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Your data matches 65 different statistics following compositions of up to 3 maps.
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Matching statistic: St000382
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [2] => 2
[2,1] => [1,1] => 1
[1,2,3] => [3] => 3
[1,3,2] => [2,1] => 2
[2,1,3] => [1,2] => 1
[2,3,1] => [2,1] => 2
[3,1,2] => [1,2] => 1
[3,2,1] => [1,1,1] => 1
[1,2,3,4] => [4] => 4
[1,2,4,3] => [3,1] => 3
[1,3,2,4] => [2,2] => 2
[1,3,4,2] => [3,1] => 3
[1,4,2,3] => [2,2] => 2
[1,4,3,2] => [2,1,1] => 2
[2,1,3,4] => [1,3] => 1
[2,1,4,3] => [1,2,1] => 1
[2,3,1,4] => [2,2] => 2
[2,3,4,1] => [3,1] => 3
[2,4,1,3] => [2,2] => 2
[2,4,3,1] => [2,1,1] => 2
[3,1,2,4] => [1,3] => 1
[3,1,4,2] => [1,2,1] => 1
[3,2,1,4] => [1,1,2] => 1
[3,2,4,1] => [1,2,1] => 1
[3,4,1,2] => [2,2] => 2
[3,4,2,1] => [2,1,1] => 2
[4,1,2,3] => [1,3] => 1
[4,1,3,2] => [1,2,1] => 1
[4,2,1,3] => [1,1,2] => 1
[4,2,3,1] => [1,2,1] => 1
[4,3,1,2] => [1,1,2] => 1
[4,3,2,1] => [1,1,1,1] => 1
[1,2,3,4,5] => [5] => 5
[1,2,3,5,4] => [4,1] => 4
[1,2,4,3,5] => [3,2] => 3
[1,2,4,5,3] => [4,1] => 4
[1,2,5,3,4] => [3,2] => 3
[1,2,5,4,3] => [3,1,1] => 3
[1,3,2,4,5] => [2,3] => 2
[1,3,2,5,4] => [2,2,1] => 2
[1,3,4,2,5] => [3,2] => 3
[1,3,4,5,2] => [4,1] => 4
[1,3,5,2,4] => [3,2] => 3
[1,3,5,4,2] => [3,1,1] => 3
[1,4,2,3,5] => [2,3] => 2
[1,4,2,5,3] => [2,2,1] => 2
[1,4,3,2,5] => [2,1,2] => 2
[1,4,3,5,2] => [2,2,1] => 2
[1,4,5,2,3] => [3,2] => 3
Description
The first part of an integer composition.
Matching statistic: St000025
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 2
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 3
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
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: St000026
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: 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] => [2] => [1,1,0,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> 3
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
Description
The position of the first return of a Dyck path.
Matching statistic: St000383
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [2] => [2] => 2
[2,1] => [1,1] => [1,1] => 1
[1,2,3] => [3] => [3] => 3
[1,3,2] => [2,1] => [1,2] => 2
[2,1,3] => [1,2] => [2,1] => 1
[2,3,1] => [2,1] => [1,2] => 2
[3,1,2] => [1,2] => [2,1] => 1
[3,2,1] => [1,1,1] => [1,1,1] => 1
[1,2,3,4] => [4] => [4] => 4
[1,2,4,3] => [3,1] => [1,3] => 3
[1,3,2,4] => [2,2] => [2,2] => 2
[1,3,4,2] => [3,1] => [1,3] => 3
[1,4,2,3] => [2,2] => [2,2] => 2
[1,4,3,2] => [2,1,1] => [1,1,2] => 2
[2,1,3,4] => [1,3] => [3,1] => 1
[2,1,4,3] => [1,2,1] => [2,1,1] => 1
[2,3,1,4] => [2,2] => [2,2] => 2
[2,3,4,1] => [3,1] => [1,3] => 3
[2,4,1,3] => [2,2] => [2,2] => 2
[2,4,3,1] => [2,1,1] => [1,1,2] => 2
[3,1,2,4] => [1,3] => [3,1] => 1
[3,1,4,2] => [1,2,1] => [2,1,1] => 1
[3,2,1,4] => [1,1,2] => [1,2,1] => 1
[3,2,4,1] => [1,2,1] => [2,1,1] => 1
[3,4,1,2] => [2,2] => [2,2] => 2
[3,4,2,1] => [2,1,1] => [1,1,2] => 2
[4,1,2,3] => [1,3] => [3,1] => 1
[4,1,3,2] => [1,2,1] => [2,1,1] => 1
[4,2,1,3] => [1,1,2] => [1,2,1] => 1
[4,2,3,1] => [1,2,1] => [2,1,1] => 1
[4,3,1,2] => [1,1,2] => [1,2,1] => 1
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => 1
[1,2,3,4,5] => [5] => [5] => 5
[1,2,3,5,4] => [4,1] => [1,4] => 4
[1,2,4,3,5] => [3,2] => [2,3] => 3
[1,2,4,5,3] => [4,1] => [1,4] => 4
[1,2,5,3,4] => [3,2] => [2,3] => 3
[1,2,5,4,3] => [3,1,1] => [1,1,3] => 3
[1,3,2,4,5] => [2,3] => [3,2] => 2
[1,3,2,5,4] => [2,2,1] => [2,1,2] => 2
[1,3,4,2,5] => [3,2] => [2,3] => 3
[1,3,4,5,2] => [4,1] => [1,4] => 4
[1,3,5,2,4] => [3,2] => [2,3] => 3
[1,3,5,4,2] => [3,1,1] => [1,1,3] => 3
[1,4,2,3,5] => [2,3] => [3,2] => 2
[1,4,2,5,3] => [2,2,1] => [2,1,2] => 2
[1,4,3,2,5] => [2,1,2] => [1,2,2] => 2
[1,4,3,5,2] => [2,2,1] => [2,1,2] => 2
[1,4,5,2,3] => [3,2] => [2,3] => 3
Description
The last part of an integer composition.
