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Your data matches 25 different statistics following compositions of up to 3 maps.
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Matching statistic: St000382
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(load all 2 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,1] => 1
[[.,.],.]
=> [1,1,0,0]
=> [2] => 2
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [3] => 3
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 3
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [4] => 4
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [4] => 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 3
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 4
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 4
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
Description
The first part of an integer composition.
Matching statistic: St000645
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1,0]
=> 0 = 1 - 1
[.,[.,.]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,.]]
=> [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 3 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [7,8,5,4,6,3,2,1] => [8,7,6,4,3,5,1,2] => ?
=> ? = 1 - 1
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [8,5,6,4,7,3,2,1] => [8,7,6,4,2,3,5,1] => ?
=> ? = 1 - 1
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,3,4,2,1] => [8,7,5,6,4,1,2,3] => ?
=> ? = 1 - 1
[.,[.,[[.,[.,.]],[[.,[.,.]],.]]]]
=> [7,6,8,4,3,5,2,1] => [8,7,5,4,6,2,1,3] => ?
=> ? = 1 - 1
[.,[.,[[[.,.],.],[.,[[.,.],.]]]]]
=> [7,8,6,3,4,5,2,1] => [8,7,4,5,6,3,1,2] => ?
=> ? = 1 - 1
[.,[.,[[.,[.,[.,.]]],[[.,.],.]]]]
=> [7,8,5,4,3,6,2,1] => [8,7,5,4,3,6,1,2] => ?
=> ? = 1 - 1
[.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [7,8,4,5,3,6,2,1] => [8,7,5,3,4,6,1,2] => ?
=> ? = 1 - 1
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> [8,4,5,6,3,7,2,1] => [8,7,5,2,3,4,6,1] => ?
=> ? = 1 - 1
[.,[.,[[[.,.],[.,[.,.]]],[.,.]]]]
=> [8,6,5,3,4,7,2,1] => [8,7,4,5,3,2,6,1] => ?
=> ? = 1 - 1
[.,[.,[[[.,.],[[.,.],.]],[.,.]]]]
=> [8,5,6,3,4,7,2,1] => [8,7,4,5,2,3,6,1] => ?
=> ? = 1 - 1
[.,[.,[[[[.,.],.],[.,.]],[.,.]]]]
=> [8,6,3,4,5,7,2,1] => [8,7,3,4,5,2,6,1] => ?
=> ? = 1 - 1
[.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> [8,4,5,3,6,7,2,1] => [8,7,4,2,3,5,6,1] => ?
=> ? = 1 - 1
[.,[.,[[[[.,.],[.,.]],.],[.,.]]]]
=> [8,5,3,4,6,7,2,1] => [8,7,3,4,2,5,6,1] => ?
=> ? = 1 - 1
[.,[.,[[[[.,[.,.]],.],.],[.,.]]]]
=> [8,4,3,5,6,7,2,1] => [8,7,3,2,4,5,6,1] => ?
=> ? = 1 - 1
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [7,5,6,4,3,8,2,1] => [8,7,5,4,2,3,1,6] => ?
=> ? = 1 - 1
[.,[.,[[.,[.,[[.,[.,.]],.]]],.]]]
=> [6,5,7,4,3,8,2,1] => [8,7,5,4,2,1,3,6] => ?
=> ? = 1 - 1
[.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> [7,4,5,6,3,8,2,1] => [8,7,5,2,3,4,1,6] => ?
=> ? = 1 - 1
[.,[.,[[[.,.],[.,[[.,.],.]]],.]]]
=> [6,7,5,3,4,8,2,1] => [8,7,4,5,3,1,2,6] => ?
=> ? = 1 - 1
[.,[.,[[[.,.],[[.,[.,.]],.]],.]]]
=> [6,5,7,3,4,8,2,1] => [8,7,4,5,2,1,3,6] => ?
=> ? = 1 - 1
[.,[.,[[[.,[.,.]],[[.,.],.]],.]]]
=> [6,7,4,3,5,8,2,1] => [8,7,4,3,5,1,2,6] => ?
=> ? = 1 - 1
[.,[.,[[[.,[[.,.],.]],[.,.]],.]]]
=> [7,4,5,3,6,8,2,1] => [8,7,4,2,3,5,1,6] => ?
=> ? = 1 - 1
[.,[.,[[[[.,[.,.]],.],[.,.]],.]]]
=> [7,4,3,5,6,8,2,1] => [8,7,3,2,4,5,1,6] => ?
=> ? = 1 - 1
[.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> [5,6,4,3,7,8,2,1] => [8,7,4,3,1,2,5,6] => ?
=> ? = 1 - 1
[.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> [6,4,5,3,7,8,2,1] => [8,7,4,2,3,1,5,6] => ?
=> ? = 1 - 1
[.,[.,[[[[.,[[.,.],.]],.],.],.]]]
=> [4,5,3,6,7,8,2,1] => [8,7,3,1,2,4,5,6] => ?
=> ? = 1 - 1
[.,[[.,.],[.,[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,4,2,3,1] => [8,6,7,5,4,1,2,3] => ?
=> ? = 1 - 1
[.,[[.,.],[.,[[.,.],[[.,.],.]]]]]
=> [7,8,5,6,4,2,3,1] => [8,6,7,5,3,4,1,2] => ?
=> ? = 1 - 1
[.,[[.,.],[.,[[[.,[.,.]],.],.]]]]
=> [6,5,7,8,4,2,3,1] => [8,6,7,5,2,1,3,4] => ?
=> ? = 1 - 1
[.,[[.,.],[[.,.],[[.,[.,.]],.]]]]
=> [7,6,8,4,5,2,3,1] => [8,6,7,4,5,2,1,3] => ?
=> ? = 1 - 1
[.,[[.,.],[[.,[.,.]],[[.,.],.]]]]
=> [7,8,5,4,6,2,3,1] => [8,6,7,4,3,5,1,2] => ?
=> ? = 1 - 1
[.,[[.,.],[[.,[[.,.],.]],[.,.]]]]
=> [8,5,6,4,7,2,3,1] => [8,6,7,4,2,3,5,1] => ?
=> ? = 1 - 1
[.,[[.,.],[[.,[.,[[.,.],.]]],.]]]
=> [6,7,5,4,8,2,3,1] => [8,6,7,4,3,1,2,5] => ?
=> ? = 1 - 1
[.,[[.,.],[[.,[[.,.],[.,.]]],.]]]
=> [7,5,6,4,8,2,3,1] => [8,6,7,4,2,3,1,5] => ?
=> ? = 1 - 1
[.,[[.,.],[[.,[[[.,.],.],.]],.]]]
=> [5,6,7,4,8,2,3,1] => [8,6,7,4,1,2,3,5] => ?
=> ? = 1 - 1
[.,[[.,.],[[[.,.],[.,[.,.]]],.]]]
=> [7,6,4,5,8,2,3,1] => [8,6,7,3,4,2,1,5] => ?
=> ? = 1 - 1
[.,[[.,[.,.]],[.,[[.,[.,.]],.]]]]
=> [7,6,8,5,3,2,4,1] => [8,6,5,7,4,2,1,3] => ?
=> ? = 1 - 1
[.,[[.,[.,.]],[[.,.],[.,[.,.]]]]]
=> [8,7,5,6,3,2,4,1] => [8,6,5,7,3,4,2,1] => ?
=> ? = 1 - 1
[.,[[[.,.],.],[.,[.,[[.,.],.]]]]]
=> [7,8,6,5,2,3,4,1] => [8,5,6,7,4,3,1,2] => ?
=> ? = 1 - 1
[.,[[[.,.],.],[.,[[.,[.,.]],.]]]]
=> [7,6,8,5,2,3,4,1] => [8,5,6,7,4,2,1,3] => ?
=> ? = 1 - 1
[.,[[[.,.],.],[[.,[[.,.],.]],.]]]
