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Your data matches 101 different statistics following compositions of up to 3 maps.
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Matching statistic: St000007
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => [1,2] => 1
[2,1] => [2,1] => [1,2] => [2,1] => 2
[1,2,3] => [1,3,2] => [2,3,1] => [2,1,3] => 1
[1,3,2] => [1,3,2] => [2,3,1] => [2,1,3] => 1
[2,1,3] => [2,1,3] => [3,1,2] => [1,3,2] => 2
[2,3,1] => [2,3,1] => [1,3,2] => [3,1,2] => 2
[3,1,2] => [3,1,2] => [2,1,3] => [2,3,1] => 2
[3,2,1] => [3,2,1] => [1,2,3] => [3,2,1] => 3
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 1
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 1
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 1
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 1
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [3,2,1,4] => 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [2,1,4,3] => 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 2
[2,3,4,1] => [2,4,3,1] => [1,3,4,2] => [4,2,1,3] => 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 2
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => [4,2,1,3] => 2
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 2
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 2
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [1,4,3,2] => 3
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => [4,1,3,2] => 3
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [3,4,1,2] => 2
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => [4,3,1,2] => 3
[4,1,2,3] => [4,1,3,2] => [2,3,1,4] => [3,2,4,1] => 2
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [3,2,4,1] => 2
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [2,4,3,1] => 3
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [4,2,3,1] => 3
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [3,4,2,1] => 3
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 4
[1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => [4,3,2,1,5] => 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern ([1],(1,1)), i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000011
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 88%●distinct values known / distinct values provided: 83%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 88%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 1
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[2,1] => [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2
[3,1,2] => [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,3,1,4] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,3,4,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,4,1,3] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,4,3,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,2,4] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1,4,2] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,2,4,1] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 3
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,1,2,3] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 2
[4,1,3,2] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 2
[4,2,1,3] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,3,1,2] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[8,6,7,4,5,3,2,1] => [8,6,7,4,5,3,2,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 6
[8,6,7,5,3,4,2,1] => [8,6,7,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 6
[8,7,5,6,3,4,2,1] => [8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 6
[8,7,4,3,5,6,2,1] => [8,7,4,3,6,5,2,1] => [[[[[[.,.],.],.],[[.,.],.]],.],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 6
[8,6,7,5,4,2,3,1] => [8,6,7,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6
[8,7,5,6,4,2,3,1] => [8,7,5,6,4,2,3,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 6
[8,7,6,4,5,2,3,1] => [8,7,6,4,5,2,3,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[8,6,5,7,3,2,4,1] => [8,6,5,7,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 6
[8,5,6,7,2,3,4,1] => [8,5,7,6,2,4,3,1] => [[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 4
[8,7,6,4,2,3,5,1] => [8,7,6,4,2,5,3,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[8,7,6,3,2,4,5,1] => [8,7,6,3,2,5,4,1] => [[[[[[.,.],.],[[.,.],.]],.],.],.]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 6
[7,8,3,4,2,5,6,1] => [7,8,3,6,2,5,4,1] => ?
=> ?
=> ? = 4
[8,6,4,3,5,2,7,1] => [8,6,4,3,7,2,5,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 6
[8,3,4,5,6,2,7,1] => [8,3,7,6,5,2,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[8,6,2,3,4,5,7,1] => [8,6,2,7,5,4,3,1] => [[[[.,.],[[[.,.],.],.]],[.,.]],.]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 4
[8,4,5,2,3,6,7,1] => [8,4,7,2,6,5,3,1] => [[[[.,.],[.,.]],[[[.,.],.],.]],.]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 4
[8,3,2,4,5,6,7,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[6,3,4,5,7,2,8,1] => [6,3,8,7,5,2,4,1] => [[[[.,.],.],[[.,.],.]],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 4
[8,7,6,4,5,3,1,2] => [8,7,6,4,5,3,1,2] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 6
[6,5,7,8,3,4,1,2] => [6,5,8,7,3,4,1,2] => [[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 4
[7,8,4,3,5,6,1,2] => [7,8,4,3,6,5,1,2] => [[[[.,[.,.]],.],[[.,.],.]],[.,.]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 4
[7,5,6,8,2,3,1,4] => [7,5,8,6,2,4,1,3] => [[[[.,.],[[.,.],.]],[.,.]],[.,.]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[6,7,5,8,3,1,2,4] => [6,8,5,7,3,1,4,2] => [[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 4
[7,8,5,6,2,1,3,4] => [7,8,5,6,2,1,4,3] => [[[[.,.],[[.,.],.]],[.,.]],[.,.]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 4
[8,7,5,3,2,1,4,6] => [8,7,5,3,2,1,6,4] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 6
[7,8,4,2,3,1,5,6] => [7,8,4,2,6,1,5,3] => [[[[.,.],[.,.]],[[.,.],.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 4
[7,8,3,2,1,4,5,6] => [7,8,3,2,1,6,5,4] => [[[[.,.],.],[[[.,.],.],.]],[.,.]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 4
[8,6,5,4,2,1,3,7] => [8,6,5,4,2,1,7,3] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 6
[8,6,2,3,4,1,5,7] => [8,6,2,7,5,1,4,3] => [[[[.,.],[[[.,.],.],.]],[.,.]],.]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 4
[7,4,5,2,3,6,1,8] => [7,4,8,2,6,5,1,3] => [[[[.,.],[.,.]],[[.,.],.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 4
[5,6,3,4,1,2,7,8] => [5,8,3,7,1,6,4,2] => [[[.,[.,.]],[.,.]],[[[.,.],.],.]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 3
