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Your data matches 14 different statistics following compositions of up to 3 maps.
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Matching statistic: St001037
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> 0
[[.,.],.]
=> [1,0,1,0]
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000386
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> ? = 2
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1
[[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 2
[[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 2
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> ? = 2
[[.,.],[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> ? = 2
[[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> ? = 2
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 2
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 2
[[.,.],[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2
[[.,.],[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 2
[[.,.],[.,[[[.,.],.],[[.,.],.]]]]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[[.,.],[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 2
[[.,.],[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 2
[[.,.],[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 2
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> [1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1
[[.,.],[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> ? = 2
[[.,.],[.,[[[.,.],[[.,.],.]],.]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 2
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 2
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[[.,.],[.,[[[[.,.],[.,.]],.],.]]]
=> [1,0,1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> ? = 2
[[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 2
[[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 3
[[.,.],[[.,[.,.]],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2
[[.,.],[[.,[.,.]],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 2
[[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 2
[[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2
[[.,.],[[.,[[.,.],.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[[.,.],[[.,[[.,.],.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 2
[[.,.],[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 2
[[.,.],[[[.,[.,.]],.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2
[[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[[.,.],[[.,[.,[[.,.],.]]],[.,.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
[[.,.],[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 3
[[.,.],[[.,[[.,[.,.]],.]],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2
[[.,.],[[.,[[[.,.],.],.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> ? = 2
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000201
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [.,.]
=> 1 = 0 + 1
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1 = 0 + 1
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,.]]],.],.]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,[.,.]],.],.],.]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[.,[.,.]],.],[.,.]]
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[.,[.,.]]],.]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[.,.]],.]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 2 = 1 + 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> ? = 1 + 1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [[[[.,.],[.,[.,[.,[.,.]]]]],.],.]
=> ? = 1 + 1
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> ? = 2 + 1
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [[[[.,.],[[.,[.,[.,.]]],.]],.],.]
=> ? = 1 + 1
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> ? = 1 + 1
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[[.,.],[[.,[.,[.,[.,.]]]],.]],.]
=> ? = 1 + 1
[[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
=> [[[.,.],[[[.,[.,[.,.]]],.],.]],.]
=> ? = 1 + 1
[[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> [[[[.,.],[[.,[.,.]],[.,.]]],.],.]
=> ? = 2 + 1
[[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[[.,.],[.,[[.,[.,[.,.]]],.]]],.]
=> ? = 1 + 1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> [[[[.,.],[.,[.,[.,.]]]],[.,.]],.]
=> ? = 2 + 1
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [[[[[.,.],[.,[.,.]]],.],[.,.]],.]
=> ? = 2 + 1
[[.,.],[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> [[[.,.],[[.,[.,[.,.]]],[.,.]]],.]
=> ? = 2 + 1
[[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> ? = 1 + 1
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [[[[[.,.],[.,.]],[.,[.,.]]],.],.]
=> ? = 2 + 1
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 2 + 1
[[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> [[[[.,.],[[.,.],[.,[.,.]]]],.],.]
=> ? = 2 + 1
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> ? = 3 + 1
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [[[[.,.],[.,.]],[.,[.,[.,.]]]],.]
=> ? = 2 + 1
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> [[[.,.],[[[.,[.,.]],[.,.]],.]],.]
=> ? = 2 + 1
[[.,.],[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> [[[.,.],[[.,[.,.]],[.,[.,.]]]],.]
=> ? = 2 + 1
[[.,.],[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> ? = 2 + 1
[[.,.],[.,[[[.,.],.],[[.,.],.]]]]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> [[[[[.,.],[.,.]],.],[.,[.,.]]],.]
=> ? = 2 + 1
[[.,.],[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [[[[.,.],[[.,[.,.]],.]],[.,.]],.]
=> ? = 2 + 1
[[.,.],[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> [[[.,.],[.,[[.,[.,.]],[.,.]]]],.]
=> ? = 2 + 1
[[.,.],[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [[[[.,.],[[.,.],[.,.]]],[.,.]],.]
=> ? = 3 + 1
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
=> [[[.,.],[[[.,[.,.]],.],[.,.]]],.]
=> ? = 2 + 1
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [[[[.,.],[.,[[.,[.,.]],.]]],.],.]
=> ? = 1 + 1
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [[[[[.,.],[[.,[.,.]],.]],.],.],.]
=> ? = 1 + 1
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [[[.,.],[[.,[[.,[.,.]],.]],.]],.]