=> [6,7,5,8,2,3,4,1] => [8,5,6,7,3,1,2,4] => ?
=> ? = 1 - 1
[.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> [7,8,6,4,3,2,5,1] => [8,6,5,4,7,3,1,2] => ?
=> ? = 1 - 1
[.,[[.,[.,[.,.]]],[[.,[.,.]],.]]]
=> [7,6,8,4,3,2,5,1] => [8,6,5,4,7,2,1,3] => ?
=> ? = 1 - 1
[.,[[.,[[.,.],.]],[[.,[.,.]],.]]]
=> [7,6,8,3,4,2,5,1] => [8,6,4,5,7,2,1,3] => ?
=> ? = 1 - 1
[.,[[[.,.],[.,.]],[[.,[.,.]],.]]]
=> [7,6,8,4,2,3,5,1] => [8,5,6,4,7,2,1,3] => ?
=> ? = 1 - 1
[.,[[[.,[.,.]],.],[[.,.],[.,.]]]]
=> [8,6,7,3,2,4,5,1] => [8,5,4,6,7,2,3,1] => ?
=> ? = 1 - 1
[.,[[[.,[.,.]],.],[[.,[.,.]],.]]]
=> [7,6,8,3,2,4,5,1] => [8,5,4,6,7,2,1,3] => ?
=> ? = 1 - 1
[.,[[[[.,.],.],.],[.,[[.,.],.]]]]
=> [7,8,6,2,3,4,5,1] => [8,4,5,6,7,3,1,2] => ?
=> ? = 1 - 1
[.,[[.,[.,[[.,.],.]]],[.,[.,.]]]]
=> [8,7,4,5,3,2,6,1] => [8,6,5,3,4,7,2,1] => ?
=> ? = 1 - 1
[.,[[.,[.,[[.,.],.]]],[[.,.],.]]]
=> [7,8,4,5,3,2,6,1] => [8,6,5,3,4,7,1,2] => ?
=> ? = 1 - 1
[.,[[.,[[.,.],[.,.]]],[.,[.,.]]]]
=> [8,7,5,3,4,2,6,1] => [8,6,4,5,3,7,2,1] => ?
=> ? = 1 - 1
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St000439
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1,0]
=> 2 = 1 + 1
[.,[.,.]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[[.,.],.]
=> [1,2] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 2 + 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> [3,6,7,5,8,4,2,1] => [1,2,4,8,5,7,6,3] => ?
=> ? = 1 + 1
[.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [3,5,7,6,8,4,2,1] => [1,2,4,8,6,7,5,3] => ?
=> ? = 1 + 1
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [3,6,5,7,8,4,2,1] => [1,2,4,8,7,5,6,3] => ?
=> ? = 1 + 1
[.,[.,[[[.,.],.],[[.,.],[.,.]]]]]
=> [3,4,6,8,7,5,2,1] => [1,2,5,7,8,6,4,3] => ?
=> ? = 1 + 1
[.,[.,[[[[.,.],.],.],[[.,.],.]]]]
=> [3,4,5,7,8,6,2,1] => [1,2,6,8,7,5,4,3] => ?
=> ? = 1 + 1
[.,[.,[[[.,.],[[.,.],.]],[.,.]]]]
=> [3,5,6,4,8,7,2,1] => [1,2,7,8,4,6,5,3] => ?
=> ? = 1 + 1
[.,[.,[[.,[.,[[.,[.,.]],.]]],.]]]
=> [6,5,7,4,3,8,2,1] => [1,2,8,3,4,7,5,6] => ?
=> ? = 1 + 1
[.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> [4,5,7,6,3,8,2,1] => [1,2,8,3,6,7,5,4] => ?
=> ? = 1 + 1
[.,[.,[[.,[[[.,.],[.,.]],.]],.]]]
=> [4,6,5,7,3,8,2,1] => [1,2,8,3,7,5,6,4] => ?
=> ? = 1 + 1
[.,[.,[[[.,.],[.,[[.,.],.]]],.]]]
=> [3,6,7,5,4,8,2,1] => [1,2,8,4,5,7,6,3] => ?
=> ? = 1 + 1
[.,[.,[[[.,.],[[.,[.,.]],.]],.]]]
=> [3,6,5,7,4,8,2,1] => [1,2,8,4,7,5,6,3] => ?
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],[[.,.],.]],.]]]
=> [4,3,6,7,5,8,2,1] => [1,2,8,5,7,6,3,4] => ?
=> ? = 1 + 1
[.,[.,[[[[.,.],.],[[.,.],.]],.]]]
=> [3,4,6,7,5,8,2,1] => [1,2,8,5,7,6,4,3] => ?
=> ? = 1 + 1
[.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> [5,4,3,7,6,8,2,1] => [1,2,8,6,7,3,4,5] => ?
=> ? = 1 + 1
[.,[.,[[[[.,.],[.,.]],[.,.]],.]]]
=> [3,5,4,7,6,8,2,1] => [1,2,8,6,7,4,5,3] => ?
=> ? = 1 + 1
[.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> [4,6,5,3,7,8,2,1] => [1,2,8,7,3,5,6,4] => ?
=> ? = 1 + 1
[.,[.,[[[.,[[.,[.,.]],.]],.],.]]]
=> [5,4,6,3,7,8,2,1] => [1,2,8,7,3,6,4,5] => ?
=> ? = 1 + 1
[.,[.,[[[[[.,.],[.,.]],.],.],.]]]
=> [3,5,4,6,7,8,2,1] => [1,2,8,7,6,4,5,3] => ?
=> ? = 1 + 1
[.,[[.,.],[.,[[.,.],[[.,.],.]]]]]
=> [2,5,7,8,6,4,3,1] => [1,3,4,6,8,7,5,2] => ?
=> ? = 1 + 1
[.,[[.,.],[.,[[.,[.,.]],[.,.]]]]]
=> [2,6,5,8,7,4,3,1] => [1,3,4,7,8,5,6,2] => ?
=> ? = 1 + 1
[.,[[.,.],[.,[[[.,.],.],[.,.]]]]]
=> [2,5,6,8,7,4,3,1] => [1,3,4,7,8,6,5,2] => ?
=> ? = 1 + 1
[.,[[.,.],[.,[[.,[[.,.],.]],.]]]]
=> [2,6,7,5,8,4,3,1] => [1,3,4,8,5,7,6,2] => ?
=> ? = 1 + 1
[.,[[.,.],[.,[[[.,[.,.]],.],.]]]]
=> [2,6,5,7,8,4,3,1] => [1,3,4,8,7,5,6,2] => ?
=> ? = 1 + 1
[.,[[.,.],[[.,.],[[.,[.,.]],.]]]]
=> [2,4,7,6,8,5,3,1] => [1,3,5,8,6,7,4,2] => ?
=> ? = 1 + 1
[.,[[.,.],[[.,[.,.]],[.,[.,.]]]]]
=> [2,5,4,8,7,6,3,1] => [1,3,6,7,8,4,5,2] => ?
=> ? = 1 + 1
[.,[[.,.],[[[.,[.,.]],.],[.,.]]]]
=> [2,5,4,6,8,7,3,1] => [1,3,7,8,6,4,5,2] => ?
=> ? = 1 + 1
[.,[[.,.],[[.,[.,[[.,.],.]]],.]]]
=> [2,6,7,5,4,8,3,1] => [1,3,8,4,5,7,6,2] => ?
=> ? = 1 + 1
[.,[[.,.],[[.,[[.,[.,.]],.]],.]]]
=> [2,6,5,7,4,8,3,1] => [1,3,8,4,7,5,6,2] => ?
=> ? = 1 + 1
[.,[[.,.],[[.,[[[.,.],.],.]],.]]]
=> [2,5,6,7,4,8,3,1] => [1,3,8,4,7,6,5,2] => ?