[5,4,6,2,1,3,7,8] => [5,4,8,2,1,7,6,3] => [[[[.,.],[.,.]],.],[[[.,.],.],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 4
[4,5,6,1,2,3,7,8] => [4,8,7,1,6,5,3,2] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[6,3,4,2,1,5,7,8] => [6,3,8,2,1,7,5,4] => [[[[.,.],.],[[.,.],.]],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 4
[8,3,2,4,6,5,7,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[8,3,2,4,6,7,5,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[8,3,2,4,7,6,5,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[6,3,2,5,4,1,7,8] => [6,3,2,8,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 4
[6,3,2,5,4,1,8,7] => [6,3,2,8,7,1,5,4] => [[[[.,.],.],[[.,.],.]],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 4
[7,3,2,5,4,6,1,8] => [7,3,2,8,6,5,1,4] => [[[[.,.],.],[[[.,.],.],.]],[.,.]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 4
[8,3,2,5,4,7,6,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[8,3,2,5,6,4,7,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[8,3,2,5,6,7,4,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[8,3,2,5,7,6,4,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[8,3,2,6,5,7,4,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[8,3,2,7,5,6,4,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[8,3,2,7,6,5,4,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[8,3,6,4,5,2,7,1] => [8,3,7,6,5,2,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[8,3,6,5,4,2,7,1] => [8,3,7,6,5,2,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 4
[8,7,4,3,6,5,2,1] => [8,7,4,3,6,5,2,1] => [[[[[[.,.],.],.],[[.,.],.]],.],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 6
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: St000439
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 83%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 2 = 1 + 1
[1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 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
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[7,8,6,5,3,4,2,1] => [7,8,6,5,3,4,2,1] => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> ? = 6 + 1
[8,6,7,5,3,4,2,1] => [8,6,7,5,3,4,2,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 6 + 1
[8,7,5,6,3,4,2,1] => [8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 6 + 1
[7,8,5,6,3,4,2,1] => [7,8,5,6,3,4,2,1] => [[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> ? = 5 + 1
[8,7,4,3,5,6,2,1] => [8,7,4,3,6,5,2,1] => [[[[[[.,.],.],.],[[.,.],.]],.],.]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 6 + 1
[8,7,6,4,5,2,3,1] => [8,7,6,4,5,2,3,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 6 + 1
[8,5,6,7,2,3,4,1] => [8,5,7,6,2,4,3,1] => [[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 4 + 1
[8,7,6,4,2,3,5,1] => [8,7,6,4,2,5,3,1] => [[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 6 + 1
[8,7,6,3,2,4,5,1] => [8,7,6,3,2,5,4,1] => [[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 6 + 1
[7,8,3,4,2,5,6,1] => [7,8,3,6,2,5,4,1] => [[[[.,[.,.]],[.,.]],[[.,.],.]],.]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 4 + 1
[8,6,4,3,5,2,7,1] => [8,6,4,3,7,2,5,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 6 + 1
[8,3,4,5,6,2,7,1] => [8,3,7,6,5,2,4,1] => [[[[.,.],[[[.,.],.],.]],[.,.]],.]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[8,6,2,3,4,5,7,1] => [8,6,2,7,5,4,3,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[8,5,4,3,2,6,7,1] => [8,5,4,3,2,7,6,1] => [[[[[[.,.],.],.],.],[[.,.],.]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? = 6 + 1
[8,4,5,2,3,6,7,1] => [8,4,7,2,6,5,3,1] => [[[[.,.],[.,.]],[[[.,.],.],.]],.]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 4 + 1
[8,3,2,4,5,6,7,1] => [8,3,2,7,6,5,4,1] => [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[6,3,4,5,7,2,8,1] => [6,3,8,7,5,2,4,1] => [[[[.,.],[[[.,.],.],.]],[.,.]],.]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[8,7,6,4,5,3,1,2] => [8,7,6,4,5,3,1,2] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 6 + 1
[8,7,6,5,3,4,1,2] => [8,7,6,5,3,4,1,2] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 6 + 1
[6,5,7,8,3,4,1,2] => [6,5,8,7,3,4,1,2] => [[[[.,.],[[.,.],.]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 4 + 1
[7,8,4,3,5,6,1,2] => [7,8,4,3,6,5,1,2] => [[[[.,[.,.]],.],[[.,.],.]],[.,.]]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
=> ? = 4 + 1
[5,4,3,6,7,8,1,2] => [5,4,3,8,7,6,1,2] => [[[[.,.],.],[[[.,.],.],.]],[.,.]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 4 + 1
[7,5,6,8,2,3,1,4] => [7,5,8,6,2,4,1,3] => [[[[.,.],[[.,.],.]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]
=> ? = 4 + 1
[6,7,5,8,3,1,2,4] => [6,8,5,7,3,1,4,2] => [[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 4 + 1
[7,8,5,6,2,1,3,4] => [7,8,5,6,2,1,4,3] => [[[[.,[.,.]],[.,.]],.],[[.,.],.]]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 4 + 1
[6,5,7,8,2,1,3,4] => [6,5,8,7,2,1,4,3] => [[[[.,.],[[.,.],.]],.],[[.,.],.]]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 4 + 1
[5,6,7,8,1,2,3,4] => [5,8,7,6,1,4,3,2] => [[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[7,8,4,2,3,1,5,6] => [7,8,4,2,6,1,5,3] => [[[[.,[.,.]],.],[.,.]],[[.,.],.]]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 4 + 1
[7,8,3,2,1,4,5,6] => [7,8,3,2,1,6,5,4] => [[[[.,[.,.]],.],.],[[[.,.],.],.]]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 + 1
[8,6,2,3,4,1,5,7] => [8,6,2,7,5,1,4,3] => [[[[.,.],.],[[.,.],.]],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 4 + 1
[7,6,4,3,5,2,1,8] => [7,6,4,3,8,2,1,5] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 6 + 1
[7,5,4,3,2,6,1,8] => [7,5,4,3,2,8,1,6] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 6 + 1
[7,4,5,2,3,6,1,8] => [7,4,8,2,6,5,1,3] => [[[[.,.],[.,.]],[[.,.],.]],[.,.]]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0]
=> ? = 4 + 1
[5,6,4,3,1,2,7,8] => [5,8,4,3,1,7,6,2] => [[[[.,[.,.]],.],.],[[[.,.],.],.]]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 + 1
[5,6,3,4,1,2,7,8] => [5,8,3,7,1,6,4,2] => [[[.,[.,.]],[.,.]],[[[.,.],.],.]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[5,4,6,2,1,3,7,8] => [5,4,8,2,1,7,6,3] => [[[[.,.],[.,.]],.],[[[.,.],.],.]]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 + 1
[4,5,6,1,2,3,7,8] => [4,8,7,1,6,5,3,2] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[6,3,4,2,1,5,7,8] => [6,3,8,2,1,7,5,4] => [[[[.,.],[.,.]],.],[[[.,.],.],.]]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 + 1
[5,4,3,2,1,6,7,8] => [5,4,3,2,1,8,7,6] => [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[5,3,2,4,1,6,7,8] => [5,3,2,8,1,7,6,4] => [[[[.,.],.],[.,.]],[[[.,.],.],.]]