=> ? = 1 + 1
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,0]
=> [[[[.,.],[.,[[.,.],[.,.]]]],.],.]
=> ? = 2 + 1
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> [1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[[.,.],[.,[.,[[.,[.,.]],.]]]],.]
=> ? = 1 + 1
[[.,.],[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [[[[[.,.],[[.,.],[.,.]]],.],.],.]
=> ? = 2 + 1
[[.,.],[.,[[[.,.],[[.,.],.]],.]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [[[[.,.],[.,.]],[[.,[.,.]],.]],.]
=> ? = 2 + 1
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
=> [[[.,.],[[.,[[.,.],[.,.]]],.]],.]
=> ? = 2 + 1
[[.,.],[.,[[[[.,.],.],[.,.]],.]]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [[[[[.,.],[[.,.],.]],[.,.]],.],.]
=> ? = 2 + 1
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> [[[[.,.],[[[.,[.,.]],.],.]],.],.]
=> ? = 1 + 1
[[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> [[[.,.],[.,[[[.,[.,.]],.],.]]],.]
=> ? = 1 + 1
[[.,.],[.,[[[[.,.],[.,.]],.],.]]]
=> [1,0,1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> [[[[.,.],[[[.,.],[.,.]],.]],.],.]
=> ? = 2 + 1
[[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [[[.,.],[[[[.,[.,.]],.],.],.]],.]
=> ? = 1 + 1
[[.,.],[.,[[[[[.,.],.],.],.],.]]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> ? = 1 + 1
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [[.,.],[[[.,[.,[.,[.,.]]]],.],.]]
=> ? = 1 + 1
[[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [[.,.],[[[[.,[.,[.,.]]],.],.],.]]
=> ? = 1 + 1
[[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [[.,.],[[.,[[.,[.,[.,.]]],.]],.]]
=> ? = 1 + 1
[[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [[.,.],[[[.,[.,[.,.]]],[.,.]],.]]
=> ? = 2 + 1
[[.,.],[[.,.],[.,[[[.,.],.],.]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> ? = 1 + 1
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [[.,.],[[[[.,[.,.]],[.,.]],.],.]]
=> ? = 2 + 1
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,[.,.]],[.,[.,.]]],.]]
=> ? = 2 + 1
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [[.,.],[[.,[[.,[.,.]],[.,.]]],.]]
=> ? = 2 + 1
[[.,.],[[.,.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [[.,.],[[[[.,[.,.]],.],[.,.]],.]]
=> ? = 2 + 1
[[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [[.,.],[[[.,[[.,[.,.]],.]],.],.]]
=> ? = 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
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [.,.]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> 0
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,.],[.,.]]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 0
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[.,[[.,[.,.]],.]],.]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,[.,.]],.],.],.]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[.,[.,.]],.],[.,.]]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[.,[.,.]]],.]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[.,.],[.,.]],.],.]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[.,.]],.]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [.,[[[.,.],.],[.,.]]]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[.,.],.],.],[.,.]]
=> 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [[[[[.,.],[.,[.,[.,.]]]],.],.],.]
=> ? = 1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [[[[.,.],[.,[.,[.,[.,.]]]]],.],.]
=> ? = 1
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> ? = 2
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [[[[.,.],[[.,[.,[.,.]]],.]],.],.]
=> ? = 1
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [[[[[[.,.],[.,[.,.]]],.],.],.],.]
=> ? = 1
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[[.,.],[[.,[.,[.,[.,.]]]],.]],.]
=> ? = 1
[[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,1,0,0,0,0]
=> [[[.,.],[[[.,[.,[.,.]]],.],.]],.]
=> ? = 1
[[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> [[[[.,.],[[.,[.,.]],[.,.]]],.],.]
=> ? = 2
[[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0]
=> [[[.,.],[.,[[.,[.,[.,.]]],.]]],.]
=> ? = 1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> [[[[.,.],[.,[.,[.,.]]]],[.,.]],.]
=> ? = 2
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [[[[[.,.],[.,[.,.]]],.],[.,.]],.]
=> ? = 2
[[.,.],[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> [[[.,.],[[.,[.,[.,.]]],[.,.]]],.]
=> ? = 2
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> ? = 2
[[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[.,.],[.,[.,[.,[.,[.,.]]]]]],.]
=> ? = 1
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [[[[[.,.],[.,.]],[.,[.,.]]],.],.]
=> ? = 2
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 2
[[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> [[[[.,.],[[.,.],[.,[.,.]]]],.],.]
