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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000386
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
St000386: 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
St000386: 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]
=> 0
[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]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 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]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 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]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[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]
=> 1
[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]
=> 1
[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]
=> 0
[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]
=> 1
[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]
=> 0
[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]
=> 1
[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]
=> 0
[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]
=> 1
[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]
=> 1
[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]
=> 1
[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]
=> 0
[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]
=> 1
[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]
=> 1
[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]
=> 0
[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]
=> 1
[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]
=> 0
[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]
=> 0
[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]
=> 1
[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]
=> 0
[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]
=> 0
[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]
=> 1
[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]
=> 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000660
(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
St000660: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●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]
=> 0
[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]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[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]
=> 0
[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]
=> 1
[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]
=> 0
[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]
=> 1
[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]
=> 1
[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]
=> 0
[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]
=> 1
[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]
=> 0
[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]
=> 0
[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]
=> 1
[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]
=> 0
[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]
=> 0
[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]
=> 0
[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]
=> 0
[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]
=> 1
[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]
=> 1
[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]
=> 0
[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]
=> 1
[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]
=> 0
[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]
=> 1
[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]
=> 0
[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]
=> 1
[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]
=> 1
[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]
=> 1
[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]
=> 0
[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]
=> 1
[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]
=> 1
[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]
=> 0
[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]
=> 1
[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]
=> 0
[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]
=> 0
[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]
=> 1
[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]
=> 0
[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]
=> 0
[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]
=> 1
[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]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> ? = 0
Description
The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St000201
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [.,.]
=> 1 = 0 + 1
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [[.,.],.]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [.,[.,.]]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [[.,.],[.,.]]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,8,7] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> [[.,[.,[[.,[[.,.],.]],.]]],[.,.]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,8,7] => [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> [[.,[[.,[.,[[.,.],.]]],[.,.]]],.]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,8,7] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [[.,[[.,[.,[[.,.],.]]],.]],[.,.]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,8,5,7] => [[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
=> [[.,[.,[[.,[.,.]],[.,.]]]],[.,.]]
=> ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,8,5,7] => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [[.,[[.,[[.,[.,.]],.]],[.,.]]],.]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,8,3,5,7] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],[.,[.,.]]]],.]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,8,3,5,7] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],[.,.]]],[.,.]]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7,8] => [[.,[.,.]],[.,[[.,.],[.,[.,.]]]]]
=> [[.,[.,[[.,[.,[.,.]]],.]]],[.,.]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7,8] => [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> [[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7,8] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
=> [[.,[[.,[.,[.,[.,.]]]],.]],[.,.]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5,8] => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [[.,[[.,[[.,[.,.]],.]],[.,.]]],.]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,6,7,8,5] => [[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [[.,[[.,[[.,.],[.,[.,.]]]],.]],.]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,7,8,5] => [[.,[.,.]],[.,[[.,[.,[.,.]]],.]]]
=> [[.,[.,[[.,.],[.,[.,.]]]]],[.,.]]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,6,3,7,8,5] => [[.,.],[[.,[.,.]],[[.,[.,.]],.]]]
=> [[.,[[.,[[.,.],[.,.]]],[.,.]]],.]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,5] => [[.,[.,.]],[[.,.],[[.,[.,.]],.]]]
=> [[.,[[.,[[.,.],[.,.]]],.]],[.,.]]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,6,7,3,8,5] => [[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [[.,[[.,[[.,.],.]],[.,[.,.]]]],.]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,6,7,3,8,5] => [[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
=> [[.,[[.,[[.,.],.]],[.,.]]],[.,.]]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,6,7,1,3,8,5] => [[.,[.,[.,[.,.]]]],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],[.,[.,[.,.]]]]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,6,7,8,3,5] => [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
=> [[.,[[.,[.,.]],[.,[.,[.,.]]]]],.]
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,6,7,8,3,5] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],[.,.]]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,8,6] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> [[.,[.,[[.,[[.,.],.]],.]]],[.,.]]
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,8,6] => [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> [[.,[[.,[.,[[.,.],.]]],[.,.]]],.]
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,8,6] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [[.,[[.,[.,[[.,.],.]]],.]],[.,.]]