=> [1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 4 + 1
[3,4,1,2,5,6,7,8] => [3,8,1,7,6,5,4,2] => [[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[3,2,1,4,5,6,7,8] => [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[3,2,1,8,7,6,5,4] => [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[3,2,1,7,8,6,5,4] => [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[3,2,1,6,8,7,5,4] => [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[3,2,1,7,6,8,5,4] => [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[3,2,1,6,7,8,5,4] => [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[3,2,1,5,8,7,6,4] => [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[3,2,1,5,7,8,6,4] => [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[3,2,1,6,5,8,7,4] => [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 83%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 83%
Values
[1] => [.,.]
=> [1,0]
=> [1] => 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,1] => 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [2] => 2
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 1
[8,5,4,6,7,3,2,1] => ?
=> ?
=> ? => ? = 6
[8,7,6,4,2,3,5,1] => ?
=> ?
=> ? => ? = 6
[4,6,7,8,5,3,2,1] => ?
=> ?
=> ? => ? = 4
[4,5,7,8,6,3,2,1] => ?
=> ?
=> ? => ? = 4
[2,6,5,7,8,4,3,1] => ?
=> ?
=> ? => ? = 2
[2,6,7,5,4,8,3,1] => ?
=> ?
=> ? => ? = 2
[2,4,7,6,5,8,3,1] => ?
=> ?
=> ? => ? = 2
[2,4,3,5,8,7,6,1] => ?
=> ?
=> ? => ? = 2
[2,5,4,6,3,7,8,1] => ?
=> ?
=> ? => ? = 2
[1,4,5,7,8,6,3,2] => ?
=> ?
=> ? => ? = 1
[1,6,5,4,8,7,3,2] => ?
=> ?
=> ? => ? = 1
[1,5,4,6,8,7,3,2] => ?
=> ?
=> ? => ? = 1
[1,3,5,7,6,8,4,2] => ?
=> ?
=> ? => ? = 1
[1,3,5,4,7,8,6,2] => ?
=> ?
=> ? => ? = 1
[1,3,5,7,6,4,8,2] => ?
=> ?
=> ? => ? = 1
[1,4,3,6,7,5,8,2] => ?
=> ?
=> ? => ? = 1
[1,6,5,4,3,7,8,2] => ?
=> ?
=> ? => ? = 1
[1,3,5,4,6,7,8,2] => ?
=> ?
=> ? => ? = 1
[2,1,4,7,6,8,5,3] => ?
=> ?
=> ? => ? = 2
[1,2,4,6,5,7,8,3] => ?
=> ?
=> ? => ? = 1
[1,3,2,5,7,6,8,4] => ?
=> ?
=> ? => ? = 1
[2,1,3,5,7,6,8,4] => ?
=> ?
=> ? => ? = 2
[1,5,4,7,6,3,2,8] => ?
=> ?
=> ? => ? = 1
[1,3,7,6,5,4,2,8] => ?
=> ?
=> ? => ? = 1
[2,1,4,6,7,5,3,8] => ?
=> ?
=> ? => ? = 2
[3,2,1,6,7,5,4,8] => ?
=> ?
=> ? => ? = 3
[3,2,1,5,6,7,4,8] => ?
=> ?
=> ? => ? = 3
[2,1,3,4,6,7,5,8] => ?
=> ?
=> ? => ? = 2
[1,2,6,4,5,8,7,3] => ?
=> ?
=> ? => ? = 1
[1,3,4,8,6,5,7,2] => ?
=> ?
=> ? => ? = 1
[1,3,4,8,6,7,5,2] => ?
=> ?
=> ? => ? = 1
[1,8,3,5,4,7,6,2] => ?
=> ?
=> ? => ? = 1
[1,7,4,5,6,3,2,8] => ?
=> ?
=> ? => ? = 1
[1,8,5,4,3,7,6,2] => ?
=> ?
=> ? => ? = 1
[2,4,3,8,6,7,5,1] => ?
=> ?
=> ? => ? = 2
[2,6,3,5,4,7,8,1] => ?
=> ?
=> ? => ? = 2
[2,6,4,3,5,7,8,1] => ?
=> ?
=> ? => ? = 2
[2,6,4,5,3,7,8,1] => ?
=> ?
=> ? => ? = 2
[2,6,4,5,3,8,7,1] => ?
=> ?
=> ? => ? = 2
[8,3,2,4,6,5,7,1] => ?
=> ?
=> ? => ? = 4
[8,3,2,4,7,6,5,1] => ?
=> ?
=> ? => ? = 4
[2,1,4,6,7,3,5,8] => ?
=> ?
=> ? => ? = 2
[2,1,6,7,3,8,4,5] => ?
=> ?
=> ? => ? = 2
[2,1,6,3,7,4,5,8] => ?
=> ?
=> ? => ? = 2
[3,4,1,7,2,5,8,6] => ?
=> ?
=> ? => ? = 2
[3,5,1,7,2,4,8,6] => ?
=> ?
=> ? => ? = 2
[1,3,5,2,4,7,8,6] => ?
=> ?
=> ? => ? = 1
[1,3,2,5,7,4,8,6] => ?
=> ?
=> ? => ? = 1
[1,3,5,2,6,7,8,4] => ?
=> ?
=> ? => ? = 1
[1,3,2,6,4,7,8,5] => ?
=> ?
=> ? => ? = 1
Description
The first part of an integer composition.