=> ? = 1 + 1
[.,[[.,.],[[[[.,.],.],[.,.]],.]]]
=> [2,4,5,7,6,8,3,1] => ? => ?
=> ? = 1 + 1
[.,[[.,.],[[[[.,[.,.]],.],.],.]]]
=> [2,5,4,6,7,8,3,1] => [1,3,8,7,6,4,5,2] => ?
=> ? = 1 + 1
[.,[[.,[.,.]],[.,[[.,.],[.,.]]]]]
=> [3,2,6,8,7,5,4,1] => [1,4,5,7,8,6,2,3] => ?
=> ? = 1 + 1
[.,[[.,[.,.]],[.,[[.,[.,.]],.]]]]
=> [3,2,7,6,8,5,4,1] => [1,4,5,8,6,7,2,3] => ?
=> ? = 1 + 1
[.,[[.,[.,.]],[[.,.],[.,[.,.]]]]]
=> [3,2,5,8,7,6,4,1] => [1,4,6,7,8,5,2,3] => ?
=> ? = 1 + 1
[.,[[.,[.,.]],[[.,.],[[.,.],.]]]]
=> [3,2,5,7,8,6,4,1] => [1,4,6,8,7,5,2,3] => ?
=> ? = 1 + 1
[.,[[.,[.,.]],[[[.,.],.],[.,.]]]]
=> [3,2,5,6,8,7,4,1] => [1,4,7,8,6,5,2,3] => ?
=> ? = 1 + 1
[.,[[.,[.,.]],[[[.,[.,.]],.],.]]]
=> [3,2,6,5,7,8,4,1] => [1,4,8,7,5,6,2,3] => ?
=> ? = 1 + 1
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [3,2,5,6,7,8,4,1] => [1,4,8,7,6,5,2,3] => ?
=> ? = 1 + 1
[.,[[[.,.],.],[.,[.,[[.,.],.]]]]]
=> [2,3,7,8,6,5,4,1] => [1,4,5,6,8,7,3,2] => ?
=> ? = 1 + 1
[.,[[[.,.],.],[[[.,[.,.]],.],.]]]
=> [2,3,6,5,7,8,4,1] => [1,4,8,7,5,6,3,2] => ?
=> ? = 1 + 1
[.,[[.,[.,[.,.]]],[[.,[.,.]],.]]]
=> [4,3,2,7,6,8,5,1] => [1,5,8,6,7,2,3,4] => ?
=> ? = 1 + 1
[.,[[.,[[.,.],.]],[.,[[.,.],.]]]]
=> [3,4,2,7,8,6,5,1] => [1,5,6,8,7,2,4,3] => ?
=> ? = 1 + 1
[.,[[.,[[.,.],.]],[[.,[.,.]],.]]]
=> [3,4,2,7,6,8,5,1] => [1,5,8,6,7,2,4,3] => ?
=> ? = 1 + 1
[.,[[[.,.],[.,.]],[.,[[.,.],.]]]]
=> [2,4,3,7,8,6,5,1] => [1,5,6,8,7,3,4,2] => ?
=> ? = 1 + 1
[.,[[[.,[.,.]],.],[[[.,.],.],.]]]
=> [3,2,4,6,7,8,5,1] => [1,5,8,7,6,4,2,3] => ?
=> ? = 1 + 1
[.,[[[[.,.],.],.],[.,[[.,.],.]]]]
=> [2,3,4,7,8,6,5,1] => [1,5,6,8,7,4,3,2] => ?
=> ? = 1 + 1
[.,[[[[.,.],.],.],[[.,.],[.,.]]]]
=> [2,3,4,6,8,7,5,1] => [1,5,7,8,6,4,3,2] => ?
=> ? = 1 + 1
[.,[[[[.,.],.],.],[[.,[.,.]],.]]]
=> [2,3,4,7,6,8,5,1] => [1,5,8,6,7,4,3,2] => ?
=> ? = 1 + 1
[.,[[.,[[.,.],[.,.]]],[[.,.],.]]]
=> [3,5,4,2,7,8,6,1] => [1,6,8,7,2,4,5,3] => ?
=> ? = 1 + 1
[.,[[.,[[[.,.],.],.]],[.,[.,.]]]]
=> [3,4,5,2,8,7,6,1] => [1,6,7,8,2,5,4,3] => ?
=> ? = 1 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000326
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => => ? = 1
[.,[.,.]]
=> [2,1] => [1,2] => 1 => 1
[[.,.],.]
=> [1,2] => [2,1] => 0 => 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 11 => 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 10 => 1
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => 01 => 2
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => 00 => 3
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 00 => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 111 => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => 110 => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 101 => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => 100 => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => 100 => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => 011 => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 010 => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => 001 => 3
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => 001 => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => 000 => 4
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => 000 => 4
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => 000 => 4
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => 000 => 4
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 000 => 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => 1110 => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => 1101 => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => 1100 => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 1100 => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => 1011 => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => 1010 => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => 1001 => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => 1001 => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => 1000 => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => 1000 => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => 1000 => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => 1000 => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => 1000 => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => 0111 => 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => 0110 => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => 0101 => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => 0100 => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => 0100 => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => 0011 => 3
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => 0010 => 3
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => 0011 => 3
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => 0010 => 3
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => 0001 => 4
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => 0001 => 4
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => 0001 => 4
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => 0001 => 4
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => 0001 => 4
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => 0000 => 5
[.,[.,[[.,.],[[[.,.],.],[.,.]]]]]
=> [8,5,6,7,3,4,2,1] => [1,2,4,3,7,6,5,8] => ? => ? = 1
[.,[.,[[.,.],[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,3,4,2,1] => [1,2,4,3,8,6,5,7] => ? => ? = 1
[.,[.,[[[.,.],.],[[.,.],[.,.]]]]]
=> [8,6,7,3,4,5,2,1] => [1,2,5,4,3,7,6,8] => ? => ? = 1
[.,[.,[[.,[[.,.],.]],[.,[.,.]]]]]
=> [8,7,4,5,3,6,2,1] => [1,2,6,3,5,4,7,8] => ? => ? = 1
[.,[.,[[.,[[.,.],.]],[[.,.],.]]]]
=> [7,8,4,5,3,6,2,1] => [1,2,6,3,5,4,8,7] => ? => ? = 1
[.,[.,[[[.,.],[.,.]],[[.,.],.]]]]
=> [7,8,5,3,4,6,2,1] => [1,2,6,4,3,5,8,7] => ? => ? = 1
[.,[.,[[[.,[.,.]],.],[[.,.],.]]]]
=> [7,8,4,3,5,6,2,1] => [1,2,6,5,3,4,8,7] => ? => ? = 1
[.,[.,[[[[.,.],.],.],[[.,.],.]]]]
=> [7,8,3,4,5,6,2,1] => [1,2,6,5,4,3,8,7] => ? => ? = 1
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> [8,4,5,6,3,7,2,1] => [1,2,7,3,6,5,4,8] => ? => ? = 1
[.,[.,[[[.,.],[[.,.],.]],[.,.]]]]
=> [8,5,6,3,4,7,2,1] => [1,2,7,4,3,6,5,8] => ? => ? = 1
[.,[.,[[[[.,.],.],[.,.]],[.,.]]]]
=> [8,6,3,4,5,7,2,1] => [1,2,7,5,4,3,6,8] => ? => ? = 1
[.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> [8,4,5,3,6,7,2,1] => [1,2,7,6,3,5,4,8] => ? => ? = 1
[.,[.,[[[[.,[.,.]],.],.],[.,.]]]]
=> [8,4,3,5,6,7,2,1] => [1,2,7,6,5,3,4,8] => ? => ? = 1
[.,[.,[[.,[.,[[.,[.,.]],.]]],.]]]
