 2
[1,2,3,5,6,4,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [7,1,2,4,3,5,6] => ? = 1
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [7,1,2,4,3,5,6] => ? = 1
[1,2,3,5,7,4,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [7,1,2,4,3,6,5] => ? = 2
[1,2,3,5,7,6,4] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [7,1,2,4,3,6,5] => ? = 2
[1,2,3,6,4,5,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [7,1,2,5,4,3,6] => ? = 1
[1,2,3,6,4,7,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [7,1,2,5,4,3,6] => ? = 1
[1,2,3,6,5,4,7] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [7,1,2,4,5,3,6] => ? = 2
[1,2,3,6,5,7,4] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [7,1,2,4,5,3,6] => ? = 2
[1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [7,1,2,5,4,3,6] => ? = 1
[1,2,3,6,7,5,4] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [7,1,2,4,5,3,6] => ? = 2
[1,2,3,7,4,5,6] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [7,1,2,6,4,5,3] => ? = 1
[1,2,3,7,4,6,5] => [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [7,1,2,6,5,4,3] => ? = 2
[1,2,3,7,5,4,6] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => [7,1,2,4,6,5,3] => ? = 2
[1,2,3,7,5,6,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => [7,1,2,4,6,5,3] => ? = 2
[1,2,3,7,6,4,5] => [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [7,1,2,5,4,6,3] => ? = 2
[1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => [7,1,2,4,5,6,3] => ? = 3
[1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [7,1,3,2,4,5,6] => ? = 1
[1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [7,1,3,2,4,6,5] => ? = 2
[1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,1,3,2,5,4,6] => ? = 2
[1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,1,3,2,5,4,6] => ? = 2
[1,2,4,3,7,5,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [7,1,3,2,6,5,4] => ? = 2
[1,2,4,3,7,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [7,1,3,2,5,6,4] => ? = 3
[1,2,4,5,3,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [7,1,3,2,4,5,6] => ? = 1
[1,2,4,5,3,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [7,1,3,2,4,6,5] => ? = 2
[1,2,4,5,6,3,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [7,1,3,2,4,5,6] => ? = 1
[1,2,4,5,6,7,3] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [7,1,3,2,4,5,6] => ? = 1
[1,2,4,5,7,3,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [7,1,3,2,4,6,5] => ? = 2
[1,2,4,5,7,6,3] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [7,1,3,2,4,6,5] => ? = 2
[1,2,4,6,3,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,1,3,2,5,4,6] => ? = 2
[1,2,4,6,3,7,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,1,3,2,5,4,6] => ? = 2
[1,2,4,6,5,3,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,1,3,2,5,4,6] => ? = 2
[1,2,4,6,5,7,3] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,1,3,2,5,4,6] => ? = 2
[1,2,4,6,7,3,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,1,3,2,5,4,6] => ? = 2
[1,2,4,6,7,5,3] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,1,3,2,5,4,6] => ? = 2
[1,2,4,7,3,5,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [7,1,3,2,6,5,4] => ? = 2
[1,2,4,7,3,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [7,1,3,2,5,6,4] => ? = 3
[1,2,4,7,5,3,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [7,1,3,2,6,5,4] => ? = 2
[1,2,4,7,5,6,3] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [7,1,3,2,6,5,4] => ? = 2
[1,2,4,7,6,3,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [7,1,3,2,5,6,4] => ? = 3
[1,2,4,7,6,5,3] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [7,1,3,2,5,6,4] => ? = 3
[1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => [7,1,4,3,2,5,6] => ? = 1
[1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [4,3,6,7,5,2,1] => [7,1,4,3,2,6,5] => ? = 2
[1,2,5,3,6,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => [7,1,4,3,2,5,6] => ? = 1
Description
The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j \geq j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000065
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [[1]]
=> 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 0
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 0
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 0
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 0
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[2,1,3,6,5,4,7] => [[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,3,6,5,7,4] => [[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,3,6,7,5,4] => [[.,.],[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,3,7,5,4,6] => [[.,.],[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,-1,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,3,7,5,6,4] => [[.,.],[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,-1,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,3,7,6,4,5] => [[.,.],[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,3,7,6,5,4] => [[.,.],[.,[[[[.,.],.],.],.]]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 3
[2,1,4,3,5,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,3,6,5,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,3,6,7,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,3,7,5,6] => [[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,3,7,6,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 3
[2,1,4,5,3,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,5,7,3,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,5,7,6,3] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,1,0,0,-1,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,6,3,5,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,6,3,7,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,6,5,3,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,6,5,7,3] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,6,7,3,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,6,7,5,3] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,7,3,5,6] => [[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,7,3,6,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 3
[2,1,4,7,5,3,6] => [[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,7,5,6,3] => [[.,.],[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,4,7,6,3,5] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 3
[2,1,4,7,6,5,3] => [[.,.],[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,-1,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 3
[2,1,5,3,4,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[2,1,5,3,4,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,5,3,6,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[2,1,5,3,6,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[2,1,5,3,7,4,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,5,3,7,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,5,4,3,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,5,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 3
[2,1,5,4,6,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,5,4,6,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,5,4,7,3,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 3
[2,1,5,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 3
[2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[2,1,5,6,4,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,5,6,4,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,5,6,7,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[2,1,5,6,7,4,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,5,7,3,4,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,5,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,5,7,4,3,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 3
[2,1,5,7,4,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 3
[2,1,5,7,6,3,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,-1,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2
Description
The number of entries equal to -1 in an alternating sign matrix.