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,7,8,5,6] => [[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
=> [[.,[.,[[.,[.,.]],[.,.]]]],[.,.]]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,7,8,3,5,6] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],[.,[.,.]]]],.]
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,8,3,5,6] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],[.,.]]],[.,.]]
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,7,8,6] => [[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
=> [[.,[[.,[.,[[.,.],[.,.]]]],.]],.]
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,7,8,6] => [[.,[.,.]],[.,[.,[[.,[.,.]],.]]]]
=> [[.,[.,[.,[[.,.],[.,.]]]]],[.,.]]
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,7,8,6] => [[.,[.,.]],[[.,.],[[.,[.,.]],.]]]
=> [[.,[[.,[[.,.],[.,.]]],.]],[.,.]]
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,7,3,8,6] => [[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [[.,[[.,[[.,.],.]],[.,[.,.]]]],.]
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,7,3,8,6] => [[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
=> [[.,[[.,[[.,.],.]],[.,.]]],[.,.]]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,4,3,8,5,6,7] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,4,1,3,8,5,6,7] => [[.,[.,.]],[.,[[.,.],[.,[.,.]]]]]
=> [[.,[.,[[.,[.,[.,.]]],.]]],[.,.]]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,4,8,3,5,6,7] => [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> [[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,4,1,8,3,5,6,7] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
=> [[.,[[.,[.,[.,[.,.]]]],.]],[.,.]]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,8,6,7] => [[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
=> [[.,[[.,[.,[[.,[.,.]],.]]],.]],.]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,1,4,5,8,3,6,7] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],[.,[.,.]]]],.]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,1,5,8,3,6,7] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],[.,.]]],[.,.]]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,8,7] => [[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
=> [[.,[[.,[.,[.,[[.,.],.]]]],.]],.]
=> ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,8,7] => [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> [[.,[[.,[.,[[.,.],.]]],[.,.]]],.]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,7] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [[.,[[.,[.,[[.,.],.]]],.]],[.,.]]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,8,7] => [[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [[.,[[.,[[.,.],.]],[.,[.,.]]]],.]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,8,7] => [[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
=> [[.,[[.,[[.,.],.]],[.,.]]],[.,.]]
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,4,5,6,1,3,8,7] => [[.,[.,[.,[.,.]]]],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],[.,[.,[.,.]]]]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,8,3,7] => [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
=> [[.,[[.,[.,.]],[.,[.,[.,.]]]]],.]
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,7] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],[.,.]]
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,7,8] => [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,4,1,3,5,6,7,8] => [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7,8] => [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> [[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> ? = 1 + 1
Description
The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size n, at least 1, with exactly one leaf node for is 2n−1, see [2].
The number of binary tree of size n, at least 3, with exactly two leaf nodes is n(n+1)2n−2, see [3].
Matching statistic: St000196
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [.,.]
=> 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [[.,.],.]
=> 0
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [.,[.,.]]
=> 0
[1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [[.,.],[.,.]]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 0
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> 0
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,8,7] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> [[.,[.,[[.,[[.,.],.]],.]]],[.,.]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,8,7] => [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> [[.,[[.,[.,[[.,.],.]]],[.,.]]],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,8,7] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [[.,[[.,[.,[[.,.],.]]],.]],[.,.]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,8,5,7] => [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [[.,[[.,[[.,[.,.]],[.,.]]],.]],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,8,5,7] => [[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
=> [[.,[.,[[.,[.,.]],[.,.]]]],[.,.]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,8,5,7] => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [[.,[[.,[[.,[.,.]],.]],[.,.]]],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,8,5,7] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,8,3,5,7] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],[.,[.,.]]]],.]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,8,3,5,7] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],[.,.]]],[.,.]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,8,1,3,5,7] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7,8] => [[.,[.,.]],[.,[[.,.],[.,[.,.]]]]]
=> [[.,[.,[[.,[.,[.,.]]],.]]],[.,.]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7,8] => [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> [[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7,8] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
=> [[.,[[.,[.,[.,[.,.]]]],.]],[.,.]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5,8] => [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [[.,[[.,[[.,[.,.]],[.,.]]],.]],.]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5,8] => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [[.,[[.,[[.,[.,.]],.]],[.,.]]],.]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5,8] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,6,7,8,5] => [[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [[.,[[.,[[.,.],[.,[.,.]]]],.]],.]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,7,8,5] => [[.,[.,.]],[.,[[.,[.,[.,.]]],.]]]