=> [6,5,7,4,3,8,2,1] => [1,2,8,3,4,7,5,6] => ? => ? = 1
[.,[.,[[[.,[.,.]],[[.,.],.]],.]]]
=> [6,7,4,3,5,8,2,1] => [1,2,8,5,3,4,7,6] => ? => ? = 1
[.,[.,[[[[.,.],.],[[.,.],.]],.]]]
=> [6,7,3,4,5,8,2,1] => [1,2,8,5,4,3,7,6] => ? => ? = 1
[.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> [7,5,4,3,6,8,2,1] => [1,2,8,6,3,4,5,7] => ? => ? = 1
[.,[.,[[[.,[[.,.],.]],[.,.]],.]]]
=> [7,4,5,3,6,8,2,1] => [1,2,8,6,3,5,4,7] => ? => ? = 1
[.,[.,[[[[.,[.,.]],.],[.,.]],.]]]
=> [7,4,3,5,6,8,2,1] => [1,2,8,6,5,3,4,7] => ? => ? = 1
[.,[.,[[[.,[[.,.],[.,.]]],.],.]]]
=> [6,4,5,3,7,8,2,1] => [1,2,8,7,3,5,4,6] => ? => ? = 1
[.,[.,[[[.,[[.,[.,.]],.]],.],.]]]
=> [5,4,6,3,7,8,2,1] => [1,2,8,7,3,6,4,5] => ? => ? = 1
[.,[.,[[[[.,.],[.,[.,.]]],.],.]]]
=> [6,5,3,4,7,8,2,1] => [1,2,8,7,4,3,5,6] => ? => ? = 1
[.,[.,[[[[.,[.,.]],[.,.]],.],.]]]
=> [6,4,3,5,7,8,2,1] => [1,2,8,7,5,3,4,6] => ? => ? = 1
[.,[[.,.],[.,[[.,[.,.]],[.,.]]]]]
=> [8,6,5,7,4,2,3,1] => [1,3,2,4,7,5,6,8] => ? => ? = 1
[.,[[.,.],[.,[[[.,[.,.]],.],.]]]]
=> [6,5,7,8,4,2,3,1] => [1,3,2,4,8,7,5,6] => ? => ? = 1
[.,[[.,.],[[.,.],[.,[[.,.],.]]]]]
=> [7,8,6,4,5,2,3,1] => [1,3,2,5,4,6,8,7] => ? => ? = 1
[.,[[.,.],[[.,[.,.]],[[.,.],.]]]]
=> [7,8,5,4,6,2,3,1] => [1,3,2,6,4,5,8,7] => ? => ? = 1
[.,[[.,.],[[[.,[.,.]],.],[.,.]]]]
=> [8,5,4,6,7,2,3,1] => [1,3,2,7,6,4,5,8] => ? => ? = 1
[.,[[.,.],[[[[.,.],.],.],[.,.]]]]
=> [8,4,5,6,7,2,3,1] => [1,3,2,7,6,5,4,8] => ? => ? = 1
[.,[[.,.],[[.,[[[.,.],.],.]],.]]]
=> [5,6,7,4,8,2,3,1] => [1,3,2,8,4,7,6,5] => ? => ? = 1
[.,[[.,.],[[[.,.],[.,[.,.]]],.]]]
=> [7,6,4,5,8,2,3,1] => [1,3,2,8,5,4,6,7] => ? => ? = 1
[.,[[.,.],[[[.,[.,.]],[.,.]],.]]]
=> [7,5,4,6,8,2,3,1] => [1,3,2,8,6,4,5,7] => ? => ? = 1
[.,[[.,[.,.]],[[[.,.],.],[.,.]]]]
=> [8,5,6,7,3,2,4,1] => [1,4,2,3,7,6,5,8] => ? => ? = 1
[.,[[[.,.],.],[.,[.,[[.,.],.]]]]]
=> [7,8,6,5,2,3,4,1] => ? => ? => ? = 1
[.,[[[.,.],.],[[.,.],[.,[.,.]]]]]
=> [8,7,5,6,2,3,4,1] => [1,4,3,2,6,5,7,8] => ? => ? = 1
[.,[[[.,.],.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,2,3,4,1] => [1,4,3,2,6,5,8,7] => ? => ? = 1
[.,[[[.,.],.],[[.,[.,.]],[.,.]]]]
=> [8,6,5,7,2,3,4,1] => [1,4,3,2,7,5,6,8] => ? => ? = 1
[.,[[[.,.],.],[[.,[[.,.],.]],.]]]
=> [6,7,5,8,2,3,4,1] => [1,4,3,2,8,5,7,6] => ? => ? = 1
[.,[[[.,.],.],[[[.,.],[.,.]],.]]]
=> [7,5,6,8,2,3,4,1] => [1,4,3,2,8,6,5,7] => ? => ? = 1
[.,[[[.,.],.],[[[.,[.,.]],.],.]]]
=> [6,5,7,8,2,3,4,1] => [1,4,3,2,8,7,5,6] => ? => ? = 1
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [8,7,6,4,2,3,5,1] => [1,5,3,2,4,6,7,8] => ? => ? = 1
[.,[[[.,[.,.]],.],[[.,.],[.,.]]]]
=> [8,6,7,3,2,4,5,1] => [1,5,4,2,3,7,6,8] => ? => ? = 1
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [8,7,6,2,3,4,5,1] => [1,5,4,3,2,6,7,8] => ? => ? = 1
[.,[[[[.,.],.],.],[.,[[.,.],.]]]]
=> [7,8,6,2,3,4,5,1] => [1,5,4,3,2,6,8,7] => ? => ? = 1
[.,[[[[.,.],.],.],[[.,.],[.,.]]]]
=> [8,6,7,2,3,4,5,1] => [1,5,4,3,2,7,6,8] => ? => ? = 1
[.,[[[[.,.],.],.],[[.,[.,.]],.]]]
=> [7,6,8,2,3,4,5,1] => [1,5,4,3,2,8,6,7] => ? => ? = 1
[.,[[.,[.,[[.,.],.]]],[.,[.,.]]]]
=> [8,7,4,5,3,2,6,1] => [1,6,2,3,5,4,7,8] => ? => ? = 1
[.,[[.,[.,[[.,.],.]]],[[.,.],.]]]
=> [7,8,4,5,3,2,6,1] => [1,6,2,3,5,4,8,7] => ? => ? = 1
[.,[[.,[[[.,.],.],.]],[.,[.,.]]]]
=> [8,7,3,4,5,2,6,1] => [1,6,2,5,4,3,7,8] => ? => ? = 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000383
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 90%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 90%
Values
[.,.]
=> [1,0]
=> [1] => [1] => 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,1] => [1,1] => 1
[[.,.],.]
