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Your data matches 14 different statistics following compositions of up to 3 maps.
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Matching statistic: St000356
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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] => 0
[2,1,3] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[2,3,1] => [[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [1,3,2] => 1
[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] => 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[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] => 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000052
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 0
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 0
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 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]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 2
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 2
[8,5,4,6,3,7,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 2
[7,6,4,5,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 2
[7,5,4,6,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 2
[6,5,4,7,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 2
[8,7,6,4,5,2,3,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[8,6,7,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[7,6,8,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[8,5,6,4,7,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[7,5,6,4,8,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[6,5,7,4,8,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[8,5,6,7,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
[6,5,7,8,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
[8,7,6,3,4,2,5,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[8,6,7,3,4,2,5,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[7,6,8,3,4,2,5,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[8,7,5,3,4,2,6,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[8,7,4,3,5,2,6,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[8,5,6,3,4,2,7,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[8,6,4,3,5,2,7,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[8,4,5,3,6,2,7,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[8,3,4,5,2,6,7,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
[7,6,5,3,4,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[7,5,6,3,4,2,8,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[6,5,7,3,4,2,8,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[7,6,4,3,5,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[7,5,4,3,6,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[7,4,5,3,6,2,8,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[6,5,4,3,7,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[6,4,5,3,7,2,8,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[5,4,6,3,7,2,8,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[7,4,5,6,2,3,8,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
[5,4,6,7,2,3,8,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
[6,3,4,5,2,7,8,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
[4,3,5,6,2,7,8,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
[5,6,7,8,3,4,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[7,8,3,4,5,6,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[4,5,6,7,3,8,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[5,6,3,4,7,8,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[4,5,3,6,7,8,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[8,7,6,5,4,1,2,3] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2
[5,6,7,8,2,3,1,4] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4
[8,7,6,5,3,1,2,4] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[8,7,6,5,2,1,3,4] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[6,7,8,5,2,1,3,4] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2
[8,7,6,5,1,2,3,4] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[7,8,5,6,1,2,3,4] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
[6,7,5,8,1,2,3,4] => [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 2
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: St001167
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 60%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 60%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 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] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 1
[8,6,7,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 1
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 1
[8,7,6,4,5,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? = 1
[8,6,7,4,5,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> ? = 2
[8,7,5,4,6,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? = 1
[8,5,6,4,7,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> ? = 2
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? = 1
[7,5,6,4,8,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> ? = 2
[6,5,7,4,8,3,2,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> ? = 2
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> ? = 2
[7,6,5,8,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> ? = 2
[8,5,4,6,3,7,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> ? = 2
[7,6,4,5,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> ? = 2
[7,5,4,6,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> ? = 2
[6,5,4,7,3,8,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> ? = 2
[8,7,6,5,4,2,3,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 1
[8,6,7,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[7,6,8,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[8,7,6,4,5,2,3,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 2
[8,6,7,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3
[7,6,8,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3
[8,5,6,4,7,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3
[7,5,6,4,8,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3
[6,5,7,4,8,2,3,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3
[8,7,6,5,3,2,4,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 1
[8,6,7,5,3,2,4,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[7,6,8,5,3,2,4,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[8,5,6,7,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 4
[6,5,7,8,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 4
[8,7,6,4,3,2,5,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 1
[7,6,8,4,3,2,5,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[8,7,6,3,4,2,5,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 2
[8,6,7,3,4,2,5,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3
[7,6,8,3,4,2,5,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3
[8,7,5,3,4,2,6,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 2
[8,7,4,3,5,2,6,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 2
[8,6,5,4,3,2,7,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 1
[8,5,6,3,4,2,7,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3
[8,6,4,3,5,2,7,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 2
[8,4,5,3,6,2,7,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3
[8,3,4,5,2,6,7,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 4
[7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 1
[6,5,7,4,3,2,8,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[7,6,5,3,4,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 2
[7,5,6,3,4,2,8,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3
[6,5,7,3,4,2,8,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3
[7,6,4,3,5,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 2
[7,5,4,3,6,2,8,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> ? = 2
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: St000065
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 90%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 90%
Values
[1] => [.,.]
=> [1,0]
=> [[1]]
=> 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 0
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 0
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 0
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [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
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [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
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [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
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [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
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [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
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [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
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [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
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [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
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [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
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [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
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [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
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [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
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [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
[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] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [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
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [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
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [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
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [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
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [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
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [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
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [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
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [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
[4,3,2,1] => [[[[.,.],.],.],.]