The number of nonzero entries, [[St000890]] is twice this number plus the dimension of the matrix.
Matching statistic: St001091
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> []
=> 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1]
=> 0
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> []
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2]
=> 0
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2]
=> 0
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> []
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 1
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 0
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? = 1
[1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? = 1
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,4,3,2,1]
=> ? = 1
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? = 1
[1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? = 2
[1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 1
[1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? = 2
[1,2,3,5,6,4,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 1
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 1
[1,2,3,5,7,4,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? = 2
[1,2,3,5,7,6,4] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2,1]
=> ? = 2
[1,2,3,6,4,5,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 1
[1,2,3,6,4,7,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 1
[1,2,3,6,5,4,7] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 2
[1,2,3,6,5,7,4] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 2
[1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 1
[1,2,3,6,7,5,4] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,2,1]
=> ? = 2
[1,2,3,7,4,5,6] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? = 1
[1,2,3,7,4,6,5] => [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> ? = 2
[1,2,3,7,5,4,6] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> ? = 2
[1,2,3,7,5,6,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> ? = 2
[1,2,3,7,6,4,5] => [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> ? = 2
[1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> ? = 3
[1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 1
[1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? = 2
[1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 2
[1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 2
[1,2,4,3,7,5,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? = 2
[1,2,4,3,7,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? = 3
[1,2,4,5,3,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 1
[1,2,4,5,3,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? = 2
[1,2,4,5,6,3,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 1
[1,2,4,5,6,7,3] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 1
[1,2,4,5,7,3,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? = 2
[1,2,4,5,7,6,3] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2,1]
=> ? = 2
[1,2,4,6,3,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 2
[1,2,4,6,3,7,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 2
[1,2,4,6,5,3,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 2
[1,2,4,6,5,7,3] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 2
[1,2,4,6,7,3,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 2
[1,2,4,6,7,5,3] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,2,1]
=> ? = 2
[1,2,4,7,3,5,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? = 2
[1,2,4,7,3,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? = 3
[1,2,4,7,5,3,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? = 2
[1,2,4,7,5,6,3] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? = 2
[1,2,4,7,6,3,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? = 3
[1,2,4,7,6,5,3] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2,2,1]
=> ? = 3
[1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 1
[1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ? = 2
Description
The number of parts in an integer partition whose next smaller part has the same size.
In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St001744
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 56%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 56%
Values
[1] => [.,.]
=> [1] => [1] => 0
[1,2] => [.,[.,.]]
=> [2,1] => [2,1] => 0
[2,1] => [[.,.],.]