Matching statistic: St000031
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 92%
Mp00069: Permutations —complement⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 92%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [2,1] => [1,2] => [1,2] => 2
[1,2,3] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[2,1,3] => [2,1,3] => [2,3,1] => [3,2,1] => 2
[2,3,1] => [2,3,1] => [2,1,3] => [2,1,3] => 2
[3,1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 2
[3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 3
[1,2,3,4] => [1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 1
[1,2,4,3] => [1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 1
[1,3,2,4] => [1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 1
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 1
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 1
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => [4,1,2,3] => 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => [4,3,2,1] => 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [4,3,2,1] => 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [4,1,3,2] => 2
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [4,1,3,2] => 2
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 2
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 2
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 2
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => [4,2,3,1] => 3
[3,2,4,1] => [3,2,4,1] => [2,3,1,4] => [3,2,1,4] => 3
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,2,1] => [3,4,2,1] => [2,1,3,4] => [2,1,3,4] => 3
[4,1,2,3] => [4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 2
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 2
[4,2,1,3] => [4,2,1,3] => [1,3,4,2] => [1,4,3,2] => 3
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 3
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 3
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 4
[1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,1,2,3,4] => [5,1,2,3,4] => 1
[2,1,3,4,5,6,7] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,4,5,7,6] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,4,6,5,7] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,4,6,7,5] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,4,7,5,6] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,4,7,6,5] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,5,4,6,7] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,5,4,7,6] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,5,6,4,7] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,5,6,7,4] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,5,7,4,6] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,5,7,6,4] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,6,4,5,7] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,6,4,7,5] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,6,5,4,7] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,6,5,7,4] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,6,7,4,5] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,6,7,5,4] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,7,4,5,6] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,7,4,6,5] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,7,5,4,6] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,7,5,6,4] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,7,6,4,5] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,3,7,6,5,4] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,3,5,6,7] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,3,5,7,6] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,3,6,5,7] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,3,6,7,5] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,3,7,5,6] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,3,7,6,5] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,5,3,6,7] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,5,3,7,6] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,5,6,3,7] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,5,6,7,3] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,5,7,3,6] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,5,7,6,3] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,6,3,5,7] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,6,3,7,5] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,6,5,3,7] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,6,5,7,3] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,6,7,3,5] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,6,7,5,3] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,7,3,5,6] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,7,3,6,5] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,7,5,3,6] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,7,5,6,3] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,7,6,3,5] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,4,7,6,5,3] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,5,3,4,6,7] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
[2,1,5,3,4,7,6] => [2,1,7,6,5,4,3] => [6,7,1,2,3,4,5] => [7,6,2,3,4,5,1] => ? = 2
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000153
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 83%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [.,.]
=> [1] => 1
[1,2] => [1,2] => [.,[.,.]]
=> [2,1] => 1
[2,1] => [2,1] => [[.,.],.]
=> [1,2] => 2
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 2
[2,3,1] => [2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => 2
[3,1,2] => [3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 2
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 3
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[2,3,1,4] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[2,3,4,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[2,4,1,3] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[2,4,3,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 2
[3,1,2,4] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[3,1,4,2] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 3
[3,2,4,1] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 3
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[3,4,2,1] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 3
[4,1,2,3] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2
[4,1,3,2] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2
[4,2,1,3] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 3
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 3
[4,3,1,2] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 3
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 4
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[2,1,3,4,5,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,4,5,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,4,6,5,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,4,6,7,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,4,7,5,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,4,7,6,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,5,4,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,5,4,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,5,6,4,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,5,6,7,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,5,7,4,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,5,7,6,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,6,4,5,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,6,4,7,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,6,5,4,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,6,5,7,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,6,7,4,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,6,7,5,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,7,4,5,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,7,4,6,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,7,5,4,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,7,5,6,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,7,6,4,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,3,7,6,5,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,3,5,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,3,5,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,3,6,5,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,3,6,7,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,3,7,5,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,3,7,6,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,5,3,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,5,3,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,5,6,3,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,5,6,7,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,5,7,3,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,5,7,6,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,6,3,5,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,6,3,7,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,6,5,3,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,6,5,7,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,6,7,3,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,6,7,5,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,7,3,5,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,7,3,6,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,7,5,3,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,7,5,6,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,7,6,3,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,4,7,6,5,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,5,3,4,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
[2,1,5,3,4,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,3,4,5,6,7,2] => ? = 2
Description
The number of adjacent cycles of a permutation.
This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St000383
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 83%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 83%
Values
[1] => [.,.]
=> [1] => [1] => 1
[1,2] => [.,[.,.]]
=> [2,1] => [1,1] => 1
[2,1] => [[.,.],.]
=> [1,2] => [2] => 2
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1] => 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [2,1] => 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [1,2] => 2
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => [1,2] => 2
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [1,2] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [3] => 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1] => 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1] => 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,1] => 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,1] => 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,1] => 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1] => 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,2] => 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2] => 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,2] => 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,2] => 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2] => 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,2] => 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,2] => 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,2] => 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,3] => 3
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,3] => 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,2] => 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,3] => 3
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,2] => 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,2] => 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3] => 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3] => 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3] => 3
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [4] => 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1] => 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,1,1] => 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,1,1] => 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,1,1] => 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1] => 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,2,1] => 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,2,1] => 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,2,1] => 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,2,1] => 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,2,1] => 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [2,2,1] => 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,2,1] => 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,2,1] => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,3,1] => 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,3,1] => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,2,1] => 1
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8] => [8] => ? = 8
[6,7,8,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [8,7,1,2,3,4,5,6] => [1,1,6] => ? = 6
[7,8,5,6,4,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [8,6,1,2,3,4,5,7] => [1,1,6] => ? = 6
[6,5,7,8,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [8,7,1,2,3,4,5,6] => [1,1,6] => ? = 6
[7,8,6,4,5,3,2,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [8,5,1,2,3,4,6,7] => [1,1,6] => ? = 6
[8,6,7,4,5,3,2,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [7,5,1,2,3,4,6,8] => [1,1,6] => ? = 6
[8,5,6,4,7,3,2,1] => [[[[[[.,.],.],.],.],[.,[.,.]]],.]
=> [7,6,1,2,3,4,5,8] => [1,1,6] => ? = 6
[8,5,4,6,7,3,2,1] => ?
=> ? => ? => ? = 6
[7,5,6,4,8,3,2,1] => [[[[[[.,.],.],.],.],[.,.]],[.,.]]