=> [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
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[2,1,3,4,6,7,5] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 1
[2,1,3,5,6,4,7] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[2,1,3,5,6,7,4] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[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,-1,1],[0,0,0,0,0,1,0]]
=> ? = 2
[2,1,3,5,7,4,6] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[2,1,3,5,7,6,4] => [[.,.],[.,[[.,[[.,.],.]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[2,1,3,6,7,4,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[2,1,4,5,6,7,3] => [[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 3
[2,3,1,4,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[2,3,1,7,6,5,4] => [[.,[.,.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[2,3,4,5,6,1,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 4
[2,3,4,5,7,1,6] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 4
[2,3,4,6,7,1,5] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 4
[2,3,5,6,7,1,4] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 4
[2,4,1,3,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[2,4,1,7,6,5,3] => [[.,[.,.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[2,4,5,6,7,1,3] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 4
[2,5,1,3,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[2,5,1,7,6,4,3] => [[.,[.,.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[2,6,1,3,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[2,6,1,7,5,4,3] => [[.,[.,.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[2,7,1,3,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[2,7,1,6,5,4,3] => [[.,[.,.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[3,1,2,4,6,7,5] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 1
[3,1,2,5,6,4,7] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[3,1,2,5,6,7,4] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[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,-1,1],[0,0,0,0,0,1,0]]
=> ? = 2
[3,1,2,5,7,4,6] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[3,1,2,5,7,6,4] => [[.,.],[.,[[.,[[.,.],.]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[3,1,2,6,7,4,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[3,1,4,5,6,7,2] => [[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 3
[3,4,1,2,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[3,4,1,7,6,5,2] => [[.,[.,.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[3,4,2,1,7,6,5] => [[[.,[.,.]],.],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[3,4,5,6,7,1,2] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 4
[3,5,1,2,4,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[3,5,1,7,6,4,2] => [[.,[.,.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[3,5,2,1,7,6,4] => [[[.,[.,.]],.],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[3,6,1,2,4,5,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[3,6,1,7,5,4,2] => [[.,[.,.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[3,6,2,1,7,5,4] => [[[.,[.,.]],.],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[3,7,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[3,7,1,6,5,4,2] => [[.,[.,.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[3,7,2,1,6,5,4] => [[[.,[.,.]],.],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[4,1,2,3,6,7,5] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 1
[4,1,2,5,6,3,7] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[4,1,2,5,6,7,3] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[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,-1,1],[0,0,0,0,0,1,0]]
=> ? = 2
[4,1,2,5,7,3,6] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[4,1,2,5,7,6,3] => [[.,.],[.,[[.,[[.,.],.]],.]]]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[4,1,2,6,7,3,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[4,1,3,5,6,7,2] => [[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> ? = 3
[4,5,1,2,3,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
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: St000371
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 90%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 90%
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] => [3,1,2] => 0
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [2,1,3] => 0
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [3,2,1] => 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => [2,1,3] => 0
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [2,3,1] => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,2,3] => 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1,2,4] => 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,1,2] => 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1,2,4] => 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,4,1,2] => 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,3,4] => 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,2,1,4] => 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,3,2,1] => 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,2,1,4] => 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,4,2,1] => 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,3,4] => 0
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,3,1,4] => 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,4,3,1] => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,2,1,4] => 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,2,4,1] => 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,3,4] => 0
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,3,1,4] => 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,4,3,1] => 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,3,1,4] => 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [2,3,4,1] => 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] => [5,1,2,3,4] => 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1,2,3,5] => 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,1,2,3] => 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1,2,3,5] => 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,5,1,2,3] => 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [3,1,2,4,5] => 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,5,1,2,4] => 0
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [4,3,1,2,5] => 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,3,1,2] => 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [4,3,1,2,5] => 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [4,5,3,1,2] => 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [3,1,2,4,5] => 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,5,1,2,4] => 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,4,1,2,5] => 0
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,5,4,1,2] => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [4,3,1,2,5] => 1
[1,3,2,7,6,5,4] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => [3,5,6,7,1,2,4] => ? = 0
[1,4,2,7,6,5,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => [3,5,6,7,1,2,4] => ? = 0
[1,4,3,2,6,5,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,4,3,2,7,5,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,4,3,2,7,6,5] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [3,4,6,7,1,2,5] => ? = 0
[1,5,2,7,6,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => [3,5,6,7,1,2,4] => ? = 0
[1,5,3,2,6,4,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,5,3,2,7,4,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,5,3,2,7,6,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [3,4,6,7,1,2,5] => ? = 0
[1,5,4,2,6,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,5,4,2,7,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,5,4,2,7,6,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [3,4,6,7,1,2,5] => ? = 0
[1,5,4,3,2,6,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [3,4,5,1,2,6,7] => ? = 0
[1,5,4,3,2,7,6] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [3,4,5,7,1,2,6] => ? = 0
[1,6,2,7,5,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => [3,5,6,7,1,2,4] => ? = 0
[1,6,3,2,5,4,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,6,3,2,7,4,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,6,3,2,7,5,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [3,4,6,7,1,2,5] => ? = 0
[1,6,4,2,5,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,6,4,2,7,3,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,6,4,2,7,5,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [3,4,6,7,1,2,5] => ? = 0
[1,6,4,3,2,5,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [3,4,5,1,2,6,7] => ? = 0
[1,6,4,3,2,7,5] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [3,4,5,7,1,2,6] => ? = 0
[1,6,5,2,4,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,6,5,2,7,3,4] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,6,5,2,7,4,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [3,4,6,7,1,2,5] => ? = 0
[1,6,5,3,2,4,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [3,4,5,1,2,6,7] => ? = 0
[1,6,5,3,2,7,4] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [3,4,5,7,1,2,6] => ? = 0
[1,6,5,4,2,3,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [3,4,5,1,2,6,7] => ? = 0
[1,6,5,4,2,7,3] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [3,4,5,7,1,2,6] => ? = 0
[1,6,5,4,3,2,7] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [2,3,4,5,7,6,1] => [3,4,5,6,1,2,7] => ? = 0
[1,7,2,6,5,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => [3,5,6,7,1,2,4] => ? = 0
[1,7,3,2,5,4,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,7,3,2,6,4,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,7,3,2,6,5,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [3,4,6,7,1,2,5] => ? = 0
[1,7,4,2,5,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,7,4,2,6,3,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,7,4,2,6,5,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [3,4,6,7,1,2,5] => ? = 0
[1,7,4,3,2,5,6] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [3,4,5,1,2,6,7] => ? = 0
[1,7,4,3,2,6,5] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [3,4,5,7,1,2,6] => ? = 0
[1,7,5,2,4,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,7,5,2,6,3,4] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,7,5,2,6,4,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [3,4,6,7,1,2,5] => ? = 0
[1,7,5,3,2,4,6] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [3,4,5,1,2,6,7] => ? = 0
[1,7,5,3,2,6,4] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [3,4,5,7,1,2,6] => ? = 0
[1,7,5,4,2,3,6] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [3,4,5,1,2,6,7] => ? = 0
[1,7,5,4,2,6,3] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [3,4,5,7,1,2,6] => ? = 0
[1,7,5,4,3,2,6] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [2,3,4,5,7,6,1] => [3,4,5,6,1,2,7] => ? = 0
[1,7,6,2,4,3,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
[1,7,6,2,5,3,4] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [3,4,6,1,2,5,7] => ? = 0
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St000711
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Values
[1] => [.,.]
