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Your data matches 28 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)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ 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
[.,[.,.]]
=> [1,1,0,0]
=> 2
[[.,.],.]
=> [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 3
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 3
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 4
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 4
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 4
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 4
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 3
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 5
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 5
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000382
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ 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
[.,[.,.]]
=> [1,1,0,0]
=> [2] => 2
[[.,.],.]
=> [1,0,1,0]
=> [1,1] => 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [3] => 3
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [3] => 3
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,2] => 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [4] => 4
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [4] => 4
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 3
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 5
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 5
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 5
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => 5
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => 5
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => 5
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 5
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => 5
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => 5
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 5
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 5
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [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
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 3
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 1
Description
The first part of an integer composition.
Matching statistic: St000505
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ 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
[.,[.,.]]
=> [1,1,0,0]
=> {{1,2}}
=> 2
[[.,.],.]
=> [1,0,1,0]
=> {{1},{2}}
=> 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 3
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{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
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 4
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 5
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 5
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 5
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 5
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 5
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 5
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 5
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 5
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 5
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 5
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 5
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{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
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
Description
The biggest entry in the block containing the 1.
Matching statistic: St000025
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ 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%
Mp00066: Permutations —inverse⟶ 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] => [2,1] => [1,1,0,0]
=> 2
[[.,.],.]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000383
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ 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%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ 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] => 1
[.,[.,.]]
=> [[.,.],.]
=> [1,1,0,0]
=> [2] => 2
[[.,.],.]
=> [.,[.,.]]
=> [1,0,1,0]
=> [1,1] => 1
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => 3
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [3] => 3
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => 1
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [4] => 4
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [4] => 4
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 3
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 5
[.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 5
[.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 5
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 5
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 5
[.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => 5
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => 5
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 5
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 5
[.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => 5
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => 5
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => 5
[.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 5
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 1
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 1
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 1
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 1
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 2
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 2
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 1
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 3
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 1
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000439
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ 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%
Mp00066: Permutations —inverse⟶ 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] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[[.,.],.]
=> [1,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000645
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ 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%
Mp00014: Binary trees —to 132-avoiding permutation⟶ 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] => [1,0]
=> 0 = 1 - 1
[.,[.,.]]
=> [[.,.],.]
=> [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[[.,.],.]
=> [.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2 = 3 - 1
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 5 - 1
[.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 5 - 1
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 4 = 5 - 1
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 4 = 5 - 1
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 4 = 5 - 1
[.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 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: St000326
Mp00017: Binary trees —to 312-avoiding permutation⟶ 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%
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
[.,[.,.]]
=> [2,1] => 0 => 2
[[.,.],.]
=> [1,2] => 1 => 1
[.,[.,[.,.]]]
=> [3,2,1] => 00 => 3
[.,[[.,.],.]]
=> [2,3,1] => 00 => 3
[[.,.],[.,.]]
=> [1,3,2] => 10 => 1
[[.,[.,.]],.]
=> [2,1,3] => 01 => 2
[[[.,.],.],.]
=> [1,2,3] => 11 => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 000 => 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 000 => 4
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 000 => 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 000 => 4
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 000 => 4
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 100 => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 100 => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 010 => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 110 => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 001 => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 001 => 3
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 101 => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 011 => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => 111 => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0000 => 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0000 => 5
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 0000 => 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0000 => 5
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0000 => 5
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 0000 => 5
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 0000 => 5
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 0000 => 5
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 0000 => 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0000 => 5
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0000 => 5
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 0000 => 5
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0000 => 5
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0000 => 5
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1000 => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 1000 => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 1000 => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 1000 => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 1000 => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 0100 => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 0100 => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1100 => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 1100 => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 0010 => 3
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 0010 => 3
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 1010 => 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 0110 => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1110 => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0001 => 4
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: St000297
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => => => ? = 1 - 1
[.,[.,.]]
=> [2,1] => 0 => 1 => 1 = 2 - 1
[[.,.],.]
=> [1,2] => 1 => 0 => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => 00 => 11 => 2 = 3 - 1
[.,[[.,.],.]]
=> [2,3,1] => 00 => 11 => 2 = 3 - 1
[[.,.],[.,.]]
=> [1,3,2] => 10 => 01 => 0 = 1 - 1
[[.,[.,.]],.]