=> ? = 2
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> ? = 3
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [[[[.,.],[.,.]],[.,[.,[.,.]]]],.]
=> ? = 2
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> [[[.,.],[[[.,[.,.]],[.,.]],.]],.]
=> ? = 2
[[.,.],[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> [[[.,.],[[.,[.,.]],[.,[.,.]]]],.]
=> ? = 2
[[.,.],[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> [[[[[[.,.],[.,.]],.],[.,.]],.],.]
=> ? = 2
[[.,.],[.,[[[.,.],.],[[.,.],.]]]]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> [[[[[.,.],[.,.]],.],[.,[.,.]]],.]
=> ? = 2
[[.,.],[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [[[[.,.],[[.,[.,.]],.]],[.,.]],.]
=> ? = 2
[[.,.],[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> [[[.,.],[.,[[.,[.,.]],[.,.]]]],.]
=> ? = 2
[[.,.],[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [[[[.,.],[[.,.],[.,.]]],[.,.]],.]
=> ? = 3
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
=> [[[.,.],[[[.,[.,.]],.],[.,.]]],.]
=> ? = 2
[[.,.],[.,[[[[.,.],.],.],[.,.]]]]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> ? = 2
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [[[[.,.],[.,[[.,[.,.]],.]]],.],.]
=> ? = 1
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [[[[[.,.],[[.,[.,.]],.]],.],.],.]
=> ? = 1
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [[[.,.],[[.,[[.,[.,.]],.]],.]],.]
=> ? = 1
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,0]
=> [[[[.,.],[.,[[.,.],[.,.]]]],.],.]
=> ? = 2
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> [1,0,1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [[[.,.],[.,[.,[[.,[.,.]],.]]]],.]
=> ? = 1
[[.,.],[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [[[[[.,.],[[.,.],[.,.]]],.],.],.]
=> ? = 2
[[.,.],[.,[[[.,.],[[.,.],.]],.]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [[[[.,.],[.,.]],[[.,[.,.]],.]],.]
=> ? = 2
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
=> [[[.,.],[[.,[[.,.],[.,.]]],.]],.]
=> ? = 2
[[.,.],[.,[[[[.,.],.],[.,.]],.]]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [[[[[.,.],[[.,.],.]],[.,.]],.],.]
=> ? = 2
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> [[[[.,.],[[[.,[.,.]],.],.]],.],.]
=> ? = 1
[[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> [[[.,.],[.,[[[.,[.,.]],.],.]]],.]
=> ? = 1
[[.,.],[.,[[[[.,.],[.,.]],.],.]]]
=> [1,0,1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> [[[[.,.],[[[.,.],[.,.]],.]],.],.]
=> ? = 2
[[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [[[.,.],[[[[.,[.,.]],.],.],.]],.]
=> ? = 1
[[.,.],[.,[[[[[.,.],.],.],.],.]]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> ? = 1
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [[.,.],[[[.,[.,[.,[.,.]]]],.],.]]
=> ? = 1
[[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [[.,.],[[[[.,[.,[.,.]]],.],.],.]]
=> ? = 1
[[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [[.,.],[[.,[[.,[.,[.,.]]],.]],.]]
=> ? = 1
[[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [[.,.],[[[.,[.,[.,.]]],[.,.]],.]]
=> ? = 2
[[.,.],[[.,.],[.,[[[.,.],.],.]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> ? = 1
[[.,.],[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [[.,.],[[[[.,[.,.]],[.,.]],.],.]]
=> ? = 2
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,[.,.]],[.,[.,.]]],.]]
=> ? = 2
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [[.,.],[[.,[[.,[.,.]],[.,.]]],.]]
=> ? = 2
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: St000159
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [[1],[]]
=> [1]
=> 1 = 0 + 1
[.,[.,.]]
=> [1,1,0,0]
=> [[2],[]]
=> [2]
=> 1 = 0 + 1
[[.,.],.]