=> [1,0]
=> [1] => ? = 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [2,1] => 0
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,2] => 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 0
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 0
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 1
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => ? = 0
[1,3,2,7,6,5,4] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 0
[1,4,2,7,6,5,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 0
[1,4,3,2,6,5,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,4,3,2,7,5,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,4,3,2,7,6,5] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 0
[1,5,2,7,6,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 0
[1,5,3,2,6,4,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,5,3,2,7,4,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,5,3,2,7,6,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 0
[1,5,4,2,6,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,5,4,2,7,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,5,4,2,7,6,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 0
[1,5,4,3,2,6,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 0
[1,5,4,3,2,7,6] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 0
[1,6,2,7,5,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 0
[1,6,3,2,5,4,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,6,3,2,7,4,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,6,3,2,7,5,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 0
[1,6,4,2,5,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,6,4,2,7,3,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,6,4,2,7,5,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 0
[1,6,4,3,2,5,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 0
[1,6,4,3,2,7,5] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 0
[1,6,5,2,4,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,6,5,2,7,3,4] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,6,5,2,7,4,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 0
[1,6,5,3,2,4,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 0
[1,6,5,3,2,7,4] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 0
[1,6,5,4,2,3,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 0
[1,6,5,4,2,7,3] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 0
[1,6,5,4,3,2,7] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 0
[1,7,2,6,5,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 0
[1,7,3,2,5,4,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,7,3,2,6,4,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,7,3,2,6,5,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 0
[1,7,4,2,5,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,7,4,2,6,3,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,7,4,2,6,5,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 0
[1,7,4,3,2,5,6] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 0
[1,7,4,3,2,6,5] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 0
[1,7,5,2,4,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,7,5,2,6,3,4] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
[1,7,5,2,6,4,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 0
[1,7,5,3,2,4,6] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 0
[1,7,5,3,2,6,4] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 0
[1,7,5,4,2,3,6] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 0
[1,7,5,4,2,6,3] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 0
[1,7,5,4,3,2,6] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 0
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: St001687
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 50%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 50%
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] => 0
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 0
[2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 1
[3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 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] => 0
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => 0
[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] => 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 0
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 0
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 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] => 0
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,1,3,2] => 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,1,3,2] => 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => 0
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,1,4,3,2] => 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,4,2,3] => 0
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,1,4,3] => 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,1,4,3] => 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,1,4,3,2] => 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,4,2,3] => 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => 0
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,1,3,2,4] => 1
[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
[1,3,2,7,6,5,4] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => [7,1,6,2,3,4,5] => ? = 0
[1,4,2,7,6,5,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => [7,1,6,2,3,4,5] => ? = 0
[1,4,3,2,6,5,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,4,3,2,7,5,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,4,3,2,7,6,5] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 0
[1,5,2,7,6,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => [7,1,6,2,3,4,5] => ? = 0
[1,5,3,2,6,4,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,5,3,2,7,4,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,5,3,2,7,6,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 0
[1,5,4,2,6,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,5,4,2,7,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,5,4,2,7,6,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 0
[1,5,4,3,2,6,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => ? = 0
[1,5,4,3,2,7,6] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 0
[1,6,2,7,5,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => [7,1,6,2,3,4,5] => ? = 0
[1,6,3,2,5,4,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,6,3,2,7,4,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,6,3,2,7,5,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 0
[1,6,4,2,5,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,6,4,2,7,3,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,6,4,2,7,5,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 0
[1,6,4,3,2,5,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => ? = 0
[1,6,4,3,2,7,5] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 0
[1,6,5,2,4,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,6,5,2,7,3,4] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,6,5,2,7,4,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 0
[1,6,5,3,2,4,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => ? = 0
[1,6,5,3,2,7,4] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 0
[1,6,5,4,2,3,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => ? = 0
[1,6,5,4,2,7,3] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 0
[1,6,5,4,3,2,7] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 0
[1,7,2,6,5,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [2,4,5,6,7,3,1] => [7,1,6,2,3,4,5] => ? = 0
[1,7,3,2,5,4,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,7,3,2,6,4,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,7,3,2,6,5,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 0
[1,7,4,2,5,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,7,4,2,6,3,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,7,4,2,6,5,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 0
[1,7,4,3,2,5,6] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => ? = 0
[1,7,4,3,2,6,5] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 0
[1,7,5,2,4,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,7,5,2,6,3,4] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
[1,7,5,2,6,4,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [2,3,5,6,7,4,1] => [7,1,2,6,3,4,5] => ? = 0
[1,7,5,3,2,4,6] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => ? = 0
[1,7,5,3,2,6,4] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 0
[1,7,5,4,2,3,6] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [2,3,4,7,6,5,1] => [7,1,2,3,6,5,4] => ? = 0
[1,7,5,4,2,6,3] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [2,3,4,6,7,5,1] => [7,1,2,3,6,4,5] => ? = 0
[1,7,5,4,3,2,6] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 0
[1,7,6,2,4,3,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [2,3,5,7,6,4,1] => [7,1,2,6,3,5,4] => ? = 0
Description
The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.