=> [2,1,3] => 01 => 10 => 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => 11 => 00 => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 000 => 111 => 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 000 => 111 => 3 = 4 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 000 => 111 => 3 = 4 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 000 => 111 => 3 = 4 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 000 => 111 => 3 = 4 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 100 => 011 => 0 = 1 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 100 => 011 => 0 = 1 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 010 => 101 => 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 110 => 001 => 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 001 => 110 => 2 = 3 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 001 => 110 => 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 101 => 010 => 0 = 1 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 011 => 100 => 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 111 => 000 => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0000 => 1111 => 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0000 => 1111 => 4 = 5 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 0000 => 1111 => 4 = 5 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0000 => 1111 => 4 = 5 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0000 => 1111 => 4 = 5 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 0000 => 1111 => 4 = 5 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 0000 => 1111 => 4 = 5 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 0000 => 1111 => 4 = 5 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 0000 => 1111 => 4 = 5 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0000 => 1111 => 4 = 5 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0000 => 1111 => 4 = 5 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 0000 => 1111 => 4 = 5 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0000 => 1111 => 4 = 5 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0000 => 1111 => 4 = 5 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1000 => 0111 => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 1000 => 0111 => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 1000 => 0111 => 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 1000 => 0111 => 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 1000 => 0111 => 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 0100 => 1011 => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 0100 => 1011 => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1100 => 0011 => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 1100 => 0011 => 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 0010 => 1101 => 2 = 3 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 0010 => 1101 => 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 1010 => 0101 => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 0110 => 1001 => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1110 => 0001 => 0 = 1 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0001 => 1110 => 3 = 4 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000054
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 1
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 2
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 3
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => [3,2,1] => 3
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 2
[[[.,.],.],.]
=> [1,2,3] => [1,3,2] => [1,3,2] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => [4,3,2,1] => 4
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,4,3,1] => [4,3,2,1] => 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => [4,2,3,1] => 4
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,4,3,1] => [4,3,2,1] => 4
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,4,3,2] => [1,4,3,2] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,4,1,3] => [3,4,1,2] => 3
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,4,3,2] => [1,4,3,2] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,4,3,2] => [1,4,3,2] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => [5,4,3,2,1] => 5
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,5,4,2,1] => [5,4,3,2,1] => 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => [5,4,3,2,1] => 5
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,5,4,2,1] => [5,4,3,2,1] => 5
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,5,4,3,1] => [5,4,3,2,1] => 5
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,5,4,3,1] => [5,4,3,2,1] => 5
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,5,4,1] => [5,2,4,3,1] => 5
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,5,4,3,1] => [5,4,3,2,1] => 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => [5,3,2,4,1] => 5
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,5,2,4,1] => [5,4,3,2,1] => 5
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,5,4,3,1] => [5,4,3,2,1] => 5
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,5,4,1] => [5,2,4,3,1] => 5
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,5,4,3,1] => [5,4,3,2,1] => 5
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 3
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,5,1,4,3] => [3,5,1,4,2] => 3
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [5,4,3,7,6,2,1] => [5,4,3,7,6,2,1] => [7,6,3,5,4,2,1] => ? = 7
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => [6,5,4,3,7,2,1] => [7,6,4,3,5,2,1] => ? = 7
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [5,4,3,6,7,2,1] => [5,4,3,7,6,2,1] => [7,6,3,5,4,2,1] => ? = 7
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [3,2,7,6,5,4,1] => [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 7
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [3,2,6,7,5,4,1] => [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 7
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [3,2,5,7,6,4,1] => [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 7
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [3,2,6,5,7,4,1] => [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 7
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [3,2,5,6,7,4,1] => [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 7
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [4,3,2,7,6,5,1] => [4,3,2,7,6,5,1] => [7,3,2,6,5,4,1] => ? = 7
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [4,3,2,6,7,5,1] => [4,3,2,7,6,5,1] => [7,3,2,6,5,4,1] => ? = 7
[.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [3,4,2,7,6,5,1] => [3,7,2,6,5,4,1] => [7,6,3,5,4,2,1] => ? = 7
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [3,4,2,6,7,5,1] => [3,7,2,6,5,4,1] => [7,6,3,5,4,2,1] => ? = 7
[.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [3,2,4,7,6,5,1] => [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 7
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [3,2,4,6,7,5,1] => [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 7
[.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [5,4,3,2,7,6,1] => [5,4,3,2,7,6,1] => [7,4,3,2,6,5,1] => ? = 7
[.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [4,5,3,2,7,6,1] => [4,7,3,2,6,5,1] => [7,6,4,3,5,2,1] => ? = 7
[.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [4,3,5,2,7,6,1] => [4,3,7,2,6,5,1] => [7,4,6,2,5,3,1] => ? = 7
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> [3,2,5,4,7,6,1] => [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 7
[.,[[[.,[.,[.,.]]],.],[.,.]]]
=> [4,3,2,5,7,6,1] => [4,3,2,7,6,5,1] => [7,3,2,6,5,4,1] => ? = 7
[.,[[[.,[[.,.],.]],.],[.,.]]]