Matching statistic: St000745
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 1
[1,2] => [2,1] => [[1],[2]]
=> 2
[2,1] => [1,2] => [[1,2]]
=> 1
[1,2,3] => [3,2,1] => [[1],[2],[3]]
=> 3
[1,3,2] => [3,1,2] => [[1,3],[2]]
=> 2
[2,1,3] => [2,3,1] => [[1,2],[3]]
=> 1
[2,3,1] => [2,1,3] => [[1,3],[2]]
=> 2
[3,1,2] => [1,3,2] => [[1,2],[3]]
=> 1
[3,2,1] => [1,2,3] => [[1,2,3]]
=> 1
[1,2,3,4] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[1,2,4,3] => [4,3,1,2] => [[1,4],[2],[3]]
=> 3
[1,3,2,4] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[1,3,4,2] => [4,2,1,3] => [[1,4],[2],[3]]
=> 3
[1,4,2,3] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2
[1,4,3,2] => [4,1,2,3] => [[1,3,4],[2]]
=> 2
[2,1,3,4] => [3,4,2,1] => [[1,2],[3],[4]]
=> 1
[2,1,4,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[2,3,1,4] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2
[2,3,4,1] => [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[2,4,1,3] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[2,4,3,1] => [3,1,2,4] => [[1,3,4],[2]]
=> 2
[3,1,2,4] => [2,4,3,1] => [[1,2],[3],[4]]
=> 1
[3,1,4,2] => [2,4,1,3] => [[1,2],[3,4]]
=> 1
[3,2,1,4] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[3,2,4,1] => [2,3,1,4] => [[1,2,4],[3]]
=> 1
[3,4,1,2] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[3,4,2,1] => [2,1,3,4] => [[1,3,4],[2]]
=> 2
[4,1,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 1
[4,1,3,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[4,2,1,3] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[4,2,3,1] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[4,3,1,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 1
[1,2,3,4,5] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 5
[1,2,3,5,4] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 4
[1,2,4,3,5] => [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 3
[1,2,4,5,3] => [5,4,2,1,3] => [[1,5],[2],[3],[4]]
=> 4
[1,2,5,3,4] => [5,4,1,3,2] => [[1,4],[2],[3],[5]]
=> 3
[1,2,5,4,3] => [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 3
[1,3,2,4,5] => [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 2
[1,3,2,5,4] => [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> 2
[1,3,4,2,5] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 3
[1,3,4,5,2] => [5,3,2,1,4] => [[1,5],[2],[3],[4]]
=> 4
[1,3,5,2,4] => [5,3,1,4,2] => [[1,4],[2,5],[3]]
=> 3
[1,3,5,4,2] => [5,3,1,2,4] => [[1,4,5],[2],[3]]
=> 3
[1,4,2,3,5] => [5,2,4,3,1] => [[1,3],[2],[4],[5]]
=> 2
[1,4,2,5,3] => [5,2,4,1,3] => [[1,3],[2,5],[4]]
=> 2
[1,4,3,2,5] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 2
[1,4,3,5,2] => [5,2,3,1,4] => [[1,3,5],[2],[4]]
=> 2
[1,4,5,2,3] => [5,2,1,4,3] => [[1,4],[2,5],[3]]
=> 3
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000439
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 2 = 1 + 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 3 = 2 + 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000011
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000297
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 => 1
[1,2] => [2,1] => [1,1] => 11 => 2
[2,1] => [1,2] => [2] => 10 => 1
[1,2,3] => [3,2,1] => [1,1,1] => 111 => 3
[1,3,2] => [3,1,2] => [1,2] => 110 => 2
[2,1,3] => [2,3,1] => [2,1] => 101 => 1
[2,3,1] => [2,1,3] => [1,2] => 110 => 2
[3,1,2] => [1,3,2] => [2,1] => 101 => 1
[3,2,1] => [1,2,3] => [3] => 100 => 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => 1111 => 4
[1,2,4,3] => [4,3,1,2] => [1,1,2] => 1110 => 3
[1,3,2,4] => [4,2,3,1] => [1,2,1] => 1101 => 2
[1,3,4,2] => [4,2,1,3] => [1,1,2] => 1110 => 3
[1,4,2,3] => [4,1,3,2] => [1,2,1] => 1101 => 2
[1,4,3,2] => [4,1,2,3] => [1,3] => 1100 => 2
[2,1,3,4] => [3,4,2,1] => [2,1,1] => 1011 => 1
[2,1,4,3] => [3,4,1,2] => [2,2] => 1010 => 1
[2,3,1,4] => [3,2,4,1] => [1,2,1] => 1101 => 2
[2,3,4,1] => [3,2,1,4] => [1,1,2] => 1110 => 3
[2,4,1,3] => [3,1,4,2] => [1,2,1] => 1101 => 2
[2,4,3,1] => [3,1,2,4] => [1,3] => 1100 => 2
[3,1,2,4] => [2,4,3,1] => [2,1,1] => 1011 => 1
[3,1,4,2] => [2,4,1,3] => [2,2] => 1010 => 1
[3,2,1,4] => [2,3,4,1] => [3,1] => 1001 => 1
[3,2,4,1] => [2,3,1,4] => [2,2] => 1010 => 1
[3,4,1,2] => [2,1,4,3] => [1,2,1] => 1101 => 2
[3,4,2,1] => [2,1,3,4] => [1,3] => 1100 => 2
[4,1,2,3] => [1,4,3,2] => [2,1,1] => 1011 => 1
[4,1,3,2] => [1,4,2,3] => [2,2] => 1010 => 1
[4,2,1,3] => [1,3,4,2] => [3,1] => 1001 => 1
[4,2,3,1] => [1,3,2,4] => [2,2] => 1010 => 1