=> [[.,[.,[[.,.],[.,[.,.]]]]],[.,.]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,6,3,7,8,5] => [[.,.],[[.,[.,.]],[[.,[.,.]],.]]]
=> [[.,[[.,[[.,.],[.,.]]],[.,.]]],.]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,5] => [[.,[.,.]],[[.,.],[[.,[.,.]],.]]]
=> [[.,[[.,[[.,.],[.,.]]],.]],[.,.]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,6,7,3,8,5] => [[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [[.,[[.,[[.,.],.]],[.,[.,.]]]],.]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,6,7,3,8,5] => [[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
=> [[.,[[.,[[.,.],.]],[.,.]]],[.,.]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,6,7,1,3,8,5] => [[.,[.,[.,[.,.]]]],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],[.,[.,[.,.]]]]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,6,7,8,3,5] => [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
=> [[.,[[.,[.,.]],[.,[.,[.,.]]]]],.]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,6,7,8,3,5] => [[.,[.,.]],[[.,[.,[.,.]]],[.,.]]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],[.,.]]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,8,6] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,8,6] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> [[.,[.,[[.,[[.,.],.]],.]]],[.,.]]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,8,6] => [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> [[.,[[.,[.,[[.,.],.]]],[.,.]]],.]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,8,6] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> [[.,[[.,[.,[[.,.],.]]],.]],[.,.]]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,7,8,5,6] => [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [[.,[[.,[[.,[.,.]],[.,.]]],.]],.]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,7,8,5,6] => [[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
=> [[.,[.,[[.,[.,.]],[.,.]]]],[.,.]]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,4,1,7,3,8,5,6] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,7,8,3,5,6] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],[.,[.,.]]]],.]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,4,1,7,8,3,5,6] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],[.,.]]],[.,.]]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,7,8,1,3,5,6] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6,8] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,7,8,6] => [[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
=> [[.,[[.,[.,[[.,.],[.,.]]]],.]],.]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,7,8,6] => [[.,[.,.]],[.,[.,[[.,[.,.]],.]]]]
=> [[.,[.,[.,[[.,.],[.,.]]]]],[.,.]]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,7,8,6] => [[.,[.,.]],[[.,.],[[.,[.,.]],.]]]
=> [[.,[[.,[[.,.],[.,.]]],.]],[.,.]]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,7,3,8,6] => [[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [[.,[[.,[[.,.],.]],[.,[.,.]]]],.]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,7,3,8,6] => [[.,[.,.]],[[.,[.,.]],[[.,.],.]]]
=> [[.,[[.,[[.,.],.]],[.,.]]],[.,.]]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,4,3,8,5,6,7] => [[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [[.,[[.,[[.,[.,[.,.]]],.]],.]],.]
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,4,1,3,8,5,6,7] => [[.,[.,.]],[.,[[.,.],[.,[.,.]]]]]
=> [[.,[.,[[.,[.,[.,.]]],.]]],[.,.]]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,4,8,3,5,6,7] => [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> [[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,4,1,8,3,5,6,7] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
=> [[.,[[.,[.,[.,[.,.]]]],.]],[.,.]]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,8,6,7] => [[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
=> [[.,[[.,[.,[[.,[.,.]],.]]],.]],.]
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,8,6,7] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> ? = 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree.
Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by 2n−2.