Matching statistic: St000373
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [1,2] => 0
[2,1] => [2,1] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [2,1,3] => 0
[2,1,3] => [2,1,3] => [3,1,2] => [1,3,2] => 0
[2,3,1] => [3,2,1] => [1,2,3] => [3,2,1] => 1
[3,1,2] => [3,1,2] => [2,1,3] => [2,3,1] => 0
[3,2,1] => [2,3,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [2,1,3,4] => 0
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [2,3,1,4] => 0
[1,4,3,2] => [1,3,4,2] => [2,4,3,1] => [3,1,2,4] => 0
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [1,2,4,3] => 0
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 0
[2,3,1,4] => [3,2,1,4] => [4,1,2,3] => [1,4,3,2] => 1
[2,3,4,1] => [4,2,3,1] => [1,3,2,4] => [4,2,3,1] => 2
[2,4,1,3] => [4,2,1,3] => [3,1,2,4] => [2,4,3,1] => 1
[2,4,3,1] => [3,2,4,1] => [1,4,2,3] => [4,1,3,2] => 1
[3,1,2,4] => [3,1,2,4] => [4,2,1,3] => [1,3,4,2] => 0
[3,1,4,2] => [4,3,1,2] => [2,1,3,4] => [3,4,2,1] => 0
[3,2,1,4] => [2,3,1,4] => [4,1,3,2] => [1,4,2,3] => 0
[3,2,4,1] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[3,4,1,2] => [4,1,3,2] => [2,3,1,4] => [3,2,4,1] => 1
[3,4,2,1] => [2,4,3,1] => [1,3,4,2] => [4,2,1,3] => 1
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [2,3,4,1] => 0
[4,1,3,2] => [3,4,1,2] => [2,1,4,3] => [3,4,1,2] => 0
[4,2,1,3] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 0
[4,2,3,1] => [3,4,2,1] => [1,2,4,3] => [4,3,1,2] => 1
[4,3,1,2] => [3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 0
[4,3,2,1] => [2,3,4,1] => [1,4,3,2] => [4,1,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,3,2,4,5] => 0
[1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => [3,2,1,4,5] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [2,3,1,4,5] => 0
[1,2,5,4,3] => [1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,2,4,3,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => 0
[1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,5,3,4,2] => [2,4,3,5,1] => [4,2,3,1,5] => 2
[1,3,5,2,4] => [1,5,3,2,4] => [4,2,3,5,1] => [2,4,3,1,5] => 1
[1,3,5,4,2] => [1,4,3,5,2] => [2,5,3,4,1] => [4,1,3,2,5] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [1,3,4,2,5] => 0
[1,4,2,5,3] => [1,5,4,2,3] => [3,2,4,5,1] => [3,4,2,1,5] => 0
[1,4,3,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [1,4,2,3,5] => 0
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,4,5,2,3] => [1,5,2,4,3] => [3,4,2,5,1] => [3,2,4,1,5] => 1
[1,3,2,7,6,5,4] => [1,3,2,5,6,7,4] => [4,7,6,5,2,3,1] => [4,1,2,3,6,5,7] => ? = 0
[1,4,2,7,6,5,3] => [1,5,4,2,6,7,3] => [3,7,6,2,4,5,1] => [5,1,2,6,4,3,7] => ? = 0
[1,4,3,2,7,5,6] => [1,3,4,2,7,5,6] => [6,5,7,2,4,3,1] => [2,3,1,6,4,5,7] => ? = 0
[1,4,3,2,7,6,5] => [1,3,4,2,6,7,5] => [5,7,6,2,4,3,1] => [3,1,2,6,4,5,7] => ? = 0
[1,5,2,7,6,4,3] => [1,4,6,5,2,7,3] => [3,7,2,5,6,4,1] => [5,1,6,3,2,4,7] => ? = 0
[1,5,3,2,7,4,6] => [1,3,7,5,2,4,6] => [6,4,2,5,7,3,1] => [2,4,6,3,1,5,7] => ? = 0
[1,5,3,2,7,6,4] => [1,3,6,5,2,7,4] => [4,7,2,5,6,3,1] => [4,1,6,3,2,5,7] => ? = 0
[1,5,4,2,6,3,7] => [1,6,4,5,2,3,7] => [7,3,2,5,4,6,1] => [1,5,6,3,4,2,7] => ? = 0
[1,5,4,2,7,3,6] => [1,7,4,5,2,3,6] => [6,3,2,5,4,7,1] => [2,5,6,3,4,1,7] => ? = 0
[1,5,4,2,7,6,3] => [1,6,4,5,2,7,3] => [3,7,2,5,4,6,1] => [5,1,6,3,4,2,7] => ? = 0
[1,6,2,7,5,4,3] => [1,4,5,7,6,2,3] => [3,2,6,7,5,4,1] => [5,6,2,1,3,4,7] => ? = 0
[1,6,3,2,7,4,5] => [1,3,7,6,2,4,5] => [5,4,2,6,7,3,1] => [3,4,6,2,1,5,7] => ? = 0
[1,6,3,2,7,5,4] => [1,3,5,7,6,2,4] => [4,2,6,7,5,3,1] => [4,6,2,1,3,5,7] => ? = 0
[1,6,4,2,5,3,7] => [1,5,4,6,2,3,7] => [7,3,2,6,4,5,1] => [1,5,6,2,4,3,7] => ? = 0
[1,6,4,2,7,3,5] => [1,7,4,6,2,3,5] => [5,3,2,6,4,7,1] => [3,5,6,2,4,1,7] => ? = 0
[1,6,4,2,7,5,3] => [1,5,4,7,6,2,3] => [3,2,6,7,4,5,1] => [5,6,2,1,4,3,7] => ? = 0
[1,6,4,3,2,7,5] => [1,3,4,7,6,2,5] => [5,2,6,7,4,3,1] => [3,6,2,1,4,5,7] => ? = 0
[1,6,5,2,4,3,7] => [1,4,5,2,6,3,7] => [7,3,6,2,5,4,1] => [1,5,2,6,3,4,7] => ? = 0
[1,6,5,2,7,3,4] => [1,7,5,2,6,3,4] => [4,3,6,2,5,7,1] => [4,5,2,6,3,1,7] => ? = 0