[4,3,1,2] => [1,2,4,3] => [3,1] => 1001 => 1
[4,3,2,1] => [1,2,3,4] => [4] => 1000 => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => 11111 => 5
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,2] => 11110 => 4
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,2,1] => 11101 => 3
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,2] => 11110 => 4
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,2,1] => 11101 => 3
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => 11100 => 3
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,1,1] => 11011 => 2
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,2] => 11010 => 2
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,2,1] => 11101 => 3
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,2] => 11110 => 4
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,2,1] => 11101 => 3
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,3] => 11100 => 3
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,1,1] => 11011 => 2
[1,4,2,5,3] => [5,2,4,1,3] => [1,2,2] => 11010 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => 11001 => 2
[1,4,3,5,2] => [5,2,3,1,4] => [1,2,2] => 11010 => 2
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,2,1] => 11101 => 3
Description
The number of leading ones in a binary word.
Matching statistic: St000505
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce 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%
Mp00231: Integer compositions —bounce 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%
Values
[1] => [1] => [1,0]
=> {{1}}
=> 1
[1,2] => [2] => [1,1,0,0]
=> {{1,2}}
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 4
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 4
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 4
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3
Description
The biggest entry in the block containing the 1.
Matching statistic: St000971
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => {{1}}
=> 1
[1,2] => [1,0,1,0]
=> [2,1] => {{1,2}}
=> 2
[2,1] => [1,1,0,0]
=> [1,2] => {{1},{2}}
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,3,1] => {{1,2,3}}
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [2,1,3] => {{1,2},{3}}
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,3,2] => {{1},{2,3}}
=> 1
[2,3,1] => [1,1,0,1,0,0]
=> [3,1,2] => {{1,3},{2}}
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> 2
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => {{1,2,3,5},{4}}
=> 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> 3
Description
The smallest closer of a set partition.
A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers.
In other words, this is the smallest among the maximal elements of the blocks.
The following 55 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001050The number of terminal closers of a set partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000678The number of up steps after the last double rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St000363The number of minimal vertex covers of a graph. St001322The size of a minimal independent dominating set in a graph. St001316The domatic number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000759The smallest missing part in an integer partition. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000054The first entry of the permutation. St000617The number of global maxima of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000501The size of the first part in the decomposition of a permutation. St000990The first ascent of a permutation. St000989The number of final rises of a permutation. St000007The number of saliances of the permutation. St000654The first descent of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000546The number of global descents of a permutation. St000740The last entry of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000738The first entry in the last row of a standard tableau. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000084The number of subtrees. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000133The "bounce" of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. St000061The number of nodes on the left branch of a binary tree. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000338The number of pixed points of a permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function.
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