=> [8,6,1,2,3,4,5,7] => [1,1,6] => ? = 6
[4,5,6,7,8,3,2,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [8,7,6,5,1,2,3,4] => [1,1,1,1,4] => ? = 4
[7,8,6,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [8,4,1,2,3,5,6,7] => [1,1,6] => ? = 6
[8,6,7,5,3,4,2,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [7,4,1,2,3,5,6,8] => [1,1,6] => ? = 6
[8,7,5,6,3,4,2,1] => [[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [6,4,1,2,3,5,7,8] => [1,1,6] => ? = 6
[7,8,5,6,3,4,2,1] => [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [8,6,4,1,2,3,5,7] => [1,1,1,5] => ? = 5
[8,7,4,3,5,6,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
=> [6,5,1,2,3,4,7,8] => [1,1,6] => ? = 6
[7,8,6,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [8,3,1,2,4,5,6,7] => [1,1,6] => ? = 6
[8,6,7,5,4,2,3,1] => [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [7,3,1,2,4,5,6,8] => [1,1,6] => ? = 6
[8,7,5,6,4,2,3,1] => [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> [6,3,1,2,4,5,7,8] => [1,1,6] => ? = 6
[8,7,6,4,5,2,3,1] => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [5,3,1,2,4,6,7,8] => [1,1,6] => ? = 6
[8,6,5,7,3,2,4,1] => [[[[[[.,.],.],[.,.]],.],[.,.]],.]
=> [7,4,1,2,3,5,6,8] => [1,1,6] => ? = 6
[8,5,6,7,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [7,6,4,3,1,2,5,8] => [1,1,1,1,4] => ? = 4
[7,6,8,4,3,2,5,1] => [[[[[[.,.],.],.],[.,.]],.],[.,.]]
=> [8,5,1,2,3,4,6,7] => [1,1,6] => ? = 6
[8,7,6,4,2,3,5,1] => ?
=> ? => ? => ? = 6
[8,7,6,3,2,4,5,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
=> [5,4,1,2,3,6,7,8] => [1,1,6] => ? = 6
[7,8,3,4,2,5,6,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [8,6,5,4,1,2,3,7] => [1,1,1,1,4] => ? = 4
[8,6,4,3,5,2,7,1] => [[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [7,5,1,2,3,4,6,8] => [1,1,6] => ? = 6
[8,3,4,5,6,2,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => [1,1,1,1,4] => ? = 4
[8,6,2,3,4,5,7,1] => [[[[.,.],[.,[.,[.,.]]]],[.,.]],.]
=> [7,5,4,3,1,2,6,8] => ? => ? = 4
[8,5,4,3,2,6,7,1] => [[[[[[.,.],.],.],.],[.,[.,.]]],.]
=> [7,6,1,2,3,4,5,8] => [1,1,6] => ? = 6
[8,4,5,2,3,6,7,1] => [[[[.,.],[.,.]],[.,[.,[.,.]]]],.]
=> [7,6,5,3,1,2,4,8] => [1,1,1,1,4] => ? = 4
[8,3,2,4,5,6,7,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
=> [7,6,5,4,1,2,3,8] => [1,1,1,1,4] => ? = 4
[6,3,4,5,7,2,8,1] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
=> [8,7,5,4,1,2,3,6] => [1,1,1,1,4] => ? = 4
[2,3,4,5,6,7,8,1] => [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => ? = 2
[7,8,6,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [8,2,1,3,4,5,6,7] => [1,1,6] => ? = 6
[8,6,7,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],[.,.]],.]
=> [7,2,1,3,4,5,6,8] => [1,1,6] => ? = 6
[8,7,5,6,4,3,1,2] => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> [6,2,1,3,4,5,7,8] => [1,1,6] => ? = 6
[8,7,6,4,5,3,1,2] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [5,2,1,3,4,6,7,8] => [1,1,6] => ? = 6
[8,7,6,5,3,4,1,2] => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [4,2,1,3,5,6,7,8] => [1,1,6] => ? = 6
[7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [8,6,4,2,1,3,5,7] => [1,1,1,1,4] => ? = 4
[6,5,7,8,3,4,1,2] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
=> [8,7,4,2,1,3,5,6] => [1,1,1,1,4] => ? = 4
[7,8,4,3,5,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
=> [8,6,5,2,1,3,4,7] => [1,1,1,1,4] => ? = 4
[5,4,3,6,7,8,1,2] => [[[[.,[.,.]],.],.],[.,[.,[.,.]]]]
=> [8,7,6,2,1,3,4,5] => [1,1,1,1,4] => ? = 4
[7,6,8,5,4,2,1,3] => [[[[[[.,.],[.,.]],.],.],.],[.,.]]
=> [8,3,1,2,4,5,6,7] => [1,1,6] => ? = 6
[8,7,6,5,4,1,2,3] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [3,2,1,4,5,6,7,8] => [1,1,6] => ? = 6
[6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
=> [8,7,3,2,1,4,5,6] => [1,1,1,1,4] => ? = 4
[7,6,5,8,3,2,1,4] => [[[[[[.,.],.],[.,.]],.],.],[.,.]]
=> [8,4,1,2,3,5,6,7] => [1,1,6] => ? = 6
[7,5,6,8,2,3,1,4] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [8,6,4,3,1,2,5,7] => [1,1,1,1,4] => ? = 4
[6,7,5,8,3,1,2,4] => [[[[.,[.,.]],[.,.]],.],[.,[.,.]]]
=> [8,7,4,2,1,3,5,6] => [1,1,1,1,4] => ? = 4
[8,7,6,5,2,1,3,4] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> [4,3,1,2,5,6,7,8] => [1,1,6] => ? = 6
[7,8,5,6,2,1,3,4] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [8,6,4,3,1,2,5,7] => [1,1,1,1,4] => ? = 4
Description
The last part of an integer composition.
Matching statistic: St000069
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 92%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 92%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 1
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1] => [2,1] => ([],2)
=> ([],2)
=> 2
[1,2,3] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2
[2,3,1] => [2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,1,2] => [3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2
[3,2,1] => [3,2,1] => ([],3)
=> ([],3)
=> 3
[1,2,3,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,2,4,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[2,3,4,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[2,4,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> 3
[3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3)],4)
=> 3
[3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
[3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[4,1,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,2),(1,3)],4)
=> 3
[4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> 3
[4,3,2,1] => [4,3,2,1] => ([],4)
=> ([],4)
=> 4
[1,2,3,4,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[7,8,6,5,4,3,2,1] => [7,8,6,5,4,3,2,1] => ([(6,7)],8)
=> ([(6,7)],8)
=> ? = 7
[6,7,8,5,4,3,2,1] => [6,8,7,5,4,3,2,1] => ([(5,6),(5,7)],8)
=> ([(5,7),(6,7)],8)
=> ? = 6
[8,5,6,4,7,3,2,1] => [8,5,7,4,6,3,2,1] => ([(4,7),(5,6),(5,7)],8)
=> ([(4,7),(5,6),(5,7)],8)
=> ? = 6
[7,5,6,4,8,3,2,1] => [7,5,8,4,6,3,2,1] => ([(3,7),(4,6),(5,6),(5,7)],8)
=> ?