Matching statistic: St000486
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [2,1] => 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,2,4,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => 0
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 0
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 0
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => 0
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,3,5,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,4,1,5,2] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,4,3,5,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,2,3,5,1] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,2,4,5,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,4,5,2,3] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,3,5,4,2] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,4,5,1,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,3,5,1,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,3,5,4,1] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2,5,4,1] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,2,4,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,3,4,1,5] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [4,5,3,1,2] => 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,7,4,6] => [1,5,3,6,2,7,4] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4,7,6] => [2,3,1,5,4,7,6] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,7,4,6] => [2,3,1,6,5,7,4] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4,7,6] => [3,4,1,5,2,7,6] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,7,4,6] => [3,5,1,6,2,7,4] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,7,2,4,6] => [3,4,1,6,5,7,2] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4,7,6] => [2,4,3,5,1,7,6] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,7,4,6] => [2,5,3,6,1,7,4] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,7,2,4,6] => [5,4,3,6,1,7,2] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,7,1,2,4,6] => [2,4,3,6,5,7,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,7,6] => [3,2,1,5,4,7,6] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,7,4,6] => [3,2,1,6,5,7,4] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4,7,6] => [4,2,3,5,1,7,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,7,4,6] => [5,2,3,6,1,7,4] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,5,7,1,4,6] => [4,2,3,6,5,7,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,7,4,6] => [2,1,3,6,5,7,4] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,7,6] => [2,1,4,5,3,7,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,7,4,6] => [2,1,5,6,3,7,4] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,1,5,7,3,4,6] => [2,1,4,6,5,7,3] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,5,1,3,4,7,6] => [3,2,4,5,1,7,6] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,7,4,6] => [3,2,5,6,1,7,4] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,5,1,7,3,4,6] => [5,2,4,6,1,7,3] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,5,7,1,3,4,6] => [3,2,4,6,5,7,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,5,2,7,3,4,6] => [1,5,4,6,2,7,3] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6] => [2,3,4,5,1,7,6] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,2,3,7,4,6] => [2,3,5,6,1,7,4] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [5,1,2,7,3,4,6] => [2,5,4,6,1,7,3] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,7,2,3,4,6] => [5,3,4,6,1,7,2] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [5,7,1,2,3,4,6] => [2,3,4,6,5,7,1] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,4,5,2,7,3,6] => [1,5,6,4,2,7,3] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [4,1,2,5,3,7,6] => [2,4,5,1,3,7,6] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [2,4,6,1,5,7,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [4,1,5,2,3,7,6] => [4,3,5,1,2,7,6] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [4,1,5,2,7,3,6] => [4,5,6,1,2,7,3] => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [4,1,5,7,2,3,6] => [4,3,6,1,5,7,2] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [4,5,1,2,3,7,6] => [2,3,5,4,1,7,6] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [4,5,1,2,7,3,6] => [2,5,6,4,1,7,3] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [4,5,1,7,2,3,6] => [5,3,6,4,1,7,2] => ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,5,7,1,2,3,6] => [2,3,6,4,5,7,1] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => [4,2,5,1,3,7,6] => ? = 0
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6] => [4,2,6,1,5,7,3] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6] => [3,2,5,4,1,7,6] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [5,2,6,4,1,7,3] => ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => [3,2,6,4,5,7,1] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [2,1,5,4,3,7,6] => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6] => [2,1,6,4,5,7,3] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6] => [2,1,4,3,5,7,6] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6] => [2,1,4,3,6,7,5] => ? = 1
Description
The number of cycles of length at least 3 of a permutation.
Matching statistic: St001086
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [2,1] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [3,1,2] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,1,3] => 0
[1,1,0,1,0,0]
=> [1,3,2] => [2,3,1] => 0
[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] => 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [4,2,1,3] => 0
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,2,1,4] => 0
[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]
=> [1,2,4,3] => [3,4,2,1] => 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]
=> [2,1,4,3,5] => [5,3,4,1,2] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [5,2,4,1,3] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [5,2,1,4,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,5,1,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,2,1,5,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [4,5,2,1,3] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [7,5,6,3,1,4,2] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7] => [7,5,3,6,4,1,2] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7] => [7,5,3,6,1,4,2] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7] => [7,5,3,1,6,4,2] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [5,7,6,3,4,1,2] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,7,5] => [5,7,6,3,1,4,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,7,5] => [5,7,3,6,4,1,2] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => [5,7,3,6,1,4,2] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5] => [5,7,3,1,6,4,2] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5] => [5,3,7,6,4,1,2] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [5,3,7,6,1,4,2] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => [5,3,7,1,6,4,2] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => [5,3,1,7,6,4,2] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6] => [6,5,7,3,1,4,2] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6] => [6,5,3,7,4,1,2] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6] => [6,5,3,7,1,4,2] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => [6,5,3,1,7,4,2] => ? = 1