[1,6,5,2,7,4,3] => [1,4,7,5,6,2,3] => [3,2,6,5,7,4,1] => [5,6,2,3,1,4,7] => ? = 0
[1,6,5,3,2,7,4] => [1,3,7,5,6,2,4] => [4,2,6,5,7,3,1] => [4,6,2,3,1,5,7] => ? = 0
[1,6,5,4,2,3,7] => [1,4,2,5,6,3,7] => [7,3,6,5,2,4,1] => [1,5,2,3,6,4,7] => ? = 0
[1,6,5,4,2,7,3] => [1,7,4,5,6,2,3] => [3,2,6,5,4,7,1] => [5,6,2,3,4,1,7] => ? = 0
[1,6,5,4,3,2,7] => [1,3,4,5,6,2,7] => [7,2,6,5,4,3,1] => [1,6,2,3,4,5,7] => ? = 0
[1,7,3,2,5,4,6] => [1,3,5,7,2,4,6] => [6,4,2,7,5,3,1] => [2,4,6,1,3,5,7] => ? = 0
[1,7,3,2,6,4,5] => [1,3,6,7,2,4,5] => [5,4,2,7,6,3,1] => [3,4,6,1,2,5,7] => ? = 0
[1,7,4,2,5,3,6] => [1,5,4,7,2,3,6] => [6,3,2,7,4,5,1] => [2,5,6,1,4,3,7] => ? = 0
[1,7,4,2,6,3,5] => [1,6,4,7,2,3,5] => [5,3,2,7,4,6,1] => [3,5,6,1,4,2,7] => ? = 0
[1,7,4,2,6,5,3] => [1,5,4,6,7,2,3] => [3,2,7,6,4,5,1] => [5,6,1,2,4,3,7] => ? = 0
[1,7,4,3,2,5,6] => [1,3,4,7,2,5,6] => [6,5,2,7,4,3,1] => [2,3,6,1,4,5,7] => ? = 0
[1,7,5,2,4,3,6] => [1,4,5,2,7,3,6] => [6,3,7,2,5,4,1] => [2,5,1,6,3,4,7] => ? = 0
[1,7,5,2,6,3,4] => [1,6,5,2,7,3,4] => [4,3,7,2,5,6,1] => [4,5,1,6,3,2,7] => ? = 0
[1,7,5,2,6,4,3] => [1,4,6,5,7,2,3] => [3,2,7,5,6,4,1] => [5,6,1,3,2,4,7] => ? = 0
[1,7,5,3,2,4,6] => [1,3,5,2,7,4,6] => [6,4,7,2,5,3,1] => [2,4,1,6,3,5,7] => ? = 0
[1,7,5,3,2,6,4] => [1,3,6,5,7,2,4] => [4,2,7,5,6,3,1] => [4,6,1,3,2,5,7] => ? = 0
[1,7,5,4,2,3,6] => [1,4,2,5,7,3,6] => [6,3,7,5,2,4,1] => [2,5,1,3,6,4,7] => ? = 0
[1,7,5,4,2,6,3] => [1,6,4,5,7,2,3] => [3,2,7,5,4,6,1] => [5,6,1,3,4,2,7] => ? = 0
[1,7,6,2,4,3,5] => [1,4,6,2,3,7,5] => [5,7,3,2,6,4,1] => [3,1,5,6,2,4,7] => ? = 0
[1,7,6,2,5,3,4] => [1,5,6,2,3,7,4] => [4,7,3,2,6,5,1] => [4,1,5,6,2,3,7] => ? = 0
[1,7,6,3,2,4,5] => [1,3,6,2,4,7,5] => [5,7,4,2,6,3,1] => [3,1,4,6,2,5,7] => ? = 0
[1,7,6,4,2,3,5] => [1,4,2,6,3,7,5] => [5,7,3,6,2,4,1] => [3,1,5,2,6,4,7] => ? = 0
[1,7,6,4,2,5,3] => [1,5,4,6,2,7,3] => [3,7,2,6,4,5,1] => [5,1,6,2,4,3,7] => ? = 0
[1,7,6,4,3,2,5] => [1,3,4,6,2,7,5] => [5,7,2,6,4,3,1] => [3,1,6,2,4,5,7] => ? = 0
[1,7,6,5,2,3,4] => [1,5,2,3,6,7,4] => [4,7,6,3,2,5,1] => [4,1,2,5,6,3,7] => ? = 0
[1,7,6,5,3,2,4] => [1,3,5,2,6,7,4] => [4,7,6,2,5,3,1] => [4,1,2,6,3,5,7] => ? = 0
[2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => [6,7,5,4,3,1,2] => [2,1,3,4,5,7,6] => ? = 0
[2,1,3,4,6,7,5] => [2,1,3,4,7,6,5] => [5,6,7,4,3,1,2] => [3,2,1,4,5,7,6] => ? = 1
[2,1,3,4,7,5,6] => [2,1,3,4,7,5,6] => [6,5,7,4,3,1,2] => [2,3,1,4,5,7,6] => ? = 0
[2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => [6,7,4,5,3,1,2] => [2,1,4,3,5,7,6] => ? = 0
[2,1,3,5,6,7,4] => [2,1,3,7,5,6,4] => [4,6,5,7,3,1,2] => [4,2,3,1,5,7,6] => ? = 2
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: St001744
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 50%
Mp00069: Permutations —complement⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [2,3,1] => [2,1,3] => [2,1,3] => 0
[2,1,3] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[2,3,1] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[3,1,2] => [2,1,3] => [2,3,1] => [1,3,2] => 0
[3,2,1] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 0
[1,3,2,4] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 1
[1,4,2,3] => [3,2,4,1] => [2,3,1,4] => [1,3,2,4] => 0
[1,4,3,2] => [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 0
[2,1,3,4] => [4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 0
[2,1,4,3] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[2,3,4,1] => [1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 2
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 1
[2,4,3,1] => [1,3,4,2] => [4,2,1,3] => [4,2,1,3] => 1
[3,1,2,4] => [4,2,1,3] => [1,3,4,2] => [1,2,4,3] => 0
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [2,1,4,3] => 0
[3,2,1,4] => [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 0
[3,2,4,1] => [1,4,2,3] => [4,1,3,2] => [4,1,3,2] => 1
[3,4,1,2] => [2,1,4,3] => [3,4,1,2] => [2,4,1,3] => 1
[3,4,2,1] => [1,2,4,3] => [4,3,1,2] => [4,3,1,2] => 1
[4,1,2,3] => [3,2,1,4] => [2,3,4,1] => [1,2,4,3] => 0
[4,1,3,2] => [2,3,1,4] => [3,2,4,1] => [2,1,4,3] => 0