=> [1,1,0,0]
=> [2] => [2] => 2
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1] => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => [1,2] => 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [3] => [3] => 3
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => [3] => 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,2,1] => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1] => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [3,1] => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => [1,3] => 3
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [4] => [4] => 4
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [4] => [4] => 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [4] => [4] => 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [4] => [4] => 4
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,1,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,2,1,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,3,1] => 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,3,1] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1] => 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1] => 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1] => 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1] => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1] => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,2,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1,2] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2] => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => 3
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [2,3] => 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [2,3] => 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,4] => 4
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,4] => 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,4] => 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,4] => 4
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,4] => 4
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,4] => [1,1,1,4,1] => ? = 1
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,4] => [1,1,1,4,1] => ? = 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,4] => [1,1,1,4,1] => ? = 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,4] => [1,1,1,4,1] => ? = 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,4] => [1,1,1,4,1] => ? = 1
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,4,1] => [1,1,4,1,1] => ? = 1
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,4,1] => [1,1,4,1,1] => ? = 1
[.,[.,[.,[[[.,.],[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,4,1] => [1,1,4,1,1] => ? = 1
[.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,4,1] => [1,1,4,1,1] => ? = 1
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [1,1,4,1,1] => ? = 1
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,4,1,1] => [1,4,1,1,1] => ? = 1
[.,[.,[[.,[[.,.],.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [1,4,1,1,1] => ? = 1
[.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,4,1,1] => [1,4,1,1,1] => ? = 1
[.,[.,[[[.,[.,.]],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [1,4,1,1,1] => ? = 1
[.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [1,4,1,1,1] => ? = 1
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,1,1,1,1,1] => [2,1,1,1,1,1,1] => ? = 1
[.,[[.,.],[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,1] => [2,4,1,1] => ? = 1
[.,[[.,.],[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,1] => [2,4,1,1] => ? = 1
[.,[[.,.],[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,4,1] => [2,4,1,1] => ? = 1
[.,[[.,.],[[[.,[.,.]],.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,4,1] => [2,4,1,1] => ? = 1
[.,[[.,.],[[[[.,.],.],.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,4,1] => [2,4,1,1] => ? = 1
[.,[[.,.],[[.,[.,[.,[.,.]]]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,5] => [2,5,1] => ? = 1
[.,[[.,.],[[.,[.,[[.,.],.]]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,5] => [2,5,1] => ? = 1
[.,[[.,.],[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,5] => [2,5,1] => ? = 1
[.,[[.,.],[[.,[[.,[.,.]],.]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,5] => [2,5,1] => ? = 1
[.,[[.,.],[[.,[[[.,.],.],.]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,5] => [2,5,1] => ? = 1
[.,[[.,.],[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5] => [2,5,1] => ? = 1
[.,[[.,.],[[[.,.],[[.,.],.]],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5] => [2,5,1] => ? = 1
[.,[[.,.],[[[.,[.,.]],[.,.]],.]]]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5] => [2,5,1] => ? = 1
[.,[[.,.],[[[[.,.],.],[.,.]],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,5] => [2,5,1] => ? = 1
[.,[[.,.],[[[.,[.,[.,.]]],.],.]]]
=> [1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5] => [2,5,1] => ? = 1
[.,[[.,.],[[[.,[[.,.],.]],.],.]]]
=> [1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5] => [2,5,1] => ? = 1
[.,[[.,.],[[[[.,.],[.,.]],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,5] => [2,5,1] => ? = 1
[.,[[.,.],[[[[.,[.,.]],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,5] => [2,5,1] => ? = 1
[.,[[.,.],[[[[[.,.],.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,5] => [2,5,1] => ? = 1
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1] => [3,1,1,1,1,1] => ? = 1
[.,[[.,[.,.]],[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,1,1,2] => [3,1,1,2,1] => ? = 1
[.,[[.,[.,.]],[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,1,2,1] => [3,1,2,1,1] => ? = 1
[.,[[.,[.,.]],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,1,1] => [3,2,1,1,1] => ? = 1
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,3,1,1,1,1] => [3,1,1,1,1,1] => ? = 1
[.,[[[.,.],.],[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,3,1,1,2] => [3,1,1,2,1] => ? = 1
[.,[[[.,.],.],[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,3,1,2,1] => [3,1,2,1,1] => ? = 1
[.,[[[.,.],.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,1,1] => [3,2,1,1,1] => ? = 1
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,4,1,1,1] => [4,1,1,1,1] => ? = 1
[.,[[.,[.,[.,.]]],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,4,2,1] => [4,2,1,1] => ? = 1
[.,[[.,[[.,.],.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,4,1,1,1] => [4,1,1,1,1] => ? = 1
[.,[[.,[[.,.],.]],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,4,2,1] => [4,2,1,1] => ? = 1
[.,[[[.,.],[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,4,1,1,1] => [4,1,1,1,1] => ? = 1
[.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,2,1] => [4,2,1,1] => ? = 1
[.,[[[.,[.,.]],.],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,4,1,1,1] => [4,1,1,1,1] => ? = 1
Description
The last part of an integer composition.
Matching statistic: St000476
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 80%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 80%
Values
[.,.]
=> [1] => [1] => [1,0]
=> ? = 1 - 1
[.,[.,.]]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 3 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [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
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [6,8,7,5,4,3,2,1] => [8,7,6,5,4,1,3,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [8,7,6,5,4,1,2,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [5,8,7,6,4,3,2,1] => [8,7,6,5,1,4,3,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [5,7,8,6,4,3,2,1] => [8,7,6,5,1,4,2,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [6,5,8,7,4,3,2,1] => [8,7,6,5,2,1,4,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [5,6,8,7,4,3,2,1] => [8,7,6,5,1,2,4,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [7,6,5,8,4,3,2,1] => [8,7,6,5,3,2,1,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> [6,7,5,8,4,3,2,1] => [8,7,6,5,3,1,2,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [5,7,6,8,4,3,2,1] => [8,7,6,5,1,3,2,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [8,7,6,5,2,1,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [8,7,6,5,1,2,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [4,8,7,6,5,3,2,1] => [8,7,6,1,5,4,3,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [4,7,8,6,5,3,2,1] => [8,7,6,1,5,4,2,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [4,6,8,7,5,3,2,1] => [8,7,6,1,5,2,4,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> [4,7,6,8,5,3,2,1] => [8,7,6,1,5,3,2,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [4,6,7,8,5,3,2,1] => [8,7,6,1,5,2,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [5,4,8,7,6,3,2,1] => [8,7,6,2,1,5,4,3] => ?
=> ? = 1 - 1
[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [5,4,7,8,6,3,2,1] => [8,7,6,2,1,5,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> [4,5,8,7,6,3,2,1] => [8,7,6,1,2,5,4,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [4,5,7,8,6,3,2,1] => [8,7,6,1,2,5,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [6,5,4,8,7,3,2,1] => [8,7,6,3,2,1,5,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [5,6,4,8,7,3,2,1] => [8,7,6,3,1,2,5,4] => ?
=> ? = 1 - 1
[.,[.,[.,[[[.,.],[.,.]],[.,.]]]]]
=> [4,6,5,8,7,3,2,1] => [8,7,6,1,3,2,5,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> [5,4,6,8,7,3,2,1] => [8,7,6,2,1,3,5,4] => ?
=> ? = 1 - 1
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> [4,5,6,8,7,3,2,1] => [8,7,6,1,2,3,5,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> [6,7,5,4,8,3,2,1] => [8,7,6,4,3,1,2,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> [5,7,6,4,8,3,2,1] => [8,7,6,4,1,3,2,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [8,7,6,4,2,1,3,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> [5,6,7,4,8,3,2,1] => [8,7,6,4,1,2,3,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [4,7,6,5,8,3,2,1] => [8,7,6,1,4,3,2,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [4,6,7,5,8,3,2,1] => [8,7,6,1,4,2,3,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [5,4,7,6,8,3,2,1] => [8,7,6,2,1,4,3,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [4,5,7,6,8,3,2,1] => [8,7,6,1,2,4,3,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [6,5,4,7,8,3,2,1] => [8,7,6,3,2,1,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
=> [5,6,4,7,8,3,2,1] => [8,7,6,3,1,2,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[[[.,.],[.,.]],.],.]]]]
=> [4,6,5,7,8,3,2,1] => [8,7,6,1,3,2,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
=> [5,4,6,7,8,3,2,1] => [8,7,6,2,1,3,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [8,7,6,1,2,3,4,5] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [3,8,7,6,5,4,2,1] => [8,7,1,6,5,4,3,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [3,7,8,6,5,4,2,1] => [8,7,1,6,5,4,2,3] => ?