=> [1,0,1,0]
=> [[1,1],[]]
=> [1,1]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> [3]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [2,2]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [3,3]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [3,2]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [3,3]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [2,2,1]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> [3,3,3]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> [4,4]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> [3,3,2]
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [3,3,3]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> [2,2,2,2]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [4,4]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [3,3,3]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [3,3,2]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [4,4]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> [4,3]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> [3,2,2]
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [4,3]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> [4,2]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> [5]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [3,3,1]
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1]
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [3,3,2]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [4,2]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [2,2,1,1]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1]
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [3,2,2]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [4,3]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [3,2,1]
=> 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [3,2,2]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4],[]]
=> [4,4,4,4]
=> ? = 0 + 1
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[5,5,5],[]]
=> [5,5,5]
=> ? = 0 + 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[4,4,4,3],[]]
=> [4,4,4,3]
=> ? = 1 + 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[4,4,4,4],[1]]
=> [4,4,4,4]
=> ? = 0 + 1
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,3],[]]
=> [3,3,3,3,3]
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[5,5,5],[1]]
=> [5,5,5]
=> ? = 0 + 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[4,4,4,4],[1,1]]
=> [4,4,4,4]
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [[4,4,4,3],[1]]
=> [4,4,4,3]
=> ? = 1 + 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [[5,5,5],[2]]
=> [5,5,5]
=> ? = 0 + 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[5,5,4],[]]
=> [5,5,4]
=> ? = 1 + 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [[4,4,3,3],[]]
=> [4,4,3,3]
=> ? = 1 + 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [[5,5,4],[1]]
=> [5,5,4]
=> ? = 1 + 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [[5,5,3],[]]
=> [5,5,3]
=> ? = 1 + 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,2],[]]
=> [4,4,4,2]
=> ? = 1 + 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,2],[]]
=> [3,3,3,3,2]
=> ? = 1 + 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4,2],[1]]
=> [4,4,4,2]
=> ? = 1 + 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [[4,4,3,2],[]]
=> [4,4,3,2]
=> ? = 2 + 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [[4,4,4,3],[1,1]]
=> [4,4,4,3]
=> ? = 1 + 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [[5,5,3],[1]]
=> [5,5,3]
=> ? = 1 + 1
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2,2],[]]
=> [3,3,3,2,2]
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [[4,4,3,3],[1]]
=> [4,4,3,3]
=> ? = 1 + 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [[5,5,4],[2]]
=> [5,5,4]
=> ? = 1 + 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3,2],[1]]
=> [4,4,3,2]
=> ? = 2 + 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [[4,4,3,3],[1,1]]
=> [4,4,3,3]
=> ? = 1 + 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[4,4,4,4],[2]]
=> [4,4,4,4]
=> ? = 0 + 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [[3,3,3,3,3],[1]]
=> [3,3,3,3,3]
=> ? = 0 + 1
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [[4,4,4,4],[2,1]]
=> [4,4,4,4]
=> ? = 0 + 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [[4,4,4,3],[2]]
=> [4,4,4,3]
=> ? = 1 + 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [[5,5,5],[3]]
=> [5,5,5]
=> ? = 0 + 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3,2],[1]]
=> [3,3,3,3,2]
=> ? = 1 + 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [[4,4,4,2],[2]]
=> [4,4,4,2]
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [[4,4,4,3],[2,1]]
=> [4,4,4,3]
=> ? = 1 + 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [[3,3,3,2,2],[1]]
=> [3,3,3,2,2]
=> ? = 1 + 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [[3,3,3,3,3],[1,1]]
=> [3,3,3,3,3]
=> ? = 0 + 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [[4,4,4,4],[2,2]]
=> [4,4,4,4]
=> ? = 0 + 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3,2],[1,1]]
=> [3,3,3,3,2]
=> ? = 1 + 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [[3,3,3,3,3],[1,1,1]]
=> [3,3,3,3,3]
=> ? = 0 + 1
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[5,5,5],[1,1]]
=> [5,5,5]
=> ? = 0 + 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [[4,4,4,4],[1,1,1]]
=> [4,4,4,4]
=> ? = 0 + 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [[5,5,5],[2,1]]
=> [5,5,5]
=> ? = 0 + 1
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [[5,5,4],[1,1]]
=> [5,5,4]
=> ? = 1 + 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [[4,4,4,3],[1,1,1]]
=> [4,4,4,3]
=> ? = 1 + 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [[5,5,3],[1,1]]
=> [5,5,3]
=> ? = 1 + 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [[5,5,4],[2,1]]
=> [5,5,4]
=> ? = 1 + 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [[4,4,3,3],[1,1,1]]
=> [4,4,3,3]
=> ? = 1 + 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [[4,4,4,4],[2,1,1]]
=> [4,4,4,4]
=> ? = 0 + 1
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [[5,5,5],[3,1]]
=> [5,5,5]
=> ? = 0 + 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [[4,4,4,3],[2,1,1]]
=> [4,4,4,3]
=> ? = 1 + 1
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [[4,4,4,4],[2,2,1]]
=> [4,4,4,4]
=> ? = 0 + 1
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [[3,3,3,3,3],[1,1,1,1]]
=> [3,3,3,3,3]
=> ? = 0 + 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000318
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [[1],[]]
=> [1]
=> 2 = 0 + 2
[.,[.,.]]