=> ? = 6
[4,5,6,7,8,3,2,1] => [4,8,7,6,5,3,2,1] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[7,8,5,6,3,4,2,1] => [7,8,5,6,3,4,2,1] => ([(2,7),(3,6),(4,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 5
[7,6,8,4,3,2,5,1] => [7,6,8,4,3,2,5,1] => ([(1,7),(2,7),(3,7),(4,6),(5,6)],8)
=> ?
=> ? = 6
[8,7,6,4,2,3,5,1] => [8,7,6,4,2,5,3,1] => ?
=> ?
=> ? = 6
[7,8,3,4,2,5,6,1] => [7,8,3,6,2,5,4,1] => ([(1,6),(1,7),(2,5),(3,4),(3,6),(3,7)],8)
=> ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5)],8)
=> ? = 4
[8,3,4,5,6,2,7,1] => [8,3,7,6,5,2,4,1] => ([(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 4
[8,6,2,3,4,5,7,1] => [8,6,2,7,5,4,3,1] => ?
=> ?
=> ? = 4
[6,3,4,5,7,2,8,1] => [6,3,8,7,5,2,4,1] => ([(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 4
[2,3,4,5,6,7,8,1] => [2,8,7,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
=> ?
=> ? = 2
[5,4,3,6,7,8,1,2] => [5,4,3,8,7,6,1,2] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ?
=> ? = 4
[8,7,6,5,4,1,2,3] => [8,7,6,5,4,1,3,2] => ([(5,6),(5,7)],8)
=> ([(5,7),(6,7)],8)
=> ? = 6
[8,7,6,1,2,3,4,5] => [8,7,6,1,5,4,3,2] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[8,7,5,3,2,1,4,6] => [8,7,5,3,2,1,6,4] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 6
[7,8,4,2,3,1,5,6] => [7,8,4,2,6,1,5,3] => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ?
=> ? = 4
[7,8,3,2,1,4,5,6] => [7,8,3,2,1,6,5,4] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4)],8)
=> ?
=> ? = 4
[8,6,5,4,2,1,3,7] => [8,6,5,4,2,1,7,3] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 6
[8,6,2,3,4,1,5,7] => [8,6,2,7,5,1,4,3] => ([(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 4
[8,1,2,3,4,5,6,7] => [8,1,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
=> ?
=> ? = 2
[7,6,5,4,3,1,2,8] => [7,6,5,4,3,1,8,2] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 6
[7,5,3,1,2,4,6,8] => [7,5,3,1,8,6,4,2] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 4
[5,6,3,4,1,2,7,8] => [5,8,3,7,1,6,4,2] => ([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,5),(2,7)],8)
=> ([(0,6),(1,5),(2,5),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 3
[6,5,4,1,2,3,7,8] => [6,5,4,1,8,7,3,2] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 4
[6,3,4,2,1,5,7,8] => [6,3,8,2,1,7,5,4] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ?
=> ? = 4
[5,4,3,2,1,6,7,8] => [5,4,3,2,1,8,7,6] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7)],8)
=> ? = 5
[5,1,2,3,4,6,7,8] => [5,1,8,7,6,4,3,2] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
=> ?
=> ? = 2
[3,2,1,4,5,6,7,8] => [3,2,1,8,7,6,5,4] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7)],8)
=> ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 3
[6,8,7,5,4,3,2,1] => [6,8,7,5,4,3,2,1] => ([(5,6),(5,7)],8)
=> ([(5,7),(6,7)],8)
=> ? = 6
[4,8,7,6,5,3,2,1] => [4,8,7,6,5,3,2,1] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[4,7,8,6,5,3,2,1] => [4,8,7,6,5,3,2,1] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[4,6,8,7,5,3,2,1] => [4,8,7,6,5,3,2,1] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[4,7,6,8,5,3,2,1] => [4,8,7,6,5,3,2,1] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[4,6,7,8,5,3,2,1] => [4,8,7,6,5,3,2,1] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[4,5,8,7,6,3,2,1] => [4,8,7,6,5,3,2,1] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[4,5,7,8,6,3,2,1] => [4,8,7,6,5,3,2,1] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[4,6,5,8,7,3,2,1] => [4,8,7,6,5,3,2,1] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[4,5,6,8,7,3,2,1] => [4,8,7,6,5,3,2,1] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[4,7,6,5,8,3,2,1] => [4,8,7,6,5,3,2,1] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[4,6,7,5,8,3,2,1] => [4,8,7,6,5,3,2,1] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[4,5,7,6,8,3,2,1] => [4,8,7,6,5,3,2,1] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[4,6,5,7,8,3,2,1] => [4,8,7,6,5,3,2,1] => ([(3,4),(3,5),(3,6),(3,7)],8)
=> ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 4
[2,8,7,6,5,4,3,1] => [2,8,7,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
=> ?
=> ? = 2
[2,7,8,6,5,4,3,1] => [2,8,7,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
=> ?
=> ? = 2
[2,6,8,7,5,4,3,1] => [2,8,7,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
=> ?
=> ? = 2
[2,7,6,8,5,4,3,1] => [2,8,7,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
=> ?
=> ? = 2
[2,6,7,8,5,4,3,1] => [2,8,7,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
=> ?
=> ? = 2
[2,5,8,7,6,4,3,1] => [2,8,7,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
=> ?
=> ? = 2
Description
The number of maximal elements of a poset.
