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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St001067
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001189
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000658
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
Description
The number of rises of length 2 of a Dyck path.
This is also the number of $(1,1)$ steps of the associated Łukasiewicz path, see [1].
A related statistic is the number of double rises in a Dyck path, [[St000024]].
Matching statistic: St000683
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000683: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000683: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> ? = 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
Description
The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps.
Matching statistic: St000932
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> ? = 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
Description
The number of occurrences of the pattern UDU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001139
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1,0]
=> ? = 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
Description
The number of occurrences of hills of size 2 in a Dyck path.
A hill of size two is a subpath beginning at height zero, consisting of two up steps followed by two down steps.
Matching statistic: St000884
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,3,1] => [3,1,2] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => [3,2,1] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [1,3,2] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => [4,1,3,2] => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [3,4,2,1] => [4,1,2,3] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,4,1] => [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,2,1,4] => [4,3,2,1] => 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => [3,2,1,4] => 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,4,3,1] => [3,1,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => [1,4,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [4,2,1,3] => [1,4,3,2] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [2,1,3,4] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [4,3,1,2] => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [4,5,2,3,1] => [5,1,4,2,3] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,2,5,3,1] => [5,3,1,2,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [4,5,2,1,3] => [5,1,4,3,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [5,3,1,4,2] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,2,1,5,3] => [5,3,2,1,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [4,5,1,3,2] => [5,1,3,2,4] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => [5,2,1,3,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [5,1,2,4,3] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [4,3,5,1,2] => [5,4,1,3,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => [5,4,2,1,3] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [4,5,3,2,1] => [5,1,2,3,4] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => [5,4,1,2,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => [5,4,3,1,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [5,4,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [3,5,2,4,1] => [4,1,5,2,3] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,2,5,4,1] => [4,3,1,2,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,5,2,1,4] => [4,1,5,3,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,2,5,1,4] => [4,3,1,5,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,2,1,5,4] => [4,3,2,1,5] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,5,1,4,2] => [4,1,3,2,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => [4,2,1,3,5] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,5,4,1,2] => [4,1,2,5,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [5,3,4,1,2] => [1,5,2,4,3] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [5,3,1,4,2] => [1,5,3,2,4] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,5,4,2,1] => [4,1,2,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [5,3,4,2,1] => [1,5,2,3,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => [1,5,4,2,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [5,3,2,1,4] => [1,5,4,3,2] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => [6,7,4,5,2,3,1] => [7,1,6,2,5,3,4] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [6,4,7,5,2,3,1] => [7,5,1,2,6,3,4] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7] => [6,7,4,2,5,3,1] => [7,1,6,4,2,3,5] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7] => [6,4,7,2,5,3,1] => [7,5,1,4,2,3,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7] => [6,4,2,7,5,3,1] => [7,5,3,1,2,4,6] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [6,7,4,5,2,1,3] => [7,1,6,2,5,4,3] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,7,5] => [6,4,7,5,2,1,3] => [7,5,1,2,6,4,3] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [6,7,4,2,5,1,3] => [7,1,6,4,2,5,3] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => [6,4,7,2,5,1,3] => [7,5,1,4,2,6,3] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5] => [6,4,2,7,5,1,3] => [7,5,3,1,2,6,4] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5] => [6,7,4,2,1,5,3] => [7,1,6,4,3,2,5] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [6,4,7,2,1,5,3] => [7,5,1,4,3,2,6] => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => [6,4,2,7,1,5,3] => [7,5,3,1,4,2,6] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => [6,4,2,1,7,5,3] => [7,5,3,2,1,4,6] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6] => [6,7,4,5,1,3,2] => [7,1,6,2,4,3,5] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6] => [6,4,7,5,1,3,2] => [7,5,1,2,4,3,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6] => [6,7,4,1,5,3,2] => [7,1,6,3,2,4,5] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6] => [6,4,7,1,5,3,2] => [7,5,1,3,2,4,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => [6,4,1,7,5,3,2] => [7,5,2,1,3,4,6] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6] => [6,7,4,5,3,1,2] => [7,1,6,2,3,5,4] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6] => [6,4,7,5,3,1,2] => [7,5,1,2,3,6,4] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [6,7,4,3,5,1,2] => [7,1,6,5,2,4,3] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => [6,4,7,3,5,1,2] => [7,5,1,6,2,4,3] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6] => [6,4,3,7,5,1,2] => [7,5,4,1,2,6,3] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6] => [6,7,4,3,1,5,2] => [7,1,6,5,3,2,4] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6] => [6,4,7,3,1,5,2] => [7,5,1,6,3,2,4] => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [6,4,3,7,1,5,2] => [7,5,4,1,3,2,6] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => [6,4,3,1,7,5,2] => [7,5,4,2,1,3,6] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [6,7,4,5,3,2,1] => [7,1,6,2,3,4,5] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7] => [6,7,4,3,5,2,1] => [7,1,6,5,2,3,4] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => [6,4,7,3,5,2,1] => [7,5,1,6,2,3,4] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,6,7] => [6,4,3,7,5,2,1] => [7,5,4,1,2,3,6] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7] => [6,7,4,3,2,5,1] => [7,1,6,5,4,2,3] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => [6,4,7,3,2,5,1] => [7,5,1,6,4,2,3] => ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,4,5,1,6,3,7] => [6,4,3,7,2,5,1] => [7,5,4,1,6,2,3] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,5,6,1,3,7] => [6,4,3,2,7,5,1] => [7,5,4,3,1,2,6] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,7,3] => [6,7,4,3,2,1,5] => [7,1,6,5,4,3,2] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => [6,4,7,3,2,1,5] => [7,5,1,6,4,3,2] => ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,6,7,3] => [6,4,3,7,2,1,5] => [7,5,4,1,6,3,2] => ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,5,6,1,7,3] => [6,4,3,2,7,1,5] => [7,5,4,3,1,6,2] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => [6,4,3,2,1,7,5] => [7,5,4,3,2,1,6] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7] => [6,7,3,5,2,4,1] => [7,1,5,2,6,3,4] => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [6,3,7,5,2,4,1] => [7,4,1,2,6,3,5] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4,7] => [6,7,3,2,5,4,1] => [7,1,5,4,2,3,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4,7] => [6,3,7,2,5,4,1] => [7,4,1,5,2,3,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4,7] => [6,3,2,7,5,4,1] => [7,4,3,1,2,5,6] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => [6,7,3,5,2,1,4] => [7,1,5,2,6,4,3] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,6,7,4] => [6,3,7,5,2,1,4] => [7,4,1,2,6,5,3] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,5,6,3,7,4] => [6,7,3,2,5,1,4] => [7,1,5,4,2,6,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,5,1,6,3,7,4] => [6,3,7,2,5,1,4] => [7,4,1,5,2,6,3] => ? = 2
Description
The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St001086
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 0
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,6,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,6,2,4,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,2,4,6,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,4,2,6,5] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,4,6,2,5] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,2,4,5,6] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,2,5,6] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,4,5,2,6] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,4,5,6,2] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,3,6] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,5,6,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,2,6,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,5,6,2,3] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,4,6,2,3,5] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,4,2,3,6,5] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,2,4,3,5,6] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,2,4,5,3,6] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,5,6,3] => 0
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,5,2,6,3,7,4] => ? = 3
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,5,6,2,3,7,4] => ? = 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,5,2,6,7,3,4] => ? = 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,5,2,6,3,4,7] => ? = 2
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,5,6,2,3,4,7] => ? = 0
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,2,3,6,4,7] => ? = 2
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,5,2,3,6,7,4] => ? = 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,2,3,7,4,6] => ? = 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,5,2,3,4,6,7] => ? = 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,2,3,4,5,7] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,7,6,8] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,4,6,8] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,7,4,6,8] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,7,2,4,6,8] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,7,8,6] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,2,4,7,8,6] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,2,5,7,4,8,6] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,5,2,7,4,8,6] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,5,7,2,4,8,6] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,2,5,7,8,4,6] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,5,2,7,8,4,6] => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,5,7,2,8,4,6] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,5,7,8,2,4,6] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4,8,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,3,2,5,8,4,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,3,5,2,8,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,3,2,5,4,6,8,7] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,3,5,2,4,6,8,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,3,2,5,6,4,8,7] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,3,5,2,6,4,8,7] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,3,5,6,2,4,8,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,3,2,5,6,8,4,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,3,5,2,6,8,4,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,5,6,2,8,4,7] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,5,6,8,2,4,7] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,3,5,2,4,6,7,8] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,3,2,5,6,4,7,8] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,3,5,2,6,4,7,8] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,3,5,6,2,4,7,8] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,3,2,5,6,7,4,8] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,5,2,6,7,4,8] => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,3,5,6,7,2,4,8] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,3,2,5,6,7,8,4] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,3,5,2,6,7,8,4] => ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,3,5,6,2,7,8,4] => ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,3,5,6,7,2,8,4] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,3,5,6,7,8,2,4] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,3,6,2,4,7,5,8] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,6,7,4,5,8] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,3,6,2,7,4,5,8] => ? = 1
Description
The number of occurrences of the consecutive pattern 132 in a permutation.