=> [3,4,2,5,7,6,1] => [3,7,2,6,5,4,1] => [7,6,3,5,4,2,1] => ? = 7
[.,[[[[.,[.,.]],.],.],[.,.]]]
=> [3,2,4,5,7,6,1] => [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 7
[.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => [6,5,4,3,2,7,1] => [7,5,4,3,2,6,1] => ? = 7
[.,[[.,[.,[[.,[.,.]],.]]],.]]
=> [5,4,6,3,2,7,1] => [5,4,7,3,2,6,1] => [7,5,6,4,2,3,1] => ? = 7
[.,[[.,[[.,[.,.]],[.,.]]],.]]
=> [4,3,6,5,2,7,1] => [4,3,7,6,2,5,1] => [7,5,6,4,2,3,1] => ? = 7
[.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => [5,4,3,7,2,6,1] => [7,5,3,6,2,4,1] => ? = 7
[.,[[.,[[[.,[.,.]],.],.]],.]]
=> [4,3,5,6,2,7,1] => [4,3,7,6,2,5,1] => [7,5,6,4,2,3,1] => ? = 7
[.,[[[.,[.,.]],[.,[.,.]]],.]]
=> [3,2,6,5,4,7,1] => [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 7
[.,[[[.,[.,.]],[[.,.],.]],.]]
=> [3,2,5,6,4,7,1] => [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 7
[.,[[[.,[.,[.,.]]],[.,.]],.]]
=> [4,3,2,6,5,7,1] => [4,3,2,7,6,5,1] => [7,3,2,6,5,4,1] => ? = 7
[.,[[[.,[[.,.],.]],[.,.]],.]]
=> [3,4,2,6,5,7,1] => [3,7,2,6,5,4,1] => [7,6,3,5,4,2,1] => ? = 7
[.,[[[[.,[.,.]],.],[.,.]],.]]
=> [3,2,4,6,5,7,1] => [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 7
[.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [5,4,3,2,6,7,1] => [5,4,3,2,7,6,1] => [7,4,3,2,6,5,1] => ? = 7
[.,[[[.,[.,[[.,.],.]]],.],.]]
=> [4,5,3,2,6,7,1] => [4,7,3,2,6,5,1] => [7,6,4,3,5,2,1] => ? = 7
[.,[[[.,[[.,[.,.]],.]],.],.]]
=> [4,3,5,2,6,7,1] => [4,3,7,2,6,5,1] => [7,4,6,2,5,3,1] => ? = 7
[.,[[[[.,[.,.]],[.,.]],.],.]]
=> [3,2,5,4,6,7,1] => [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 7
[.,[[[[.,[.,[.,.]]],.],.],.]]
=> [4,3,2,5,6,7,1] => [4,3,2,7,6,5,1] => [7,3,2,6,5,4,1] => ? = 7
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => [3,7,2,6,5,4,1] => [7,6,3,5,4,2,1] => ? = 7
[.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => [3,2,7,6,5,4,1] => [7,2,6,5,4,3,1] => ? = 7
[[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 3
[[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [3,2,1,6,7,5,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 3
[[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [3,2,1,5,7,6,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 3
[[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [3,2,1,6,5,7,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 3
[[.,[.,[.,.]]],[[[.,.],.],.]]
=> [3,2,1,5,6,7,4] => [3,2,1,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 3
[[.,[[.,.],.]],[.,[.,[.,.]]]]
=> [2,3,1,7,6,5,4] => [2,7,1,6,5,4,3] => [3,7,1,6,5,4,2] => ? = 3
[[.,[[.,.],.]],[.,[[.,.],.]]]
=> [2,3,1,6,7,5,4] => [2,7,1,6,5,4,3] => [3,7,1,6,5,4,2] => ? = 3
[[.,[[.,.],.]],[[.,.],[.,.]]]
=> [2,3,1,5,7,6,4] => [2,7,1,6,5,4,3] => [3,7,1,6,5,4,2] => ? = 3
[[.,[[.,.],.]],[[.,[.,.]],.]]
=> [2,3,1,6,5,7,4] => [2,7,1,6,5,4,3] => [3,7,1,6,5,4,2] => ? = 3
[[.,[[.,.],.]],[[[.,.],.],.]]
=> [2,3,1,5,6,7,4] => [2,7,1,6,5,4,3] => [3,7,1,6,5,4,2] => ? = 3
[[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => ? = 4
[[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [4,3,2,1,6,7,5] => [4,3,2,1,7,6,5] => [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
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000066The column of the unique '1' in the first row of the alternating sign matrix. St000501The size of the first part in the decomposition of a permutation. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St000051The size of the left subtree of a binary tree. St000335The difference of lower and upper interactions. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000133The "bounce" of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001645The pebbling number of a connected graph. 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. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation.
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