=> ? = 1 - 1
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [3,6,8,7,5,4,2,1] => [8,7,1,6,5,2,4,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [3,7,6,8,5,4,2,1] => [8,7,1,6,5,3,2,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [3,6,7,8,5,4,2,1] => [8,7,1,6,5,2,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [3,5,8,7,6,4,2,1] => [8,7,1,6,2,5,4,3] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
[.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> [3,5,7,8,6,4,2,1] => [8,7,1,6,2,5,3,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 - 1
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St000505
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => {{1}}
=> 1
[.,[.,.]]
=> [2,1] => [1,2] => {{1},{2}}
=> 1
[[.,.],.]
=> [1,2] => [2,1] => {{1,2}}
=> 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => {{1},{2},{3}}
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => {{1},{2,3}}
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => {{1,2},{3}}
=> 2
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => {{1,2,3}}
=> 3
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => {{1,3},{2}}
=> 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => {{1,2,3},{4}}
=> 3
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => {{1,3},{2},{4}}
=> 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => {{1,2,3,4}}
=> 4
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => {{1,2,4},{3}}
=> 4
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => {{1,3,4},{2}}
=> 4
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => {{1,2,3,4}}
=> 4
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => {{1,4},{2,3}}
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => {{1},{2,3,5},{4}}
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => {{1},{2,4,5},{3}}
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => {{1},{2,3,4,5}}
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 3
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 3
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 3
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 4
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => {{1,2,4},{3},{5}}
=> 4
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => {{1,3,4},{2},{5}}
=> 4
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => {{1,2,3,4},{5}}
=> 4
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 4
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [8,6,7,5,4,3,2,1] => [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [1,2,3,4,5,8,7,6] => {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? = 1
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [8,7,5,6,4,3,2,1] => [1,2,3,4,6,5,7,8] => {{1},{2},{3},{4},{5,6},{7},{8}}
=> ? = 1
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => {{1},{2},{3},{4},{5,6},{7,8}}
=> ? = 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [8,6,5,7,4,3,2,1] => [1,2,3,4,7,5,6,8] => {{1},{2},{3},{4},{5,6,7},{8}}
=> ? = 1
[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [8,5,6,7,4,3,2,1] => [1,2,3,4,7,6,5,8] => {{1},{2},{3},{4},{5,7},{6},{8}}
=> ? = 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [7,6,5,8,4,3,2,1] => [1,2,3,4,8,5,6,7] => {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 1
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> [6,7,5,8,4,3,2,1] => [1,2,3,4,8,5,7,6] => {{1},{2},{3},{4},{5,6,8},{7}}
=> ? = 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [7,5,6,8,4,3,2,1] => [1,2,3,4,8,6,5,7] => {{1},{2},{3},{4},{5,7,8},{6}}
=> ? = 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [1,2,3,4,8,7,5,6] => {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => {{1},{2},{3},{4},{5,8},{6,7}}
=> ? = 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [8,7,6,4,5,3,2,1] => [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => {{1},{2},{3},{4,5},{6},{7,8}}
=> ? = 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 1
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> [7,6,8,4,5,3,2,1] => [1,2,3,5,4,8,6,7] => {{1},{2},{3},{4,5},{6,7,8}}
=> ? = 1
[.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [6,7,8,4,5,3,2,1] => [1,2,3,5,4,8,7,6] => {{1},{2},{3},{4,5},{6,8},{7}}
=> ? = 1
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [8,7,5,4,6,3,2,1] => [1,2,3,6,4,5,7,8] => {{1},{2},{3},{4,5,6},{7},{8}}
=> ? = 1
[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [7,8,5,4,6,3,2,1] => [1,2,3,6,4,5,8,7] => {{1},{2},{3},{4,5,6},{7,8}}
=> ? = 1
[.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> [8,7,4,5,6,3,2,1] => [1,2,3,6,5,4,7,8] => {{1},{2},{3},{4,6},{5},{7},{8}}
=> ? = 1
[.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [7,8,4,5,6,3,2,1] => [1,2,3,6,5,4,8,7] => {{1},{2},{3},{4,6},{5},{7,8}}
=> ? = 1
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [8,6,5,4,7,3,2,1] => [1,2,3,7,4,5,6,8] => {{1},{2},{3},{4,5,6,7},{8}}
=> ? = 1
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [8,5,6,4,7,3,2,1] => [1,2,3,7,4,6,5,8] => {{1},{2},{3},{4,5,7},{6},{8}}
=> ? = 1
[.,[.,[.,[[[.,.],[.,.]],[.,.]]]]]
=> [8,6,4,5,7,3,2,1] => [1,2,3,7,5,4,6,8] => {{1},{2},{3},{4,6,7},{5},{8}}
=> ? = 1
[.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> [8,5,4,6,7,3,2,1] => [1,2,3,7,6,4,5,8] => {{1},{2},{3},{4,5,6,7},{8}}
=> ? = 1
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> [8,4,5,6,7,3,2,1] => [1,2,3,7,6,5,4,8] => {{1},{2},{3},{4,7},{5,6},{8}}
=> ? = 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 1
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> [6,7,5,4,8,3,2,1] => [1,2,3,8,4,5,7,6] => {{1},{2},{3},{4,5,6,8},{7}}
=> ? = 1
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> [7,5,6,4,8,3,2,1] => [1,2,3,8,4,6,5,7] => {{1},{2},{3},{4,5,7,8},{6}}
=> ? = 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 1
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> [5,6,7,4,8,3,2,1] => [1,2,3,8,4,7,6,5] => {{1},{2},{3},{4,5,8},{6,7}}
=> ? = 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [7,6,4,5,8,3,2,1] => [1,2,3,8,5,4,6,7] => {{1},{2},{3},{4,6,7,8},{5}}
=> ? = 1
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [6,7,4,5,8,3,2,1] => [1,2,3,8,5,4,7,6] => {{1},{2},{3},{4,6,8},{5},{7}}
=> ? = 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [7,5,4,6,8,3,2,1] => [1,2,3,8,6,4,5,7] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 1
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [7,4,5,6,8,3,2,1] => [1,2,3,8,6,5,4,7] => {{1},{2},{3},{4,7,8},{5,6}}
=> ? = 1
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [6,5,4,7,8,3,2,1] => [1,2,3,8,7,4,5,6] => {{1},{2},{3},{4,6,8},{5,7}}
=> ? = 1
[.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
=> [5,6,4,7,8,3,2,1] => [1,2,3,8,7,4,6,5] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 1
[.,[.,[.,[[[[.,.],[.,.]],.],.]]]]
=> [6,4,5,7,8,3,2,1] => [1,2,3,8,7,5,4,6] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 1
[.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
=> [5,4,6,7,8,3,2,1] => [1,2,3,8,7,6,4,5] => {{1},{2},{3},{4,5,7,8},{6}}
=> ? = 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [1,2,3,8,7,6,5,4] => {{1},{2},{3},{4,8},{5,7},{6}}
=> ? = 1
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [8,7,6,5,3,4,2,1] => [1,2,4,3,5,6,7,8] => {{1},{2},{3,4},{5},{6},{7},{8}}
=> ? = 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 1
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => {{1},{2},{3,4},{5},{6,7},{8}}
=> ? = 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,3,4,2,1] => [1,2,4,3,5,8,6,7] => {{1},{2},{3,4},{5},{6,7,8}}
=> ? = 1
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,3,4,2,1] => [1,2,4,3,5,8,7,6] => {{1},{2},{3,4},{5},{6,8},{7}}
=> ? = 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 1
[.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> [7,8,5,6,3,4,2,1] => [1,2,4,3,6,5,8,7] => {{1},{2},{3,4},{5,6},{7,8}}
=> ? = 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [8,6,5,7,3,4,2,1] => [1,2,4,3,7,5,6,8] => {{1},{2},{3,4},{5,6,7},{8}}
=> ? = 1
Description
The biggest entry in the block containing the 1.
Matching statistic: St000054
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 1
[.,[.,.]]
=> [2,1] => [1,2] => [1,2] => 1
[[.,.],.]
