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Your data matches 38 different statistics following compositions of up to 3 maps.
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Matching statistic: St000727
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(load all 58 compositions to match this statistic)
St000727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 2
[2,1] => 1
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 3
[2,3,1] => 3
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 4
[1,2,4,3] => 3
[1,3,2,4] => 4
[1,3,4,2] => 4
[1,4,2,3] => 3
[1,4,3,2] => 2
[2,1,3,4] => 4
[2,1,4,3] => 3
[2,3,1,4] => 4
[2,3,4,1] => 4
[2,4,1,3] => 3
[2,4,3,1] => 3
[3,1,2,4] => 4
[3,1,4,2] => 4
[3,2,1,4] => 4
[3,2,4,1] => 4
[3,4,1,2] => 4
[3,4,2,1] => 4
[4,1,2,3] => 3
[4,1,3,2] => 2
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 2
[4,3,2,1] => 1
[1,2,3,4,5] => 5
[1,2,3,5,4] => 4
[1,2,4,3,5] => 5
[1,2,4,5,3] => 5
[1,2,5,3,4] => 4
[1,2,5,4,3] => 3
[1,3,2,4,5] => 5
[1,3,2,5,4] => 4
[1,3,4,2,5] => 5
[1,3,4,5,2] => 5
[1,3,5,2,4] => 4
[1,3,5,4,2] => 4
[1,4,2,3,5] => 5
[1,4,2,5,3] => 5
[1,4,3,2,5] => 5
[1,4,3,5,2] => 5
[1,4,5,2,3] => 5
[1,4,5,3,2] => 5
Description
The largest label of a leaf in the binary search tree associated with the permutation.
Alternatively, this is 1 plus the position of the last descent of the inverse of the reversal of the permutation, and 1 if there is no descent.
Matching statistic: St000013
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 2
[2,1] => [[.,.],.]
=> [1,2] => [1,0,1,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000025
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 2
[2,1] => [[.,.],.]
=> [1,2] => [1,0,1,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000444
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 2
[2,1] => [[.,.],.]
=> [1,2] => [1,0,1,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000738
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => [[1],[2]]
=> 2
[2,1] => [[.,.],.]
=> [1,2] => [[1,2]]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [[1],[2],[3]]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [[1,3],[2]]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [[1,2],[3]]
=> 3
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => [[1,2],[3]]
=> 3
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [[1,3],[2]]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [[1,2,3]]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[1,4],[2],[3]]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[1,4],[2],[3]]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[1,2],[3],[4]]
=> 4
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[1,2],[3],[4]]
=> 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[1,2],[3],[4]]
=> 4
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 3
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[1,3],[2],[4]]
=> 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[1,3],[2],[4]]
=> 4
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 4
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[1,3],[2],[4]]
=> 4
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[1,5],[2],[3],[4]]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[1,4],[2],[3],[5]]
=> 5
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[1,4],[2],[3],[5]]
=> 5
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[1,5],[2],[3],[4]]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[1,4,5],[2],[3]]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[1,3],[2],[4],[5]]
=> 5
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[1,3],[2,5],[4]]
=> 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[1,3],[2],[4],[5]]
=> 5
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[1,3],[2],[4],[5]]
=> 5
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[1,3],[2,5],[4]]
=> 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[1,3],[2,5],[4]]
=> 4
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 5
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 5
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000740
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => [1,2] => 2
[2,1] => [[.,.],.]
=> [1,2] => [2,1] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => 3
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 2
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => 3
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => 3
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [3,2,1] => 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => 4
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => 4
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => 3
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => 4
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => 4
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => 4
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => 5
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => 5
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => 5
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => 5
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => 5
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => 4
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => 5
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => 5
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => 5
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => 5
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001007
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 2
[2,1] => [[.,.],.]
=> [1,2] => [1,0,1,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001497
(load all 3 compositions to match this statistic)
(load all 3 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
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001497: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001497: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,2] => 2
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [2,1] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 3
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1,3] => 3
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 4
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 4
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 3
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 4
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 5
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 5
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 5
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 5
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 5
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 5
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 5
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 5
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 5
Description
The position of the largest weak excedence of a permutation.
Matching statistic: St001809
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001809: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001809: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 2
[2,1] => [[.,.],.]
=> [1,2] => [1,0,1,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
Description
The index of the step at the first peak of maximal height in a Dyck path.
Matching statistic: St000024
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 1 = 2 - 1
[2,1] => [[.,.],.]
=> [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 3 = 4 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 3 = 4 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 3 = 4 - 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 3 = 4 - 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 3 = 4 - 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
The following 28 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000439The position of the first down step of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000730The maximal arc length of a set partition. St000874The position of the last double rise in a Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000653The last descent of a permutation. St000147The largest part of an integer partition. St000054The first entry of the permutation. St000141The maximum drop size of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000051The size of the left subtree of a binary tree. St000216The absolute length of a permutation. St000809The reduced reflection length of the permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001480The number of simple summands of the module J^2/J^3. St000840The number of closers smaller than the largest opener in a perfect matching. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001877Number of indecomposable injective modules with projective dimension 2.
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