=> [1,1,0,0]
=> [[2],[]]
=> [2]
=> 2 = 0 + 2
[[.,.],.]
=> [1,0,1,0]
=> [[1,1],[]]
=> [1,1]
=> 2 = 0 + 2
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2]
=> 2 = 0 + 2
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> [3]
=> 2 = 0 + 2
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1]
=> 3 = 1 + 2
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [2,2]
=> 2 = 0 + 2
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1]
=> 2 = 0 + 2
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3]
=> 2 = 0 + 2
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2]
=> 2 = 0 + 2
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [3,3]
=> 2 = 0 + 2
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2]
=> 3 = 1 + 2
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4]
=> 2 = 0 + 2
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1]
=> 3 = 1 + 2
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1]
=> 3 = 1 + 2
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [3,2]
=> 3 = 1 + 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1]
=> 3 = 1 + 2
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 2 = 0 + 2
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [3,3]
=> 2 = 0 + 2
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [2,2,1]
=> 3 = 1 + 2
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 2 = 0 + 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 2 = 0 + 2
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> [3,3,3]
=> 2 = 0 + 2
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> [4,4]
=> 2 = 0 + 2
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> [3,3,2]
=> 3 = 1 + 2
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [3,3,3]
=> 2 = 0 + 2
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> [2,2,2,2]
=> 2 = 0 + 2
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [4,4]
=> 2 = 0 + 2
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [3,3,3]
=> 2 = 0 + 2
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [3,3,2]
=> 3 = 1 + 2
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [4,4]
=> 2 = 0 + 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> [4,3]
=> 3 = 1 + 2
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> [3,2,2]
=> 3 = 1 + 2
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [4,3]
=> 3 = 1 + 2
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> [4,2]
=> 3 = 1 + 2
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> [5]
=> 2 = 0 + 2
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1]
=> 3 = 1 + 2
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1]
=> 3 = 1 + 2
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [3,3,1]
=> 3 = 1 + 2
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1]
=> 4 = 2 + 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1]
=> 3 = 1 + 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [3,3,2]
=> 3 = 1 + 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [4,2]
=> 3 = 1 + 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [2,2,1,1]
=> 3 = 1 + 2
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1]
=> 3 = 1 + 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [3,2,2]
=> 3 = 1 + 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [4,3]
=> 3 = 1 + 2
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [3,2,1]
=> 4 = 2 + 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [3,2,2]
=> 3 = 1 + 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1]
=> 3 = 1 + 2
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> [4,4,4]
=> ? = 0 + 2
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3],[]]
=> [3,3,3,3]
=> ? = 0 + 2
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4],[1]]
=> [4,4,4]
=> ? = 0 + 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [3,3,3,3]
=> ? = 0 + 2
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [[4,4,4],[2]]
=> [4,4,4]
=> ? = 0 + 2
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [3,3,3,3]
=> ? = 0 + 2
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> [4,4,4]
=> ? = 0 + 2
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[4,4,4],[2,1]]
=> [4,4,4]
=> ? = 0 + 2