Matching statistic: St000925
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 58% ●values known / values provided: 62%●distinct values known / distinct values provided: 58%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 58% ●values known / values provided: 62%●distinct values known / distinct values provided: 58%
Values
[1] => [1] => [1,0]
=> {{1}}
=> ? = 1
[1,2] => [2,1] => [1,1,0,0]
=> {{1,2}}
=> 1
[2,1] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 3
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 3
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 3
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,4,5,3,2] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 8
[7,8,6,5,4,3,2,1] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 7
[6,7,8,5,4,3,2,1] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 6
[7,8,5,6,4,3,2,1] => [2,1,4,3,5,6,7,8] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 6
[6,5,7,8,4,3,2,1] => [3,4,2,1,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,4},{3},{5},{6},{7},{8}}
=> ? = 6
[7,8,6,4,5,3,2,1] => [2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4,5},{6},{7},{8}}
=> ? = 6
[8,6,7,4,5,3,2,1] => [1,3,2,5,4,6,7,8] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 6
[8,5,6,4,7,3,2,1] => [1,4,3,5,2,6,7,8] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> {{1},{2,5},{3,4},{6},{7},{8}}
=> ? = 6
[8,5,4,6,7,3,2,1] => [1,4,5,3,2,6,7,8] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> {{1},{2,3,5},{4},{6},{7},{8}}
=> ? = 6
[7,5,6,4,8,3,2,1] => [2,4,3,5,1,6,7,8] => [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> {{1,5},{2},{3,4},{6},{7},{8}}
=> ? = 6
[4,5,6,7,8,3,2,1] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 4
[7,8,6,5,3,4,2,1] => [2,1,3,4,6,5,7,8] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4},{5,6},{7},{8}}
=> ? = 6
[8,6,7,5,3,4,2,1] => [1,3,2,4,6,5,7,8] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5,6},{7},{8}}
=> ? = 6
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 6
[7,8,5,6,3,4,2,1] => [2,1,4,3,6,5,7,8] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 5
[8,7,4,3,5,6,2,1] => [1,2,5,6,4,3,7,8] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> {{1},{2},{3,4,6},{5},{7},{8}}
=> ? = 6
[7,8,6,5,4,2,3,1] => [2,1,3,4,5,7,6,8] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4},{5},{6,7},{8}}
=> ? = 6
[8,6,7,5,4,2,3,1] => [1,3,2,4,5,7,6,8] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4},{5},{6,7},{8}}
=> ? = 6
[8,7,5,6,4,2,3,1] => [1,2,4,3,5,7,6,8] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5},{6,7},{8}}
=> ? = 6
[8,7,6,4,5,2,3,1] => [1,2,3,5,4,7,6,8] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 6
[8,6,5,7,3,2,4,1] => [1,3,4,2,6,7,5,8] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5,7},{6},{8}}
=> ? = 6
[8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5,6,7},{8}}
=> ? = 4
[7,6,8,4,3,2,5,1] => [2,3,1,5,6,7,4,8] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> {{1,3},{2},{4,7},{5},{6},{8}}
=> ? = 6
[8,7,6,4,2,3,5,1] => [1,2,3,5,7,6,4,8] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> {{1},{2},{3},{4,6,7},{5},{8}}
=> ? = 6
[8,7,6,3,2,4,5,1] => [1,2,3,6,7,5,4,8] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> {{1},{2},{3},{4,5,7},{6},{8}}
=> ? = 6
[7,8,3,4,2,5,6,1] => [2,1,6,5,7,4,3,8] => ?
=> ?
=> ? = 4
[8,6,4,3,5,2,7,1] => [1,3,5,6,4,7,2,8] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> {{1},{2,7},{3},{4,6},{5},{8}}
=> ? = 6
[8,3,4,5,6,2,7,1] => [1,6,5,4,3,7,2,8] => ?
=> ?
=> ? = 4
[8,6,2,3,4,5,7,1] => [1,3,7,6,5,4,2,8] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> {{1},{2,4,5,6,7},{3},{8}}
=> ? = 4
[8,5,4,3,2,6,7,1] => [1,4,5,6,7,3,2,8] => ?
=> ?
=> ? = 6
[8,4,5,2,3,6,7,1] => [1,5,4,7,6,3,2,8] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> {{1},{2,3,6,7},{4,5},{8}}
=> ? = 4
[8,3,2,4,5,6,7,1] => [1,6,7,5,4,3,2,8] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> {{1},{2,3,4,5,7},{6},{8}}
=> ? = 4
[6,3,4,5,7,2,8,1] => [3,6,5,4,2,7,1,8] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
=> {{1,7},{2,4,5,6},{3},{8}}
=> ? = 4
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5},{6},{7,8}}
=> ? = 6
[8,6,7,5,4,3,1,2] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2,3},{4},{5},{6},{7,8}}
=> ? = 6
[8,7,5,6,4,3,1,2] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 6
[8,7,6,4,5,3,1,2] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5},{6},{7,8}}
=> ? = 6
[8,7,6,5,3,4,1,2] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> {{1},{2},{3},{4},{5,6},{7,8}}
=> ? = 6
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 4
[6,5,7,8,3,4,1,2] => [3,4,2,1,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> {{1,2,4},{3},{5,6},{7,8}}
=> ? = 4
[7,8,4,3,5,6,1,2] => [2,1,5,6,4,3,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> {{1,2},{3,4,6},{5},{7,8}}
=> ? = 4
[5,4,3,6,7,8,1,2] => [4,5,6,3,2,1,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,6},{4},{5},{7,8}}
=> ? = 4
[7,6,8,5,4,2,1,3] => [2,3,1,4,5,7,8,6] => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> {{1,3},{2},{4},{5},{6,8},{7}}
=> ? = 6
[8,7,6,5,4,1,2,3] => [1,2,3,4,5,8,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 6
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> {{1,2,3},{4},{5},{6,7,8}}
=> ? = 4
[7,6,5,8,3,2,1,4] => [2,3,4,1,6,7,8,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3},{5,8},{6},{7}}
=> ? = 6
[7,5,6,8,2,3,1,4] => [2,4,3,1,7,6,8,5] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> {{1,3,4},{2},{5,8},{6,7}}
=> ? = 4
[6,7,5,8,3,1,2,4] => [3,2,4,1,6,8,7,5] => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> {{1,4},{2,3},{5,7,8},{6}}
=> ? = 4
[8,7,6,5,2,1,3,4] => [1,2,3,4,7,8,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> {{1},{2},{3},{4},{5,6,8},{7}}
=> ? = 6
Description
The number of topologically connected components of a set partition.