=> [1,2] => [1,2] => 0
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 0
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 0
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 0
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 0
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 0
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 0
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,1,3,2] => 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,1,3,2] => 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,1,4,3,2] => 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,4,2,3] => 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,1,4,3,2] => 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,1,4,3,2] => 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,4,2,3] => 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,4,2,3] => 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,1,4,3] => 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,1,4,3] => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,1,4,3] => 1
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => [7,6,5,4,3,1,2] => ? = 1
[1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => [7,6,5,4,1,3,2] => ? = 1
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [5,7,6,4,3,2,1] => [7,6,5,4,1,3,2] => ? = 1
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => [7,6,5,4,2,1,3] => ? = 1
[1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => [7,6,5,4,1,2,3] => ? = 2
[1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [7,6,5,1,4,3,2] => ? = 1
[1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [7,6,5,1,4,2,3] => ? = 2
[1,2,3,5,6,4,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [7,6,5,1,4,3,2] => ? = 1
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [4,7,6,5,3,2,1] => [7,6,5,1,4,3,2] => ? = 1
[1,2,3,5,7,4,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [7,6,5,1,4,2,3] => ? = 2
[1,2,3,5,7,6,4] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [4,6,7,5,3,2,1] => [7,6,5,1,4,2,3] => ? = 2
[1,2,3,6,4,5,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [7,6,5,2,1,4,3] => ? = 1
[1,2,3,6,4,7,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [7,6,5,2,1,4,3] => ? = 1
[1,2,3,6,5,4,7] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [7,6,5,1,2,4,3] => ? = 2
[1,2,3,6,5,7,4] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [7,6,5,1,2,4,3] => ? = 2
[1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [5,4,7,6,3,2,1] => [7,6,5,2,1,4,3] => ? = 1
[1,2,3,6,7,5,4] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [4,5,7,6,3,2,1] => [7,6,5,1,2,4,3] => ? = 2
[1,2,3,7,4,5,6] => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => [7,6,5,3,2,1,4] => ? = 1
[1,2,3,7,4,6,5] => [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [7,6,5,3,1,2,4] => ? = 2
[1,2,3,7,5,4,6] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => [7,6,5,1,3,2,4] => ? = 2
[1,2,3,7,5,6,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [4,6,5,7,3,2,1] => [7,6,5,1,3,2,4] => ? = 2
[1,2,3,7,6,4,5] => [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [5,4,6,7,3,2,1] => [7,6,5,2,1,3,4] => ? = 2
[1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => ? = 3
[1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [7,6,1,5,4,3,2] => ? = 1
[1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [7,6,1,5,4,2,3] => ? = 2
[1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,6,1,5,2,4,3] => ? = 2
[1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,6,1,5,2,4,3] => ? = 2
[1,2,4,3,7,5,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [7,6,1,5,3,2,4] => ? = 2
[1,2,4,3,7,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [7,6,1,5,2,3,4] => ? = 3
[1,2,4,5,3,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [7,6,1,5,4,3,2] => ? = 1
[1,2,4,5,3,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [7,6,1,5,4,2,3] => ? = 2
[1,2,4,5,6,3,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [7,6,1,5,4,3,2] => ? = 1
[1,2,4,5,6,7,3] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [3,7,6,5,4,2,1] => [7,6,1,5,4,3,2] => ? = 1
[1,2,4,5,7,3,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [7,6,1,5,4,2,3] => ? = 2
[1,2,4,5,7,6,3] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [3,6,7,5,4,2,1] => [7,6,1,5,4,2,3] => ? = 2
[1,2,4,6,3,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,6,1,5,2,4,3] => ? = 2
[1,2,4,6,3,7,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,6,1,5,2,4,3] => ? = 2
[1,2,4,6,5,3,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,6,1,5,2,4,3] => ? = 2
[1,2,4,6,5,7,3] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,6,1,5,2,4,3] => ? = 2
[1,2,4,6,7,3,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,6,1,5,2,4,3] => ? = 2
[1,2,4,6,7,5,3] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [3,5,7,6,4,2,1] => [7,6,1,5,2,4,3] => ? = 2
[1,2,4,7,3,5,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [7,6,1,5,3,2,4] => ? = 2
[1,2,4,7,3,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [7,6,1,5,2,3,4] => ? = 3
[1,2,4,7,5,3,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [7,6,1,5,3,2,4] => ? = 2
[1,2,4,7,5,6,3] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [3,6,5,7,4,2,1] => [7,6,1,5,3,2,4] => ? = 2
[1,2,4,7,6,3,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [7,6,1,5,2,3,4] => ? = 3
[1,2,4,7,6,5,3] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [3,5,6,7,4,2,1] => [7,6,1,5,2,3,4] => ? = 3
[1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [4,3,7,6,5,2,1] => [7,6,2,1,5,4,3] => ? = 1
[1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [4,3,6,7,5,2,1] => [7,6,2,1,5,3,4] => ? = 2
Description
The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
Matching statistic: St000711
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 56%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 56%
Values
[1] => [.,.]