This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Matching statistic: St000441
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1] => 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,2] => 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [7,5,6,4,2,1,3] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [7,6,4,3,5,1,2] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,2,1,3] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,7,5] => [[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [6,5,7,4,2,1,3] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [6,7,4,3,5,1,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [6,7,4,5,2,1,3] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [6,7,5,3,2,1,4] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [7,5,4,3,6,1,2] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [7,5,4,6,2,1,3] => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [7,5,6,3,2,1,4] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [7,6,4,3,2,1,5] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [7,5,6,4,2,1,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [7,6,4,3,5,1,2] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,2,1,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6] => [[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [6,7,5,4,2,1,3] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [6,7,4,3,5,1,2] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [6,7,4,5,2,1,3] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [6,7,5,3,2,1,4] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [7,5,4,3,6,1,2] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [7,5,4,6,2,1,3] => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [7,5,6,3,2,1,4] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [7,6,4,3,2,1,5] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [7,6,4,3,5,1,2] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,2,1,3] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,6,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [7,5,4,3,6,1,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [7,5,4,6,2,1,3] => ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,4,5,1,6,3,7] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [7,5,6,3,2,1,4] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,5,6,1,3,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [7,6,4,3,2,1,5] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,7,3] => [[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [6,5,4,3,7,1,2] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => [[.,[.,.]],[[.,[.,[.,.]]],.]]
=> [6,5,4,7,2,1,3] => ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,6,7,3] => [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [6,5,7,3,2,1,4] => ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,5,6,1,7,3] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [6,7,4,3,2,1,5] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [7,5,4,3,2,1,6] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [7,5,6,4,2,1,3] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [7,6,4,3,5,1,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,2,1,3] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,6,7,4] => [[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [6,5,7,4,2,1,3] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [6,7,4,3,5,1,2] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,5,1,6,3,7,4] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [6,7,4,5,2,1,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,5,6,1,3,7,4] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [6,7,5,3,2,1,4] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,6,7,3,4] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [7,5,4,3,6,1,2] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,5,1,6,7,3,4] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [7,5,4,6,2,1,3] => ? = 0
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,5,6,1,7,3,4] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [7,5,6,3,2,1,4] => ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,5,6,7,1,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [7,6,4,3,2,1,5] => ? = 0
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000665
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000665: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1] => 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,2] => 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [7,5,6,4,2,1,3] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [7,6,4,3,5,1,2] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,2,1,3] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,7,5] => [[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [6,5,7,4,2,1,3] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [6,7,4,3,5,1,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [6,7,4,5,2,1,3] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [6,7,5,3,2,1,4] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [7,5,4,3,6,1,2] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [7,5,4,6,2,1,3] => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [7,5,6,3,2,1,4] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [7,6,4,3,2,1,5] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [7,5,6,4,2,1,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [7,6,4,3,5,1,2] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,2,1,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6] => [[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [6,7,5,4,2,1,3] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [6,7,4,3,5,1,2] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [6,7,4,5,2,1,3] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [6,7,5,3,2,1,4] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [7,5,4,3,6,1,2] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [7,5,4,6,2,1,3] => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [7,5,6,3,2,1,4] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [7,6,4,3,2,1,5] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [7,6,5,4,2,1,3] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [7,6,4,3,5,1,2] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,2,1,3] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,6,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [7,5,4,3,6,1,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [7,5,4,6,2,1,3] => ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,4,5,1,6,3,7] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [7,5,6,3,2,1,4] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,5,6,1,3,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [7,6,4,3,2,1,5] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,7,3] => [[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [6,5,4,3,7,1,2] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => [[.,[.,.]],[[.,[.,[.,.]]],.]]
=> [6,5,4,7,2,1,3] => ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,6,7,3] => [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [6,5,7,3,2,1,4] => ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,5,6,1,7,3] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [6,7,4,3,2,1,5] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [7,5,4,3,2,1,6] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [7,5,6,4,2,1,3] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
=> [7,6,4,3,5,1,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [7,6,4,5,2,1,3] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [7,6,5,3,2,1,4] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,6,7,4] => [[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [6,5,7,4,2,1,3] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,5,6,3,7,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
=> [6,7,4,3,5,1,2] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,5,1,6,3,7,4] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [6,7,4,5,2,1,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,5,6,1,3,7,4] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [6,7,5,3,2,1,4] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,6,7,3,4] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [7,5,4,3,6,1,2] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,5,1,6,7,3,4] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [7,5,4,6,2,1,3] => ? = 0
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,5,6,1,7,3,4] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [7,5,6,3,2,1,4] => ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,5,6,7,1,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [7,6,4,3,2,1,5] => ? = 0
Description
The number of rafts of a permutation.
Let $\pi$ be a permutation of length $n$. A small ascent of $\pi$ is an index $i$ such that $\pi(i+1)= \pi(i)+1$, see [[St000441]], and a raft of $\pi$ is a non-empty maximal sequence of consecutive small ascents.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. 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. St001948The number of augmented double ascents of a permutation. St001621The number of atoms of a lattice.
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