For example, the set partition {{1,5},{2,3},{4,6}} has the two connected components {1,4,5,6} and {2,3}.
The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St000340
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 67%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[2,1] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[7,8,6,5,4,3,2,1] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 7 - 1
[6,7,8,5,4,3,2,1] => [3,2,1,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 6 - 1
[7,8,5,6,4,3,2,1] => [2,1,4,3,5,6,7,8] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 6 - 1
[6,5,7,8,4,3,2,1] => [3,4,2,1,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 6 - 1
[7,8,6,4,5,3,2,1] => [2,1,3,5,4,6,7,8] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 1
[7,5,6,4,8,3,2,1] => [2,4,3,5,1,6,7,8] => [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 6 - 1
[4,5,6,7,8,3,2,1] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[7,8,6,5,3,4,2,1] => [2,1,3,4,6,5,7,8] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 6 - 1
[7,8,5,6,3,4,2,1] => [2,1,4,3,6,5,7,8] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[7,8,6,5,4,2,3,1] => [2,1,3,4,5,7,6,8] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[7,6,8,4,3,2,5,1] => [2,3,1,5,6,7,4,8] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 6 - 1
[7,8,3,4,2,5,6,1] => [2,1,6,5,7,4,3,8] => ?
=> ?
=> ? = 4 - 1
[8,3,4,5,6,2,7,1] => [1,6,5,4,3,7,2,8] => ?
=> ?
=> ? = 4 - 1
[8,5,4,3,2,6,7,1] => [1,4,5,6,7,3,2,8] => ?
=> ?
=> ? = 6 - 1
[6,3,4,5,7,2,8,1] => [3,6,5,4,2,7,1,8] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
[2,3,4,5,6,7,8,1] => [7,6,5,4,3,2,1,8] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 - 1
[7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[8,6,7,5,4,3,1,2] => [1,3,2,4,5,6,8,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 6 - 1
[8,7,5,6,4,3,1,2] => [1,2,4,3,5,6,8,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 6 - 1
[8,7,6,4,5,3,1,2] => [1,2,3,5,4,6,8,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 6 - 1
[8,7,6,5,3,4,1,2] => [1,2,3,4,6,5,8,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 6 - 1
[7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[6,5,7,8,3,4,1,2] => [3,4,2,1,6,5,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> ? = 4 - 1
[7,8,4,3,5,6,1,2] => [2,1,5,6,4,3,8,7] => [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 4 - 1
[5,4,3,6,7,8,1,2] => [4,5,6,3,2,1,8,7] => [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 4 - 1
[7,6,8,5,4,2,1,3] => [2,3,1,4,5,7,8,6] => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 6 - 1
[8,7,6,5,4,1,2,3] => [1,2,3,4,5,8,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 6 - 1
[6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 4 - 1
[7,6,5,8,3,2,1,4] => [2,3,4,1,6,7,8,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 6 - 1
[7,5,6,8,2,3,1,4] => [2,4,3,1,7,6,8,5] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[6,7,5,8,3,1,2,4] => [3,2,4,1,6,8,7,5] => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> ? = 4 - 1
[8,7,6,5,2,1,3,4] => [1,2,3,4,7,8,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 6 - 1
[7,8,5,6,2,1,3,4] => [2,1,4,3,7,8,6,5] => [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 4 - 1
[6,5,7,8,2,1,3,4] => [3,4,2,1,7,8,6,5] => [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[8,7,6,1,2,3,4,5] => [1,2,3,8,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[8,7,5,3,2,1,4,6] => [1,2,4,6,7,8,5,3] => [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 6 - 1
[7,8,4,2,3,1,5,6] => [2,1,5,7,6,8,4,3] => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 4 - 1
[7,8,3,2,1,4,5,6] => [2,1,6,7,8,5,4,3] => [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 4 - 1
[8,6,5,4,2,1,3,7] => [1,3,4,5,7,8,6,2] => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 6 - 1
[8,6,2,3,4,1,5,7] => [1,3,7,6,5,8,4,2] => ?
=> ?
=> ? = 4 - 1
[8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[6,3,4,2,1,5,7,8] => [3,6,5,7,8,4,2,1] => ?
=> ?
=> ? = 4 - 1
[6,8,7,5,4,3,2,1] => [3,1,2,4,5,6,7,8] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 6 - 1
[6,5,8,7,4,3,2,1] => [3,4,1,2,5,6,7,8] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 6 - 1
[4,8,7,6,5,3,2,1] => [5,1,2,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[4,7,8,6,5,3,2,1] => [5,2,1,3,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[4,6,8,7,5,3,2,1] => [5,3,1,2,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[4,7,6,8,5,3,2,1] => [5,2,3,1,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
[4,6,7,8,5,3,2,1] => [5,3,2,1,4,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 4 - 1
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns 110 and 001.
The following 91 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000010The length of the partition. St000288The number of ones in a binary word. St000678The number of up steps after the last double rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000912The number of maximal antichains in a poset. St000676The number of odd rises of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000068The number of minimal elements in a poset. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000167The number of leaves of an ordered tree. St000971The smallest closer of a set partition. St000054The first entry of the permutation. St000237The number of small exceedances. St000025The number of initial rises of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St000234The number of global ascents of a permutation. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000546The number of global descents of a permutation. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000354The number of recoils 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. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000990The first ascent of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001461The number of topologically connected components of the chord diagram of a permutation. St000702The number of weak deficiencies of a permutation. St000989The number of final rises of a permutation. St000843The decomposition number of a perfect matching. St000297The number of leading ones in a binary word. St000738The first entry in the last row of a standard tableau. St000308The height of the tree associated to a permutation. St000654The first descent of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St000015The number of peaks of a Dyck path. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000213The number of weak exceedances (also weak excedences) of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000287The number of connected components of a graph. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001201The grade of the simple module S0 in the special CNakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000203The number of external nodes of a binary tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St000083The number of left oriented leafs of a binary tree except the first one. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001889The size of the connectivity set of a signed permutation. St001863The number of weak excedances of a signed permutation. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001712The number of natural descents of a standard Young tableau. St001621The number of atoms of a lattice. St001200The number of simple modules in eAe with projective dimension at most 2 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St000454The largest eigenvalue of a graph if it is integral. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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