=> [1,0]
=> [1] => ? = 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [1,2] => 0
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [2,1] => 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 1
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 2
[1,2,3,4,5,7,6] => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => ? = 1
[1,2,3,4,6,5,7] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5,7] => ? = 1
[1,2,3,4,6,7,5] => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5,7] => ? = 1
[1,2,3,4,7,5,6] => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => ? = 1
[1,2,3,4,7,6,5] => [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [6,7,1,2,3,4,5] => ? = 2
[1,2,3,5,4,6,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,3,5,4,7,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [5,7,1,2,3,4,6] => ? = 2
[1,2,3,5,6,4,7] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,3,5,6,7,4] => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [5,1,2,3,4,6,7] => ? = 1
[1,2,3,5,7,4,6] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [5,7,1,2,3,4,6] => ? = 2
[1,2,3,5,7,6,4] => [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [5,7,1,2,3,4,6] => ? = 2
[1,2,3,6,4,5,7] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5,7] => ? = 1
[1,2,3,6,4,7,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5,7] => ? = 1
[1,2,3,6,5,4,7] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,1,2,3,4,7] => ? = 2
[1,2,3,6,5,7,4] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,1,2,3,4,7] => ? = 2
[1,2,3,6,7,4,5] => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5,7] => ? = 1
[1,2,3,6,7,5,4] => [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [5,6,1,2,3,4,7] => ? = 2
[1,2,3,7,4,6,5] => [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [6,1,7,2,3,4,5] => ? = 2
[1,2,3,7,5,4,6] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [5,1,7,2,3,4,6] => ? = 2
[1,2,3,7,5,6,4] => [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [5,1,7,2,3,4,6] => ? = 2
[1,2,3,7,6,4,5] => [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,6,7,2,3,4,5] => ? = 2
[1,2,3,7,6,5,4] => [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 3
[1,2,4,3,5,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [4,1,2,3,5,6,7] => ? = 1
[1,2,4,3,5,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [4,7,1,2,3,5,6] => ? = 2
[1,2,4,3,6,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5,7] => ? = 2
[1,2,4,3,6,7,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5,7] => ? = 2
[1,2,4,3,7,5,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,1,7,2,3,5,6] => ? = 2
[1,2,4,3,7,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,6,7,1,2,3,5] => ? = 3
[1,2,4,5,3,6,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [4,1,2,3,5,6,7] => ? = 1
[1,2,4,5,3,7,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [4,7,1,2,3,5,6] => ? = 2
[1,2,4,5,6,3,7] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [4,1,2,3,5,6,7] => ? = 1
[1,2,4,5,6,7,3] => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [4,1,2,3,5,6,7] => ? = 1
[1,2,4,5,7,3,6] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [4,7,1,2,3,5,6] => ? = 2
[1,2,4,5,7,6,3] => [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [4,7,1,2,3,5,6] => ? = 2
[1,2,4,6,3,5,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5,7] => ? = 2
[1,2,4,6,3,7,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5,7] => ? = 2
[1,2,4,6,5,3,7] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5,7] => ? = 2
[1,2,4,6,5,7,3] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5,7] => ? = 2
[1,2,4,6,7,3,5] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5,7] => ? = 2
[1,2,4,6,7,5,3] => [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5,7] => ? = 2
[1,2,4,7,3,5,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,1,7,2,3,5,6] => ? = 2
[1,2,4,7,3,6,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,6,7,1,2,3,5] => ? = 3
[1,2,4,7,5,3,6] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,1,7,2,3,5,6] => ? = 2
[1,2,4,7,5,6,3] => [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,1,7,2,3,5,6] => ? = 2
[1,2,4,7,6,3,5] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,6,7,1,2,3,5] => ? = 3
[1,2,4,7,6,5,3] => [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,6,7,1,2,3,5] => ? = 3
[1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,5,2,3,4,6,7] => ? = 1
[1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,5,7,2,3,4,6] => ? = 2
[1,2,5,3,6,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,5,2,3,4,6,7] => ? = 1
Description
The number of big exceedences of a permutation.
A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$.
This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Matching statistic: St001083
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => [1] => 0
[1,2] => [.,[.,.]]
=> [2,1] => [1,2] => 0
[2,1] => [[.,.],.]
=> [1,2] => [2,1] => 0
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 0
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => 0
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => 0
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => 0
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,4,2] => 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,4,2] => 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => 0
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => 0
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => 0
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => 0
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => 0
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,4,5,3] => 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,4,5,3] => 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,4,5,2,3] => 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,4,5,2,3] => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,4,5,3,2] => 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,4,5,3,2] => 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,4,5,2,3] => 1
[1,4,2,3,6,5,7] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,2,3,6,7,5] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,2,3,7,5,6] => [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => [1,4,7,5,6,2,3] => ? = 2
[1,4,2,3,7,6,5] => [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => [1,4,7,6,5,2,3] => ? = 3
[1,4,2,6,3,5,7] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,2,6,3,7,5] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,2,6,5,3,7] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,2,6,5,7,3] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,2,6,7,3,5] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,2,6,7,5,3] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,2,7,3,5,6] => [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => [1,4,7,5,6,2,3] => ? = 2
[1,4,2,7,3,6,5] => [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => [1,4,7,6,5,2,3] => ? = 3
[1,4,2,7,5,3,6] => [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => [1,4,7,5,6,2,3] => ? = 2
[1,4,2,7,5,6,3] => [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => [1,4,7,5,6,2,3] => ? = 2
[1,4,2,7,6,3,5] => [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => [1,4,7,6,5,2,3] => ? = 3
[1,4,2,7,6,5,3] => [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => [1,4,7,6,5,2,3] => ? = 3
[1,4,3,2,7,5,6] => [.,[[[.,.],.],[[.,[.,.]],.]]]