[4,2,1,3] => [3,1,2,4] => [2,4,3,1] => [1,4,3,2] => 0
[4,2,3,1] => [1,3,2,4] => [4,2,3,1] => [4,1,3,2] => 1
[4,3,1,2] => [2,1,3,4] => [3,4,2,1] => [1,4,3,2] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,2,4,3,5] => [5,3,4,2,1] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => [3,1,2,4,5] => 1
[1,2,5,3,4] => [4,3,5,2,1] => [2,3,1,4,5] => [1,3,2,4,5] => 0
[1,2,5,4,3] => [3,4,5,2,1] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,3,4,2,5] => [5,2,4,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [4,1,2,3,5] => [4,1,2,3,5] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => [2,4,1,3,5] => 1
[1,3,5,4,2] => [2,4,5,3,1] => [4,2,1,3,5] => [4,2,1,3,5] => 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,3,4,2,5] => [1,2,4,3,5] => 0
[1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => [2,1,4,3,5] => 0
[1,4,3,2,5] => [5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,3,5,2] => [2,5,3,4,1] => [4,1,3,2,5] => [4,1,3,2,5] => 1
[1,4,5,2,3] => [3,2,5,4,1] => [3,4,1,2,5] => [2,4,1,3,5] => 1
[1,3,2,7,6,5,4] => [4,5,6,7,2,3,1] => [4,3,2,1,6,5,7] => [4,3,2,1,6,5,7] => ? = 0
[1,4,2,7,6,5,3] => [3,5,6,7,2,4,1] => [5,3,2,1,6,4,7] => [4,3,2,1,6,5,7] => ? = 0
[1,4,3,2,7,6,5] => [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => [3,2,1,6,5,4,7] => ? = 0
[1,5,2,7,6,4,3] => [3,4,6,7,2,5,1] => [5,4,2,1,6,3,7] => [4,3,2,1,6,5,7] => ? = 0
[1,5,3,2,7,6,4] => [4,6,7,2,3,5,1] => [4,2,1,6,5,3,7] => [3,2,1,6,5,4,7] => ? = 0
[1,5,4,2,7,6,3] => [3,6,7,2,4,5,1] => [5,2,1,6,4,3,7] => [3,2,1,6,5,4,7] => ? = 0
[1,5,4,3,2,7,6] => [6,7,2,3,4,5,1] => [2,1,6,5,4,3,7] => [2,1,6,5,4,3,7] => ? = 0
[1,6,2,7,5,4,3] => [3,4,5,7,2,6,1] => [5,4,3,1,6,2,7] => [4,3,2,1,6,5,7] => ? = 0
[1,6,3,2,7,5,4] => [4,5,7,2,3,6,1] => [4,3,1,6,5,2,7] => [3,2,1,6,5,4,7] => ? = 0
[1,6,4,2,7,5,3] => [3,5,7,2,4,6,1] => [5,3,1,6,4,2,7] => [3,2,1,6,5,4,7] => ? = 0
[1,6,4,3,2,7,5] => [5,7,2,3,4,6,1] => [3,1,6,5,4,2,7] => [2,1,6,5,4,3,7] => ? = 0
[1,6,5,2,7,4,3] => [3,4,7,2,5,6,1] => [5,4,1,6,3,2,7] => [3,2,1,6,5,4,7] => ? = 0
[1,6,5,3,2,7,4] => [4,7,2,3,5,6,1] => [4,1,6,5,3,2,7] => [2,1,6,5,4,3,7] => ? = 0
[1,6,5,4,2,7,3] => [3,7,2,4,5,6,1] => [5,1,6,4,3,2,7] => [2,1,6,5,4,3,7] => ? = 0
[1,6,5,4,3,2,7] => [7,2,3,4,5,6,1] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 0
[1,7,2,6,5,4,3] => [3,4,5,6,2,7,1] => [5,4,3,2,6,1,7] => [4,3,2,1,6,5,7] => ? = 0
[1,7,3,2,6,5,4] => [4,5,6,2,3,7,1] => [4,3,2,6,5,1,7] => [3,2,1,6,5,4,7] => ? = 0
[1,7,4,2,6,5,3] => [3,5,6,2,4,7,1] => [5,3,2,6,4,1,7] => [3,2,1,6,5,4,7] => ? = 0
[1,7,4,3,2,6,5] => [5,6,2,3,4,7,1] => [3,2,6,5,4,1,7] => [2,1,6,5,4,3,7] => ? = 0
[1,7,5,2,6,4,3] => [3,4,6,2,5,7,1] => [5,4,2,6,3,1,7] => [3,2,1,6,5,4,7] => ? = 0
[1,7,5,3,2,6,4] => [4,6,2,3,5,7,1] => [4,2,6,5,3,1,7] => [2,1,6,5,4,3,7] => ? = 0
[1,7,5,4,2,6,3] => [3,6,2,4,5,7,1] => [5,2,6,4,3,1,7] => [2,1,6,5,4,3,7] => ? = 0
[1,7,5,4,3,2,6] => [6,2,3,4,5,7,1] => [2,6,5,4,3,1,7] => [1,6,5,4,3,2,7] => ? = 0
[1,7,6,2,5,4,3] => [3,4,5,2,6,7,1] => [5,4,3,6,2,1,7] => [3,2,1,6,5,4,7] => ? = 0
[1,7,6,3,2,5,4] => [4,5,2,3,6,7,1] => [4,3,6,5,2,1,7] => [2,1,6,5,4,3,7] => ? = 0
[1,7,6,4,2,5,3] => [3,5,2,4,6,7,1] => [5,3,6,4,2,1,7] => [2,1,6,5,4,3,7] => ? = 0
[1,7,6,4,3,2,5] => [5,2,3,4,6,7,1] => [3,6,5,4,2,1,7] => [1,6,5,4,3,2,7] => ? = 0
[1,7,6,5,2,4,3] => [3,4,2,5,6,7,1] => [5,4,6,3,2,1,7] => [2,1,6,5,4,3,7] => ? = 0
[1,7,6,5,3,2,4] => [4,2,3,5,6,7,1] => [4,6,5,3,2,1,7] => [1,6,5,4,3,2,7] => ? = 0
[1,7,6,5,4,2,3] => [3,2,4,5,6,7,1] => [5,6,4,3,2,1,7] => [1,6,5,4,3,2,7] => ? = 0
[1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 0
[2,1,3,4,5,7,6] => [6,7,5,4,3,1,2] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 0
[2,1,3,4,6,7,5] => [5,7,6,4,3,1,2] => [3,1,2,4,5,7,6] => [3,1,2,4,5,7,6] => ? = 1
[2,1,3,4,7,6,5] => [5,6,7,4,3,1,2] => [3,2,1,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 0
[2,1,3,5,4,7,6] => [6,7,4,5,3,1,2] => [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 0