=> [1,2] => [2,1] => [2,1] => 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,3,2] => 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,3,2] => 1
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => [2,1,3] => 2
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [3,1,2] => 3
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [3,2,1] => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,4,3,2] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,4,3,2] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => [1,4,3,2] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,4,3,2] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,4,3] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => [3,1,4,2] => 3
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [4,1,3,2] => 4
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 4
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 4
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 4
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,5,4,3,2] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,5,4,3,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => [1,5,4,3,2] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,5,4,3,2] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,5,4,3,2] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => [1,5,4,3,2] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => [1,5,4,3,2] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => [1,5,4,3,2] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => [1,5,4,3,2] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,5,4,3,2] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,5,4,3,2] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => [1,5,4,3,2] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,3,2] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [2,1,5,4,3] => 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => [2,1,5,4,3] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [2,1,5,4,3] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,5,4,3] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [3,1,5,4,2] => 3
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => [3,1,5,4,2] => 3
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,5,4] => 3
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => 3
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [4,1,5,3,2] => 4
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => [4,1,5,3,2] => 4
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,5,3] => 4
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,5,2] => 4
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => 4
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => [3,1,2,4,5,6,7] => [3,1,7,6,5,4,2] => ? = 3
[[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [6,7,5,4,2,1,3] => [3,1,2,4,5,7,6] => [3,1,7,6,5,4,2] => ? = 3
[[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [7,5,6,4,2,1,3] => [3,1,2,4,6,5,7] => [3,1,7,6,5,4,2] => ? = 3
[[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [6,5,7,4,2,1,3] => [3,1,2,4,7,5,6] => [3,1,7,6,5,4,2] => ? = 3
[[.,[.,.]],[.,[[[.,.],.],.]]]
=> [5,6,7,4,2,1,3] => [3,1,2,4,7,6,5] => [3,1,7,6,5,4,2] => ? = 3
[[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,2,1,3] => [3,1,2,5,4,6,7] => [3,1,7,6,5,4,2] => ? = 3
[[.,[.,.]],[[.,.],[[.,.],.]]]
=> [6,7,4,5,2,1,3] => [3,1,2,5,4,7,6] => [3,1,7,6,5,4,2] => ? = 3
[[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [7,5,4,6,2,1,3] => [3,1,2,6,4,5,7] => [3,1,7,6,5,4,2] => ? = 3
[[.,[.,.]],[[[.,.],.],[.,.]]]
=> [7,4,5,6,2,1,3] => [3,1,2,6,5,4,7] => [3,1,7,6,5,4,2] => ? = 3
[[.,[.,.]],[[.,[.,[.,.]]],.]]
=> [6,5,4,7,2,1,3] => [3,1,2,7,4,5,6] => [3,1,7,6,5,4,2] => ? = 3
[[.,[.,.]],[[.,[[.,.],.]],.]]
=> [5,6,4,7,2,1,3] => [3,1,2,7,4,6,5] => [3,1,7,6,5,4,2] => ? = 3
[[.,[.,.]],[[[.,.],[.,.]],.]]
=> [6,4,5,7,2,1,3] => [3,1,2,7,5,4,6] => [3,1,7,6,5,4,2] => ? = 3
[[.,[.,.]],[[[.,[.,.]],.],.]]
=> [5,4,6,7,2,1,3] => [3,1,2,7,6,4,5] => [3,1,7,6,5,4,2] => ? = 3
[[.,[.,.]],[[[[.,.],.],.],.]]
=> [4,5,6,7,2,1,3] => [3,1,2,7,6,5,4] => [3,1,7,6,5,4,2] => ? = 3
[[[.,.],.],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,1,2,3] => [3,2,1,4,5,6,7] => [3,2,1,7,6,5,4] => ? = 3
[[[.,.],.],[.,[.,[[.,.],.]]]]
=> [6,7,5,4,1,2,3] => [3,2,1,4,5,7,6] => [3,2,1,7,6,5,4] => ? = 3
[[[.,.],.],[.,[[.,.],[.,.]]]]
=> [7,5,6,4,1,2,3] => [3,2,1,4,6,5,7] => [3,2,1,7,6,5,4] => ? = 3
[[[.,.],.],[.,[[.,[.,.]],.]]]
=> [6,5,7,4,1,2,3] => [3,2,1,4,7,5,6] => [3,2,1,7,6,5,4] => ? = 3
[[[.,.],.],[.,[[[.,.],.],.]]]
=> [5,6,7,4,1,2,3] => [3,2,1,4,7,6,5] => [3,2,1,7,6,5,4] => ? = 3
[[[.,.],.],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,1,2,3] => [3,2,1,5,4,6,7] => [3,2,1,7,6,5,4] => ? = 3
[[[.,.],.],[[.,.],[[.,.],.]]]
=> [6,7,4,5,1,2,3] => [3,2,1,5,4,7,6] => [3,2,1,7,6,5,4] => ? = 3
[[[.,.],.],[[.,[.,.]],[.,.]]]
=> [7,5,4,6,1,2,3] => [3,2,1,6,4,5,7] => [3,2,1,7,6,5,4] => ? = 3
[[[.,.],.],[[[.,.],.],[.,.]]]
=> [7,4,5,6,1,2,3] => [3,2,1,6,5,4,7] => [3,2,1,7,6,5,4] => ? = 3
[[[.,.],.],[[.,[.,[.,.]]],.]]
=> [6,5,4,7,1,2,3] => [3,2,1,7,4,5,6] => [3,2,1,7,6,5,4] => ? = 3
[[[.,.],.],[[.,[[.,.],.]],.]]
=> [5,6,4,7,1,2,3] => [3,2,1,7,4,6,5] => [3,2,1,7,6,5,4] => ? = 3
[[[.,.],.],[[[.,.],[.,.]],.]]
=> [6,4,5,7,1,2,3] => [3,2,1,7,5,4,6] => [3,2,1,7,6,5,4] => ? = 3
[[[.,.],.],[[[.,[.,.]],.],.]]
=> [5,4,6,7,1,2,3] => [3,2,1,7,6,4,5] => [3,2,1,7,6,5,4] => ? = 3
[[[.,.],.],[[[[.,.],.],.],.]]
=> [4,5,6,7,1,2,3] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 3
[[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => [4,1,2,3,5,6,7] => [4,1,7,6,5,3,2] => ? = 4
[[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [6,7,5,3,2,1,4] => [4,1,2,3,5,7,6] => [4,1,7,6,5,3,2] => ? = 4
[[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [7,5,6,3,2,1,4] => [4,1,2,3,6,5,7] => [4,1,7,6,5,3,2] => ? = 4
[[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [6,5,7,3,2,1,4] => [4,1,2,3,7,5,6] => [4,1,7,6,5,3,2] => ? = 4
[[.,[.,[.,.]]],[[[.,.],.],.]]
=> [5,6,7,3,2,1,4] => [4,1,2,3,7,6,5] => [4,1,7,6,5,3,2] => ? = 4
[[.,[[.,.],.]],[.,[.,[.,.]]]]
=> [7,6,5,2,3,1,4] => [4,1,3,2,5,6,7] => [4,1,7,6,5,3,2] => ? = 4
[[.,[[.,.],.]],[.,[[.,.],.]]]
=> [6,7,5,2,3,1,4] => [4,1,3,2,5,7,6] => [4,1,7,6,5,3,2] => ? = 4
[[.,[[.,.],.]],[[.,.],[.,.]]]
=> [7,5,6,2,3,1,4] => [4,1,3,2,6,5,7] => [4,1,7,6,5,3,2] => ? = 4
[[.,[[.,.],.]],[[.,[.,.]],.]]
=> [6,5,7,2,3,1,4] => [4,1,3,2,7,5,6] => [4,1,7,6,5,3,2] => ? = 4
[[.,[[.,.],.]],[[[.,.],.],.]]