[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] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6] => [6,7,5,3,1,4,2] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [6,7,3,5,4,1,2] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => [6,7,3,5,1,4,2] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6] => [6,7,3,1,5,4,2] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6] => [6,3,7,5,4,1,2] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6] => [6,3,7,5,1,4,2] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [6,3,7,1,5,4,2] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => [6,3,1,7,5,4,2] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,6,7] => [7,6,5,3,1,4,2] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,6,7] => [7,6,3,5,4,1,2] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => [7,6,3,5,1,4,2] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,6,7] => [7,6,3,1,5,4,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7] => [7,3,6,5,4,1,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => [7,3,6,5,1,4,2] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,4,5,1,6,3,7] => [7,3,6,1,5,4,2] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,5,6,1,3,7] => [7,3,1,6,5,4,2] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,7,3] => [3,7,6,5,4,1,2] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => [3,7,6,5,1,4,2] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,6,7,3] => [3,7,6,1,5,4,2] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,5,6,1,7,3] => [3,7,1,6,5,4,2] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => [3,1,7,6,5,4,2] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7] => [7,4,6,3,5,1,2] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [7,4,6,3,1,5,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4,7] => [7,4,3,6,5,1,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4,7] => [7,4,3,6,1,5,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4,7] => [7,4,3,1,6,5,2] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => [4,7,6,3,5,1,2] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,6,7,4] => [4,7,6,3,1,5,2] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,5,6,3,7,4] => [4,7,3,6,5,1,2] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,5,1,6,3,7,4] => [4,7,3,6,1,5,2] => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,5,6,1,3,7,4] => [4,7,3,1,6,5,2] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,6,7,3,4] => [4,3,7,6,5,1,2] => ? = 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: St000353
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1] => ? = 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 0
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 0
[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] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 0
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 0
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 0
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 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]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [2,1,5,7,6,4,3] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,7,6,5,3] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,7,5] => [[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [2,1,6,5,7,4,3] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [2,1,4,6,7,5,3] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,7,5] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [3,2,1,6,7,5,4] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [1,5,4,3,7,6,2] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,7,3,5] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [2,1,5,4,7,6,3] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,7,3,5] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [3,2,1,5,7,6,4] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [2,1,5,7,6,4,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,7,3,5,6] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,7,6,5,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6] => [[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [2,1,6,7,5,4,3] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [2,1,4,6,7,5,3] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,7,6] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [3,2,1,6,7,5,4] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,5,7,3,6] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [1,5,4,3,7,6,2] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,7,3,6] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [2,1,5,4,7,6,3] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [3,2,1,5,7,6,4] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,5,7,1,3,6] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,6,7] => [[.,[.,.]],[.,[.,[.,[.,.]]]]]
=> [2,1,7,6,5,4,3] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,7,6,5,3] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,6,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,5,6,3,7] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [1,5,4,3,7,6,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,5,6,3,7] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [2,1,5,4,7,6,3] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,4,5,1,6,3,7] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [3,2,1,5,7,6,4] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,5,6,1,3,7] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,7,3] => [[.,.],[[.,[.,[.,[.,.]]]],.]]
=> [1,6,5,4,3,7,2] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => [[.,[.,.]],[[.,[.,[.,.]]],.]]
=> [2,1,6,5,4,7,3] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,5,1,6,7,3] => [[.,[.,[.,.]]],[[.,[.,.]],.]]
=> [3,2,1,6,5,7,4] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,5,6,1,7,3] => [[.,[.,[.,[.,.]]]],[[.,.],.]]
=> [4,3,2,1,6,7,5] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
=> [5,4,3,2,1,7,6] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [2,1,5,7,6,4,3] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4,7] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,7,6,5,3] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4,7] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,6,7,4] => [[.,[.,.]],[.,[[.,[.,.]],.]]]
=> [2,1,6,5,7,4,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,5,1,6,3,7,4] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [2,1,4,6,7,5,3] => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,5,6,1,3,7,4] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [3,2,1,6,7,5,4] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,6,7,3,4] => [[.,.],[[.,[.,[.,.]]],[.,.]]]