[.,[[.,.],[[[.,.],.],.]]]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1,1]]
=> [3,3,3,3]
=> ? = 0 + 2
[.,[[[.,.],.],[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[4,4,4],[2,2]]
=> [4,4,4]
=> ? = 0 + 2
[[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> [3,3,3,3]
=> ? = 0 + 2
[[.,[.,[.,[[.,.],.]]]],.]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> [4,4,4]
=> ? = 0 + 2
[[.,[.,[[.,[.,.]],.]]],.]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> [3,3,3,3]
=> ? = 0 + 2
[[.,[[.,.],[.,[.,.]]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> [4,4,4]
=> ? = 0 + 2
[[.,[[.,.],[[.,.],.]]],.]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> [3,3,3,3]
=> ? = 0 + 2
[[.,[[[.,.],.],[.,.]]],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> [4,4,4]
=> ? = 0 + 2
[[[.,[.,[.,[.,.]]]],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> [3,3,3,3]
=> ? = 0 + 2
[[[.,[[.,.],[.,.]]],.],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2,1]]
=> [3,3,3,3]
=> ? = 0 + 2
[[[.,[[[.,.],.],.]],.],.]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [[4,4,4],[3,3]]
=> [4,4,4]
=> ? = 0 + 2
[[[[.,[[.,.],.]],.],.],.]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> ? = 0 + 2
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4],[]]
=> [4,4,4,4]
=> ? = 0 + 2
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[5,5,5],[]]
=> [5,5,5]
=> ? = 0 + 2
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[4,4,4,3],[]]
=> [4,4,4,3]
=> ? = 1 + 2
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[4,4,4,4],[1]]
=> [4,4,4,4]
=> ? = 0 + 2
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,3],[]]
=> [3,3,3,3,3]
=> ? = 0 + 2
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[5,5,5],[1]]
=> [5,5,5]
=> ? = 0 + 2
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[4,4,4,4],[1,1]]
=> [4,4,4,4]
=> ? = 0 + 2
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [[4,4,4,3],[1]]
=> [4,4,4,3]
=> ? = 1 + 2
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [[5,5,5],[2]]
=> [5,5,5]
=> ? = 0 + 2
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[5,5,4],[]]
=> [5,5,4]
=> ? = 1 + 2
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [[4,4,3,3],[]]
=> [4,4,3,3]
=> ? = 1 + 2
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [[5,5,4],[1]]
=> [5,5,4]
=> ? = 1 + 2
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [[5,5,3],[]]
=> [5,5,3]
=> ? = 1 + 2
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [[6,6],[]]
=> [6,6]
=> ? = 0 + 2
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,2],[]]
=> [4,4,4,2]
=> ? = 1 + 2
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,2],[]]
=> [3,3,3,3,2]
=> ? = 1 + 2
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4,2],[1]]
=> [4,4,4,2]
=> ? = 1 + 2
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [[5,5,2],[]]
=> [5,5,2]
=> ? = 1 + 2
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [[4,4,4,3],[1,1]]
=> [4,4,4,3]
=> ? = 1 + 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [[5,5,3],[1]]
=> [5,5,3]
=> ? = 1 + 2
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2,2],[]]
=> [3,3,3,2,2]
=> ? = 1 + 2
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [[4,4,3,3],[1]]
=> [4,4,3,3]
=> ? = 1 + 2
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [[5,5,4],[2]]
=> [5,5,4]
=> ? = 1 + 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [[4,4,3,3],[1,1]]
=> [4,4,3,3]
=> ? = 1 + 2
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [[3,3,2,2,2],[]]
=> [3,3,2,2,2]
=> ? = 1 + 2
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[4,4,4,4],[2]]
=> [4,4,4,4]
=> ? = 0 + 2
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [[3,3,3,3,3],[1]]
=> [3,3,3,3,3]
=> ? = 0 + 2
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [[4,4,4,4],[2,1]]
=> [4,4,4,4]
=> ? = 0 + 2
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [[4,4,4,3],[2]]
=> [4,4,4,3]
=> ? = 1 + 2
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [[5,5,5],[3]]
=> [5,5,5]
=> ? = 0 + 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St001124
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [[1],[]]
=> [1]
=> ? = 0
[.,[.,.]]
=> [1,1,0,0]
=> [[2],[]]
=> [2]
=> 0
[[.,.],.]
