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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St001067
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ 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%
Mp00072: Permutations —binary search tree: left to right⟶ 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]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 0
[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]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001189
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
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
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 0
[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]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 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 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000658: 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]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[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]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
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
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
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
St000683: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 0
[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]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
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
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ 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%
Mp00072: Permutations —binary search tree: left to right⟶ 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]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 0
[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]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
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
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
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
St001139: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 0
[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]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
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: St000441
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ 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%
Mp00072: Permutations —binary search tree: left to right⟶ 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]
=> [2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[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]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,6,7,1,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,5,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,1,7,5,6] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [6,5,7,4,3,1,2] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [6,5,7,4,3,1,2] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,7,4,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,7,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,5,1,6,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,7,4,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,4,7,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,6,1,4,7,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [6,5,7,4,3,1,2] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [6,5,4,7,3,1,2] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [6,5,4,7,3,1,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,4,5,7,3,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,4,1,5,3,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,4,1,5,7,3,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,4,5,1,3,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,4,5,1,7,3,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,3,5,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,1,3,5,7,4,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
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
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ 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%
Mp00072: Permutations —binary search tree: left to right⟶ 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]
=> [2,3,1] => [[.,.],[.,.]]
=> [3,1,2] => 1
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[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]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,6,7,1,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,5,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,1,7,5,6] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [6,5,7,4,3,1,2] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [6,5,7,4,3,1,2] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,7,4,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,7,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,5,1,6,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,7,4,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,3,1,2] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,4,7,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,6,1,4,7,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,3,1,2] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [6,5,7,4,3,1,2] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [6,5,4,7,3,1,2] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [6,5,4,7,3,1,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,4,5,7,3,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,4,1,5,3,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,4,1,5,7,3,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,4,5,1,3,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,4,5,1,7,3,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,4,1,3,5,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [6,7,5,3,4,1,2] => ? = 3
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,3,1,2] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,1,3,5,7,4,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => ? = 3
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.
Matching statistic: St001222
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St001222: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 1
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 3
Description
Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module.
Matching statistic: St001223
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●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]
=> [2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 0
[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]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,1,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,4,6,1,7,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,6,7,1,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,5,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,1,7,5,6] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,7,1,5,6] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,1,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,7,4,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,3,5,1,6,7,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,3,5,1,6,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,3,5,6,1,7,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,7,4,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,7,1,4,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => [[.,.],[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => [[.,.],[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,4,7,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,3,6,1,4,7,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,6,1,7,4,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,3,6,7,1,4,5] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => [[.,.],[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,7,5,6] => [[.,.],[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,3,1,7,4,5,6] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,3,7,1,4,5,6] => [[.,.],[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,4,5,7,3,6] => [[.,.],[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
Description
Number 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.
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