=> [5,6,7,2,3,1,4] => [4,1,3,2,7,6,5] => [4,1,7,6,5,3,2] => ? = 4
[[[.,.],[.,.]],[.,[.,[.,.]]]]
=> [7,6,5,3,1,2,4] => [4,2,1,3,5,6,7] => [4,2,1,7,6,5,3] => ? = 4
[[[.,.],[.,.]],[.,[[.,.],.]]]
=> [6,7,5,3,1,2,4] => [4,2,1,3,5,7,6] => [4,2,1,7,6,5,3] => ? = 4
[[[.,.],[.,.]],[[.,.],[.,.]]]
=> [7,5,6,3,1,2,4] => [4,2,1,3,6,5,7] => [4,2,1,7,6,5,3] => ? = 4
[[[.,.],[.,.]],[[.,[.,.]],.]]
=> [6,5,7,3,1,2,4] => [4,2,1,3,7,5,6] => [4,2,1,7,6,5,3] => ? = 4
[[[.,.],[.,.]],[[[.,.],.],.]]
=> [5,6,7,3,1,2,4] => [4,2,1,3,7,6,5] => [4,2,1,7,6,5,3] => ? = 4
[[[.,[.,.]],.],[.,[.,[.,.]]]]
=> [7,6,5,2,1,3,4] => [4,3,1,2,5,6,7] => [4,3,1,7,6,5,2] => ? = 4
[[[.,[.,.]],.],[.,[[.,.],.]]]
=> [6,7,5,2,1,3,4] => [4,3,1,2,5,7,6] => [4,3,1,7,6,5,2] => ? = 4
[[[.,[.,.]],.],[[.,.],[.,.]]]
=> [7,5,6,2,1,3,4] => [4,3,1,2,6,5,7] => [4,3,1,7,6,5,2] => ? = 4
[[[.,[.,.]],.],[[.,[.,.]],.]]
=> [6,5,7,2,1,3,4] => [4,3,1,2,7,5,6] => [4,3,1,7,6,5,2] => ? = 4
[[[.,[.,.]],.],[[[.,.],.],.]]
=> [5,6,7,2,1,3,4] => [4,3,1,2,7,6,5] => [4,3,1,7,6,5,2] => ? = 4
[[[[.,.],.],.],[.,[.,[.,.]]]]
=> [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => [4,3,2,1,7,6,5] => ? = 4
[[[[.,.],.],.],[.,[[.,.],.]]]
=> [6,7,5,1,2,3,4] => [4,3,2,1,5,7,6] => [4,3,2,1,7,6,5] => ? = 4
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000026
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 70%
St000026: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 70%
Values
[.,.]
=> [1,0]
=> 1
[.,[.,.]]
=> [1,0,1,0]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> 2
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 4
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 4
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 1
[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1
[.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 1
[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1
[.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 1
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[.,[.,[.,[[[.,.],[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 1
[.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 1
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 1
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 1
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 1
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 1
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 1
[.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[[.,.],[.,.]],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1
[.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 1
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000025
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 70%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 70%
Values
[.,.]
=> [1] => [1] => [1,0]
=> 1
[.,[.,.]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
[[.,.],.]
=> [1,2] => [2,1] => [1,1,0,0]
=> 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [1,2,3,4,5,6,8,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [6,8,7,5,4,3,2,1] => [1,2,3,4,5,7,8,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [1,2,3,4,5,8,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [5,8,7,6,4,3,2,1] => [1,2,3,4,6,7,8,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [5,7,8,6,4,3,2,1] => [1,2,3,4,6,8,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [6,5,8,7,4,3,2,1] => [1,2,3,4,7,8,5,6] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 1
[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [5,6,8,7,4,3,2,1] => [1,2,3,4,7,8,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [7,6,5,8,4,3,2,1] => [1,2,3,4,8,5,6,7] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> [6,7,5,8,4,3,2,1] => [1,2,3,4,8,5,7,6] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [5,7,6,8,4,3,2,1] => [1,2,3,4,8,6,7,5] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [1,2,3,4,8,7,5,6] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [4,8,7,6,5,3,2,1] => [1,2,3,5,6,7,8,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [4,7,8,6,5,3,2,1] => [1,2,3,5,6,8,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [4,6,8,7,5,3,2,1] => [1,2,3,5,7,8,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 1
[.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> [4,7,6,8,5,3,2,1] => [1,2,3,5,8,6,7,4] => [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[[.,.],[[[.,.],.],.]]]]]
=> [4,6,7,8,5,3,2,1] => [1,2,3,5,8,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [5,4,8,7,6,3,2,1] => [1,2,3,6,7,8,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 1
[.,[.,[.,[[.,[.,.]],[[.,.],.]]]]]
=> [5,4,7,8,6,3,2,1] => [1,2,3,6,8,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> [4,5,8,7,6,3,2,1] => [1,2,3,6,7,8,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 1
[.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [4,5,7,8,6,3,2,1] => [1,2,3,6,8,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [6,5,4,8,7,3,2,1] => [1,2,3,7,8,4,5,6] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[[.,[[.,.],.]],[.,.]]]]]
=> [5,6,4,8,7,3,2,1] => [1,2,3,7,8,4,6,5] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[.,.],[.,.]],[.,.]]]]]
=> [4,6,5,8,7,3,2,1] => [1,2,3,7,8,5,6,4] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[.,[.,.]],.],[.,.]]]]]
=> [5,4,6,8,7,3,2,1] => [1,2,3,7,8,6,4,5] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
=> [4,5,6,8,7,3,2,1] => [1,2,3,7,8,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> [6,7,5,4,8,3,2,1] => [1,2,3,8,4,5,7,6] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> [5,7,6,4,8,3,2,1] => [1,2,3,8,4,6,7,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[.,[[.,[[[.,.],.],.]],.]]]]
=> [5,6,7,4,8,3,2,1] => [1,2,3,8,4,7,6,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [4,7,6,5,8,3,2,1] => [1,2,3,8,5,6,7,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [4,6,7,5,8,3,2,1] => [1,2,3,8,5,7,6,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [5,4,7,6,8,3,2,1] => [1,2,3,8,6,7,4,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [4,5,7,6,8,3,2,1] => [1,2,3,8,6,7,5,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [6,5,4,7,8,3,2,1] => [1,2,3,8,7,4,5,6] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
=> [5,6,4,7,8,3,2,1] => [1,2,3,8,7,4,6,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[[.,.],[.,.]],.],.]]]]
=> [4,6,5,7,8,3,2,1] => [1,2,3,8,7,5,6,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
=> [5,4,6,7,8,3,2,1] => [1,2,3,8,7,6,4,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [1,2,3,8,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [3,8,7,6,5,4,2,1] => [1,2,4,5,6,7,8,3] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [3,7,8,6,5,4,2,1] => [1,2,4,5,6,8,7,3] => [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [3,6,8,7,5,4,2,1] => [1,2,4,5,7,8,6,3] => [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [3,7,6,8,5,4,2,1] => [1,2,4,5,8,6,7,3] => [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> [3,6,7,8,5,4,2,1] => [1,2,4,5,8,7,6,3] => [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [3,5,8,7,6,4,2,1] => [1,2,4,6,7,8,5,3] => [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1
[.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> [3,5,7,8,6,4,2,1] => [1,2,4,6,8,7,5,3] => [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [3,6,5,8,7,4,2,1] => [1,2,4,7,8,5,6,3] => [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000171The degree of the graph. St001725The harmonious chromatic number of a graph. St000740The last entry of a permutation. St000501The size of the first part in the decomposition of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000051The size of the left subtree of a binary tree. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001481The minimal height of a peak of a Dyck path. St000133The "bounce" of a permutation. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000060The greater neighbor of the maximum. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St001882The number of occurrences of a type-B 231 pattern in a signed permutation.
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