[2,1,3,5,6,7,4] => [4,7,6,5,3,1,2] => [4,1,2,3,5,7,6] => [4,1,2,3,5,7,6] => ? = 2
[2,1,3,5,7,4,6] => [6,4,7,5,3,1,2] => [2,4,1,3,5,7,6] => [2,4,1,3,5,7,6] => ? = 1
[2,1,3,5,7,6,4] => [4,6,7,5,3,1,2] => [4,2,1,3,5,7,6] => [4,2,1,3,5,7,6] => ? = 1
[2,1,3,6,4,7,5] => [5,7,4,6,3,1,2] => [3,1,4,2,5,7,6] => [2,1,4,3,5,7,6] => ? = 0
[2,1,3,6,7,4,5] => [5,4,7,6,3,1,2] => [3,4,1,2,5,7,6] => [2,4,1,3,5,7,6] => ? = 1
[2,1,3,7,4,6,5] => [5,6,4,7,3,1,2] => [3,2,4,1,5,7,6] => [2,1,4,3,5,7,6] => ? = 0
[2,1,4,5,6,7,3] => [3,7,6,5,4,1,2] => [5,1,2,3,4,7,6] => [5,1,2,3,4,7,6] => ? = 3
[2,1,7,6,5,4,3] => [3,4,5,6,7,1,2] => [5,4,3,2,1,7,6] => [5,4,3,2,1,7,6] => ? = 0
[2,3,1,7,6,5,4] => [4,5,6,7,1,3,2] => [4,3,2,1,7,5,6] => [4,3,2,1,7,5,6] => ? = 1
[2,3,4,5,6,1,7] => [7,1,6,5,4,3,2] => [1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 4
[2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 5
[2,3,4,5,7,1,6] => [6,1,7,5,4,3,2] => [2,7,1,3,4,5,6] => [2,7,1,3,4,5,6] => ? = 4
[2,3,4,5,7,6,1] => [1,6,7,5,4,3,2] => [7,2,1,3,4,5,6] => [7,2,1,3,4,5,6] => ? = 4
[2,3,4,6,5,7,1] => [1,7,5,6,4,3,2] => [7,1,3,2,4,5,6] => [7,1,3,2,4,5,6] => ? = 4
[2,3,4,6,7,1,5] => [5,1,7,6,4,3,2] => [3,7,1,2,4,5,6] => [3,7,1,2,4,5,6] => ? = 4
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: St000317
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%
Mp00069: Permutations —complement⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [2,3,1] => [2,1,3] => [2,1,3] => 0
[2,1,3] => [3,1,2] => [1,3,2] => [1,3,2] => 0
[2,3,1] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[3,1,2] => [2,1,3] => [2,3,1] => [3,2,1] => 0
[3,2,1] => [1,2,3] => [3,2,1] => [2,3,1] => 0
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 0
[1,3,2,4] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 1
[1,4,2,3] => [3,2,4,1] => [2,3,1,4] => [3,2,1,4] => 0
[1,4,3,2] => [2,3,4,1] => [3,2,1,4] => [2,3,1,4] => 0
[2,1,3,4] => [4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 0
[2,1,4,3] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[2,3,4,1] => [1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 2
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[2,4,3,1] => [1,3,4,2] => [4,2,1,3] => [2,4,1,3] => 1
[3,1,2,4] => [4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 0
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 0
[3,2,1,4] => [4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 0
[3,2,4,1] => [1,4,2,3] => [4,1,3,2] => [4,3,1,2] => 1
[3,4,1,2] => [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 1
[3,4,2,1] => [1,2,4,3] => [4,3,1,2] => [3,1,4,2] => 1
[4,1,2,3] => [3,2,1,4] => [2,3,4,1] => [4,2,3,1] => 0
[4,1,3,2] => [2,3,1,4] => [3,2,4,1] => [4,3,2,1] => 0
[4,2,1,3] => [3,1,2,4] => [2,4,3,1] => [3,2,4,1] => 0
[4,2,3,1] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1
[4,3,1,2] => [2,1,3,4] => [3,4,2,1] => [2,4,3,1] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[1,2,4,3,5] => [5,3,4,2,1] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => [3,1,2,4,5] => 1
[1,2,5,3,4] => [4,3,5,2,1] => [2,3,1,4,5] => [3,2,1,4,5] => 0
[1,2,5,4,3] => [3,4,5,2,1] => [3,2,1,4,5] => [2,3,1,4,5] => 0
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,3,4,2,5] => [5,2,4,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [4,1,2,3,5] => [4,1,2,3,5] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => [4,2,1,3,5] => 1
[1,3,5,4,2] => [2,4,5,3,1] => [4,2,1,3,5] => [2,4,1,3,5] => 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,3,4,2,5] => [1,4,3,2,5] => 0
[1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => [3,4,1,2,5] => 0
[1,4,3,2,5] => [5,2,3,4,1] => [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,4,3,5,2] => [2,5,3,4,1] => [4,1,3,2,5] => [4,3,1,2,5] => 1
[1,4,5,2,3] => [3,2,5,4,1] => [3,4,1,2,5] => [4,1,3,2,5] => 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,3,2,7,6,5,4] => [4,5,6,7,2,3,1] => [4,3,2,1,6,5,7] => [2,3,4,1,6,5,7] => ? = 0
[1,4,2,7,6,5,3] => [3,5,6,7,2,4,1] => [5,3,2,1,6,4,7] => [2,3,5,6,1,4,7] => ? = 0
[1,4,3,2,6,5,7] => [7,5,6,2,3,4,1] => [1,3,2,6,5,4,7] => [1,3,2,5,6,4,7] => ? = 0