=> [1,5,4,3,7,6,2] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,5,1,6,7,3,4] => [[.,[.,.]],[[.,[.,.]],[.,.]]]
=> [2,1,5,4,7,6,3] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,5,6,1,7,3,4] => [[.,[.,[.,.]]],[[.,.],[.,.]]]
=> [3,2,1,5,7,6,4] => ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,5,6,7,1,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,.]]]
=> [4,3,2,1,7,6,5] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,7,4,6] => [[.,[.,.]],[.,[[.,.],[.,.]]]]
=> [2,1,5,7,6,4,3] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,5,1,7,3,4,6] => [[.,[.,.]],[[.,.],[.,[.,.]]]]
=> [2,1,4,7,6,5,3] => ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,5,7,1,3,4,6] => [[.,[.,[.,.]]],[.,[.,[.,.]]]]
=> [3,2,1,7,6,5,4] => ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,5,1,3,4,7,6] => [[.,[.,.]],[.,[.,[[.,.],.]]]]
=> [2,1,6,7,5,4,3] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => [[.,.],[.,[[.,.],[[.,.],.]]]]
=> [1,4,6,7,5,3,2] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,7,6] => [[.,[.,.]],[[.,.],[[.,.],.]]]
=> [2,1,4,6,7,5,3] => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4,7,6] => [[.,[.,[.,.]]],[.,[[.,.],.]]]
=> [3,2,1,6,7,5,4] => ? = 1
Description
The number of inner valleys of a permutation.
The number of valleys including the boundary is [[St000099]].
Matching statistic: St000455
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> ? = 0 - 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> -1 = 0 - 1
[1,1,0,0]
=> [2] => ([],2)
=> ([],1)
=> ? = 0 - 1
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> -1 = 0 - 1
[1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> -1 = 0 - 1
[1,1,1,0,0,0]
=> [3] => ([],3)
=> ([],1)
=> ? = 0 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 - 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> -1 = 0 - 1
[1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> -1 = 0 - 1
[1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> -1 = 0 - 1
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ([],1)
=> ? = 0 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 0 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> -1 = 0 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> -1 = 0 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> -1 = 0 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> -1 = 0 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? = 0 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1 = 0 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1 = 0 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000125
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000125: Binary trees ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000125: Binary trees ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 0
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],.],.]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [.,[[[.,.],[.,[.,.]]],.]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [.,[[.,[[[.,.],.],.]],.]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[[.,.],.]],.]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],[.,.]],.]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [.,[[[.,[[.,.],.]],.],.]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[[[[.,.],[.,.]],.],.]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [.,[[[.,[.,[.,.]]],.],.]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [.,[[[[.,[.,.]],.],.],.]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [.,[[.,.],[[.,[.,.]],.]]]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[[[[.,.],.],.],[.,.]]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> ? = 0
[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]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [.,[[[.,.],[.,[[.,[.,.]],.]]],.]]
=> ? = 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]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> ? = 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]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [.,[[[.,.],[[.,.],[.,[.,.]]]],.]]
=> ? = 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]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [.,[[[[.,[.,.]],.],[.,[.,.]]],.]]
=> ? = 1
[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]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> ? = 1
[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]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [.,[[[.,.],[.,[[[.,.],.],.]]],.]]
=> ? = 2
[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]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [.,[[.,[[[.,.],[[.,.],.]],.]],.]]
=> ? = 1
[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]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[[.,.],[[.,.],.]]],.]]
=> ? = 1
[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]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [.,[[[[.,[.,.]],.],[[.,.],.]],.]]
=> ? = 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]
=> [1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,[[[[.,.],.],[.,.]],.]],.]]
=> ? = 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]
=> [1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [.,[[[.,.],[[[.,.],.],[.,.]]],.]]
=> ? = 2
[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]
=> [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [.,[[[[.,[[.,.],.]],.],[.,.]],.]]
=> ? = 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]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [.,[[[[[.,.],[.,.]],.],[.,.]],.]]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> ? = 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]
=> [1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [.,[[[.,.],[.,[[.,.],[.,.]]]],.]]
=> ? = 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]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> ? = 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]
=> [1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [.,[[[.,.],[[.,[.,.]],[.,.]]],.]]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [.,[[[[.,[.,[.,.]]],.],[.,.]],.]]