=> [2,3,6,5,7,4,1] => [1,4,7,5,6,3,2] => ? = 3
[1,4,3,7,2,5,6] => [.,[[[.,.],.],[[.,[.,.]],.]]]
=> [2,3,6,5,7,4,1] => [1,4,7,5,6,3,2] => ? = 3
[1,4,3,7,5,2,6] => [.,[[[.,.],.],[[.,[.,.]],.]]]
=> [2,3,6,5,7,4,1] => [1,4,7,5,6,3,2] => ? = 3
[1,4,3,7,5,6,2] => [.,[[[.,.],.],[[.,[.,.]],.]]]
=> [2,3,6,5,7,4,1] => [1,4,7,5,6,3,2] => ? = 3
[1,4,6,2,3,5,7] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,6,2,3,7,5] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,6,2,5,3,7] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,6,2,5,7,3] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,6,2,7,3,5] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,6,2,7,5,3] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,6,5,2,3,7] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,6,5,2,7,3] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,6,5,7,2,3] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,6,7,2,3,5] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,6,7,2,5,3] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,6,7,5,2,3] => [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [1,4,6,7,5,2,3] => ? = 2
[1,4,7,2,3,5,6] => [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => [1,4,7,5,6,2,3] => ? = 2
[1,4,7,2,3,6,5] => [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => [1,4,7,6,5,2,3] => ? = 3
[1,4,7,2,5,3,6] => [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => [1,4,7,5,6,2,3] => ? = 2
[1,4,7,2,5,6,3] => [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => [1,4,7,5,6,2,3] => ? = 2
[1,4,7,2,6,3,5] => [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => [1,4,7,6,5,2,3] => ? = 3
[1,4,7,2,6,5,3] => [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => [1,4,7,6,5,2,3] => ? = 3
[1,4,7,3,2,5,6] => [.,[[[.,.],.],[[.,[.,.]],.]]]
=> [2,3,6,5,7,4,1] => [1,4,7,5,6,3,2] => ? = 3
[1,4,7,3,5,2,6] => [.,[[[.,.],.],[[.,[.,.]],.]]]
=> [2,3,6,5,7,4,1] => [1,4,7,5,6,3,2] => ? = 3
[1,4,7,3,5,6,2] => [.,[[[.,.],.],[[.,[.,.]],.]]]
=> [2,3,6,5,7,4,1] => [1,4,7,5,6,3,2] => ? = 3
[1,4,7,5,2,3,6] => [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => [1,4,7,5,6,2,3] => ? = 2
[1,4,7,5,2,6,3] => [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => [1,4,7,5,6,2,3] => ? = 2
[1,4,7,5,3,2,6] => [.,[[[.,.],.],[[.,[.,.]],.]]]
=> [2,3,6,5,7,4,1] => [1,4,7,5,6,3,2] => ? = 3
[1,4,7,5,3,6,2] => [.,[[[.,.],.],[[.,[.,.]],.]]]
=> [2,3,6,5,7,4,1] => [1,4,7,5,6,3,2] => ? = 3
[1,4,7,5,6,2,3] => [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => [1,4,7,5,6,2,3] => ? = 2
[1,4,7,5,6,3,2] => [.,[[[.,.],.],[[.,[.,.]],.]]]
=> [2,3,6,5,7,4,1] => [1,4,7,5,6,3,2] => ? = 3
[1,4,7,6,2,3,5] => [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => [1,4,7,6,5,2,3] => ? = 3
[1,4,7,6,2,5,3] => [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => [1,4,7,6,5,2,3] => ? = 3
[1,4,7,6,5,2,3] => [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => [1,4,7,6,5,2,3] => ? = 3
Description
The number of boxed occurrences of 132 in a permutation.
This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000358The number of occurrences of the pattern 31-2. St000317The cycle descent number of a permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation.
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