=> [1,0,1,0]
=> [[1,1],[]]
=> [1,1]
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> [2,2]
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> [3]
=> 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [2,2]
=> 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> [3,3]
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2]
=> 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [3,3]
=> 0
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> [3,2]
=> 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> [4]
=> 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [3,2]
=> 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [3,3]
=> 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [2,2,1]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> [3,3,3]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> [4,4]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> [3,3,2]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [3,3,3]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> [2,2,2,2]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [4,4]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [3,3,3]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [3,3,2]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [4,4]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> [4,3]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> [3,2,2]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [4,3]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> [4,2]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> [5]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [3,3,1]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [3,3,2]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [4,2]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [2,2,1,1]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [3,2,2]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [4,3]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [3,2,1]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [3,2,2]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [3,3,3]
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> [4,4,4]
=> ? = 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3],[]]
=> [3,3,3,3]
=> ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4],[1]]
=> [4,4,4]
=> ? = 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3],[1]]
=> [3,3,3,3]
=> ? = 0
[.,[.,[[.,[[.,.],.]],.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [[4,4,4],[2]]
=> [4,4,4]
=> ? = 0
[.,[.,[[[.,[.,.]],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1]]
=> [3,3,3,3]
=> ? = 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [[4,4,4],[1,1]]
=> [4,4,4]
=> ? = 0
[.,[[.,.],[[.,[.,.]],.]]]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [[4,4,4],[2,1]]
=> [4,4,4]
=> ? = 0
[.,[[.,.],[[[.,.],.],.]]]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [[3,3,3,3],[1,1,1]]
=> [3,3,3,3]
=> ? = 0
[.,[[[.,.],.],[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[4,4,4],[2,2]]
=> [4,4,4]
=> ? = 0
[[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> [3,3,3,3]
=> ? = 0
[[.,[.,[.,[[.,.],.]]]],.]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> [4,4,4]
=> ? = 0
[[.,[.,[[.,[.,.]],.]]],.]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> [3,3,3,3]
=> ? = 0
[[.,[[.,.],[.,[.,.]]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> [4,4,4]
=> ? = 0
[[.,[[.,.],[[.,.],.]]],.]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> [3,3,3,3]
=> ? = 0
[[.,[[[.,.],.],[.,.]]],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> [4,4,4]
=> ? = 0
[[[.,[.,[.,[.,.]]]],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> [3,3,3,3]
=> ? = 0
[[[.,[[.,.],[.,.]]],.],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2,1]]
=> [3,3,3,3]
=> ? = 0
[[[.,[[[.,.],.],.]],.],.]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [[4,4,4],[3,3]]
=> [4,4,4]
=> ? = 0
[[[[.,[[.,.],.]],.],.],.]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> ? = 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4],[]]
=> [4,4,4,4]
=> ? = 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[5,5,5],[]]
=> [5,5,5]
=> ? = 0
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[4,4,4,3],[]]
=> [4,4,4,3]
=> ? = 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[4,4,4,4],[1]]
=> [4,4,4,4]
=> ? = 0
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,3],[]]
=> [3,3,3,3,3]
=> ? = 0
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[5,5,5],[1]]
=> [5,5,5]
=> ? = 0
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[4,4,4,4],[1,1]]
=> [4,4,4,4]
=> ? = 0
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [[4,4,4,3],[1]]
=> [4,4,4,3]
=> ? = 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [[5,5,5],[2]]
=> [5,5,5]
=> ? = 0
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[5,5,4],[]]
=> [5,5,4]
=> ? = 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [[4,4,3,3],[]]
=> [4,4,3,3]
=> ? = 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [[5,5,4],[1]]
=> [5,5,4]
=> ? = 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [[5,5,3],[]]
=> [5,5,3]
=> ? = 1
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [[6,6],[]]
=> [6,6]
=> ? = 0
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,2],[]]
=> [4,4,4,2]
=> ? = 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,2],[]]
=> [3,3,3,3,2]
=> ? = 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4,2],[1]]
=> [4,4,4,2]
=> ? = 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [[5,5,2],[]]
=> [5,5,2]
=> ? = 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [[4,4,4,3],[1,1]]
=> [4,4,4,3]
=> ? = 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [[5,5,3],[1]]
=> [5,5,3]
=> ? = 1
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2,2],[]]
=> [3,3,3,2,2]
=> ? = 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [[4,4,3,3],[1]]
=> [4,4,3,3]
=> ? = 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [[5,5,4],[2]]
=> [5,5,4]
=> ? = 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [[4,4,3,3],[1,1]]
=> [4,4,3,3]
=> ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [[3,3,2,2,2],[]]
=> [3,3,2,2,2]
=> ? = 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[4,4,4,4],[2]]
=> [4,4,4,4]
=> ? = 0
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [[3,3,3,3,3],[1]]
=> [3,3,3,3,3]
=> ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [[4,4,4,4],[2,1]]
=> [4,4,4,4]
=> ? = 0
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [[4,4,4,3],[2]]
=> [4,4,4,3]
=> ? = 1
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity gλμ,ν of the Specht module Sλ in Sμ⊗Sν:
Sμ⊗Sν=⨁λgλμ,νSλ
This statistic records the Kronecker coefficient g(n−1)1λ,λ, for λ⊢n>1. For n≤1 the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000069
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [[1],[]]
=> ([],1)
=> 1 = 0 + 1
[.,[.,.]]
