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Your data matches 25 different statistics following compositions of up to 3 maps.
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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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
Description
The position of the first return of a Dyck path.
Matching statistic: St000382
(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
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: St000505
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ 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},{2}}
=> 1
[[.,.],.]
=> [1,1,0,0]
=> {{1,2}}
=> 2
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 3
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 4
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 4
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 3
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 4
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 4
Description
The biggest entry in the block containing the 1.
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
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
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [1] => 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1] => 1
[[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2] => 2
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 3
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 3
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 4
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [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]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 4
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 4
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
Description
The last part of an integer composition.
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: 100% ●values known / values provided: 100%●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: 100% ●values known / values provided: 100%●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
Description
The position of the first down step of a Dyck path.
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: 100% ●values known / values provided: 100%●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: 100% ●values known / values provided: 100%●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
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: St000987
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => ([],1)
=> 0 = 1 - 1
[.,[.,.]]
=> [1,1,0,0]
=> [2] => ([],2)
=> 0 = 1 - 1
[[.,.],.]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 0 = 1 - 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> 0 = 1 - 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 3 - 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 3 - 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
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: 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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
$$
\sum_v (j_v-i_v)/2.
$$
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000171The degree of the graph. St001725The harmonious chromatic number of a graph. St000054The first entry of the permutation. 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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