=> ? = 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]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[[.,[.,[[.,.],.]]],.]],.]]
=> ? = 0
[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]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [.,[[[.,.],[.,[.,[[.,.],.]]]],.]]
=> ? = 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]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [.,[[.,[[[.,[[.,.],.]],.],.]],.]]
=> ? = 1
[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]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [.,[[[.,.],[[.,[[.,.],.]],.]],.]]
=> ? = 1
[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]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [.,[[[[.,.],.],[.,[[.,.],.]]],.]]
=> ? = 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]
=> [1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [.,[[.,[[[[.,.],[.,.]],.],.]],.]]
=> ? = 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]
=> [1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],[[[.,.],[.,.]],.]],.]]
=> ? = 2
[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]
=> [1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [.,[[[[.,.],.],[[.,.],[.,.]]],.]]
=> ? = 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]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> ? = 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]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> ? = 1
[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]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [.,[[.,[[[.,[.,[.,.]]],.],.]],.]]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [.,[[[.,.],[[.,[.,[.,.]]],.]],.]]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [.,[[[[.,.],.],[.,[.,[.,.]]]],.]]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [.,[[.,[[[[.,[.,.]],.],.],.]],.]]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [.,[[[[.,.],.],[[.,[.,.]],.]],.]]
=> ? = 1
[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]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [.,[[[[[.,.],.],.],[.,[.,.]]],.]]
=> ? = 1
[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]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> ? = 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]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [.,[[[.,.],[[[[.,.],.],.],.]],.]]
=> ? = 2
[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]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [.,[[[[.,.],.],[[[.,.],.],.]],.]]
=> ? = 2
[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]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [.,[[[[[.,.],.],.],[[.,.],.]],.]]
=> ? = 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]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [.,[[.,[[.,.],[[.,[.,.]],.]]],.]]
=> ? = 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]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> ? = 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]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[[.,[[[.,.],.],[.,[.,.]]]],.]]
=> ? = 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]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [.,[[[.,[[.,.],.]],[.,[.,.]]],.]]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [.,[[[[.,.],[.,[.,.]]],[.,.]],.]]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[.,[[.,.],[[[.,.],.],.]]],.]]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [.,[[[.,[.,.]],[[[.,.],.],.]],.]]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[.,[[[.,.],.],[[.,.],.]]],.]]
=> ? = 1
Description
The number of occurrences of the contiguous pattern {{{[.,[[[.,.],.],.]]}}} in a binary tree.
[[oeis:A005773]] counts binary trees avoiding this pattern.
Matching statistic: St000663
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000663: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000663: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [1] => 0
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [3,1,2] => 0
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [3,1,2] => 0
[1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 0
[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] => 0
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> [3,4,1,2] => 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 0
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 0
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 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]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[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] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 0
[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] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[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] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => ? = 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] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [[.,.],[[.,.],[[.,[.,.]],.]]]
=> [6,5,7,3,4,1,2] => ? = 1
[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] => ? = 2
[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] => ? = 1
[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] => ? = 1
[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] => ? = 2
[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] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => ? = 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] => ? = 1
[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] => ? = 0
[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] => ? = 1
[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] => ? = 1
[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] => ? = 2
[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] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => [[.,.],[[.,.],[.,[.,[.,.]]]]]
=> [7,6,5,3,4,1,2] => ? = 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] => ? = 1
[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] => ? = 1
[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] => ? = 2
[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] => ? = 1
[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] => ? = 2
[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] => ? = 2
[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] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7] => [[.,.],[[.,.],[[.,.],[.,.]]]]
=> [7,5,6,3,4,1,2] => ? = 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] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => [[.,.],[[.,.],[[.,[.,.]],.]]]
=> [6,5,7,3,4,1,2] => ? = 1
[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] => ? = 2
[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] => ? = 1
Description
The number of right floats of a permutation.
Let π be a permutation of length n. A raft of π is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right float is a large ascent not consecutive to any raft of π.
See Definition 3.10 and Example 3.11 in [1].
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000092The number of outer peaks of a permutation. St000646The number of big ascents of a permutation. St000837The number of ascents of distance 2 of a permutation. St001330The hat guessing number of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af.
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