[1,4,3,2,7,5,6] => [6,5,7,2,3,4,1] => [2,3,1,6,5,4,7] => [3,2,1,5,6,4,7] => ? = 0
[1,4,3,2,7,6,5] => [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => [2,3,1,5,6,4,7] => ? = 0
[1,5,2,7,6,4,3] => [3,4,6,7,2,5,1] => [5,4,2,1,6,3,7] => [2,4,5,6,3,1,7] => ? = 0
[1,5,3,2,6,4,7] => [7,4,6,2,3,5,1] => [1,4,2,6,5,3,7] => [1,4,5,2,6,3,7] => ? = 0
[1,5,3,2,7,4,6] => [6,4,7,2,3,5,1] => [2,4,1,6,5,3,7] => [4,2,5,1,6,3,7] => ? = 0
[1,5,3,2,7,6,4] => [4,6,7,2,3,5,1] => [4,2,1,6,5,3,7] => [2,4,5,1,6,3,7] => ? = 0
[1,5,4,2,6,3,7] => [7,3,6,2,4,5,1] => [1,5,2,6,4,3,7] => [1,5,4,6,2,3,7] => ? = 0
[1,5,4,2,7,3,6] => [6,3,7,2,4,5,1] => [2,5,1,6,4,3,7] => [5,2,4,6,1,3,7] => ? = 0
[1,5,4,2,7,6,3] => [3,6,7,2,4,5,1] => [5,2,1,6,4,3,7] => [2,5,4,6,1,3,7] => ? = 0
[1,5,4,3,2,6,7] => [7,6,2,3,4,5,1] => [1,2,6,5,4,3,7] => [1,2,4,5,6,3,7] => ? = 0
[1,5,4,3,2,7,6] => [6,7,2,3,4,5,1] => [2,1,6,5,4,3,7] => [2,1,4,5,6,3,7] => ? = 0
[1,6,2,7,5,4,3] => [3,4,5,7,2,6,1] => [5,4,3,1,6,2,7] => [3,4,5,6,1,2,7] => ? = 0
[1,6,3,2,5,4,7] => [7,4,5,2,3,6,1] => [1,4,3,6,5,2,7] => [1,5,4,3,6,2,7] => ? = 0
[1,6,3,2,7,4,5] => [5,4,7,2,3,6,1] => [3,4,1,6,5,2,7] => [4,5,3,1,6,2,7] => ? = 0
[1,6,3,2,7,5,4] => [4,5,7,2,3,6,1] => [4,3,1,6,5,2,7] => [3,4,5,2,6,1,7] => ? = 0
[1,6,4,2,5,3,7] => [7,3,5,2,4,6,1] => [1,5,3,6,4,2,7] => [1,4,6,5,2,3,7] => ? = 0
[1,6,4,2,7,3,5] => [5,3,7,2,4,6,1] => [3,5,1,6,4,2,7] => [5,4,3,6,1,2,7] => ? = 0
[1,6,4,2,7,5,3] => [3,5,7,2,4,6,1] => [5,3,1,6,4,2,7] => [3,5,4,6,2,1,7] => ? = 0
[1,6,4,3,2,5,7] => [7,5,2,3,4,6,1] => [1,3,6,5,4,2,7] => [1,4,3,5,6,2,7] => ? = 0
[1,6,4,3,2,7,5] => [5,7,2,3,4,6,1] => [3,1,6,5,4,2,7] => [3,4,1,5,6,2,7] => ? = 0
[1,6,5,2,4,3,7] => [7,3,4,2,5,6,1] => [1,5,4,6,3,2,7] => [1,3,6,5,4,2,7] => ? = 0
[1,6,5,2,7,3,4] => [4,3,7,2,5,6,1] => [4,5,1,6,3,2,7] => [5,3,6,4,1,2,7] => ? = 0
[1,6,5,2,7,4,3] => [3,4,7,2,5,6,1] => [5,4,1,6,3,2,7] => [4,3,5,6,2,1,7] => ? = 0
[1,6,5,3,2,4,7] => [7,4,2,3,5,6,1] => [1,4,6,5,3,2,7] => [1,3,5,4,6,2,7] => ? = 0
[1,6,5,3,2,7,4] => [4,7,2,3,5,6,1] => [4,1,6,5,3,2,7] => [4,3,5,1,6,2,7] => ? = 0
[1,6,5,4,2,3,7] => [7,3,2,4,5,6,1] => [1,5,6,4,3,2,7] => [1,3,4,6,5,2,7] => ? = 0
[1,6,5,4,2,7,3] => [3,7,2,4,5,6,1] => [5,1,6,4,3,2,7] => [5,3,4,6,1,2,7] => ? = 0
[1,6,5,4,3,2,7] => [7,2,3,4,5,6,1] => [1,6,5,4,3,2,7] => [1,3,4,5,6,2,7] => ? = 0
[1,7,2,6,5,4,3] => [3,4,5,6,2,7,1] => [5,4,3,2,6,1,7] => [6,3,4,5,2,1,7] => ? = 0
[1,7,3,2,5,4,6] => [6,4,5,2,3,7,1] => [2,4,3,6,5,1,7] => [5,2,4,3,6,1,7] => ? = 0
[1,7,3,2,6,4,5] => [5,4,6,2,3,7,1] => [3,4,2,6,5,1,7] => [5,4,3,2,6,1,7] => ? = 0
[1,7,3,2,6,5,4] => [4,5,6,2,3,7,1] => [4,3,2,6,5,1,7] => [5,3,4,2,6,1,7] => ? = 0
[1,7,4,2,5,3,6] => [6,3,5,2,4,7,1] => [2,5,3,6,4,1,7] => [4,2,6,5,1,3,7] => ? = 0
[1,7,4,2,6,3,5] => [5,3,6,2,4,7,1] => [3,5,2,6,4,1,7] => [4,6,3,5,1,2,7] => ? = 0
[1,7,4,2,6,5,3] => [3,5,6,2,4,7,1] => [5,3,2,6,4,1,7] => [4,3,6,5,1,2,7] => ? = 0
[1,7,4,3,2,5,6] => [6,5,2,3,4,7,1] => [2,3,6,5,4,1,7] => [4,2,3,5,6,1,7] => ? = 0
[1,7,4,3,2,6,5] => [5,6,2,3,4,7,1] => [3,2,6,5,4,1,7] => [4,3,2,5,6,1,7] => ? = 0
[1,7,5,2,4,3,6] => [6,3,4,2,5,7,1] => [2,5,4,6,3,1,7] => [3,2,6,5,4,1,7] => ? = 0
[1,7,5,2,6,3,4] => [4,3,6,2,5,7,1] => [4,5,2,6,3,1,7] => [3,6,5,4,1,2,7] => ? = 0
[1,7,5,2,6,4,3] => [3,4,6,2,5,7,1] => [5,4,2,6,3,1,7] => [3,6,4,5,1,2,7] => ? = 0
[1,7,5,3,2,4,6] => [6,4,2,3,5,7,1] => [2,4,6,5,3,1,7] => [3,2,5,4,6,1,7] => ? = 0
[1,7,5,3,2,6,4] => [4,6,2,3,5,7,1] => [4,2,6,5,3,1,7] => [3,5,4,1,6,2,7] => ? = 0
[1,7,5,4,2,3,6] => [6,3,2,4,5,7,1] => [2,5,6,4,3,1,7] => [3,2,4,6,5,1,7] => ? = 0
[1,7,5,4,2,6,3] => [3,6,2,4,5,7,1] => [5,2,6,4,3,1,7] => [3,4,6,5,2,1,7] => ? = 0
[1,7,5,4,3,2,6] => [6,2,3,4,5,7,1] => [2,6,5,4,3,1,7] => [3,2,4,5,6,1,7] => ? = 0
[1,7,6,2,4,3,5] => [5,3,4,2,6,7,1] => [3,5,4,6,2,1,7] => [2,6,3,5,4,1,7] => ? = 0
Description
The cycle descent number of a permutation.
Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.
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