=> [1,1,0,0]
=> [[2],[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> [1,0,1,0]
=> [[1,1],[]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(5,4),(6,4),(7,5),(7,6)],8)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ([(0,6),(1,3),(1,6),(2,4),(3,2),(3,5),(5,4),(6,5)],7)
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ([(0,2),(0,3),(1,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ([(0,3),(0,6),(1,2),(1,6),(2,4),(3,5),(6,4),(6,5)],7)
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[4,4,4,3],[]]
=> ([(0,5),(0,6),(1,4),(1,14),(2,3),(2,13),(3,9),(4,10),(5,2),(5,11),(6,1),(6,11),(9,7),(10,8),(11,13),(11,14),(12,7),(12,8),(13,9),(13,12),(14,10),(14,12)],15)
=> ? = 1 + 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[4,4,4,4],[1]]
=> ([(0,5),(0,14),(1,4),(1,14),(2,9),(3,10),(4,2),(4,11),(5,3),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(11,13),(12,10),(12,13),(13,7),(13,8),(14,11),(14,12)],15)
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[5,5,5],[1]]
=> ([(0,5),(0,13),(1,4),(1,13),(2,3),(2,12),(3,10),(4,7),(5,2),(5,11),(7,8),(8,9),(9,6),(10,6),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11)],14)
=> ? = 0 + 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[4,4,4,4],[1,1]]
=> ([(0,4),(0,5),(1,3),(1,13),(2,9),(3,10),(4,2),(4,12),(5,12),(5,13),(7,6),(8,6),(9,7),(10,8),(11,7),(11,8),(12,9),(12,11),(13,10),(13,11)],14)
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [[4,4,4,3],[1]]
=> ([(0,5),(0,13),(1,4),(1,13),(2,8),(3,9),(4,2),(4,10),(5,3),(5,11),(8,6),(9,7),(10,8),(10,12),(11,9),(11,12),(12,6),(12,7),(13,10),(13,11)],14)
=> ? = 1 + 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [[5,5,5],[2]]
=> ([(0,4),(0,5),(1,2),(1,11),(2,3),(2,12),(3,9),(4,10),(5,10),(5,11),(7,8),(8,6),(9,6),(10,7),(11,7),(11,12),(12,8),(12,9)],13)
=> ? = 0 + 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[5,5,4],[]]
=> ([(0,5),(0,6),(1,10),(2,3),(2,13),(3,4),(3,12),(4,7),(5,1),(5,11),(6,2),(6,11),(9,8),(10,9),(11,10),(11,13),(12,7),(12,8),(13,9),(13,12)],14)
=> ? = 1 + 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [[4,4,3,3],[]]
=> ([(0,5),(0,6),(1,4),(1,13),(2,3),(2,12),(3,10),(4,7),(5,1),(5,11),(6,2),(6,11),(9,8),(10,8),(11,12),(11,13),(12,9),(12,10),(13,7),(13,9)],14)
=> ? = 1 + 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [[5,5,4],[1]]
=> ([(0,5),(0,12),(1,4),(1,12),(2,3),(2,11),(3,7),(4,8),(5,2),(5,10),(8,9),(9,6),(10,9),(10,11),(11,6),(11,7),(12,8),(12,10)],13)
=> ? = 1 + 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [[5,5,3],[]]
=> ([(0,5),(0,6),(1,9),(2,8),(3,4),(3,11),(4,2),(4,10),(5,3),(5,7),(6,1),(6,7),(7,9),(7,11),(9,12),(10,8),(11,10),(11,12)],13)
=> ? = 1 + 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,2],[]]
=> ([(0,5),(0,6),(1,4),(1,13),(2,3),(2,12),(3,10),(4,7),(5,1),(5,11),(6,2),(6,11),(9,8),(10,8),(11,12),(11,13),(12,9),(12,10),(13,7),(13,9)],14)
=> ? = 1 + 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,2],[]]
=> ([(0,5),(0,6),(1,10),(2,3),(2,13),(3,4),(3,12),(4,7),(5,1),(5,11),(6,2),(6,11),(9,8),(10,9),(11,10),(11,13),(12,7),(12,8),(13,9),(13,12)],14)
=> ? = 1 + 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4,2],[1]]
=> ([(0,4),(0,12),(1,5),(1,12),(2,9),(3,7),(4,2),(4,10),(5,3),(5,11),(8,6),(9,6),(10,8),(10,9),(11,7),(11,8),(12,10),(12,11)],13)
=> ? = 1 + 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [[4,4,3,2],[]]
=> ([(0,5),(0,6),(1,9),(2,8),(3,2),(3,10),(4,1),(4,11),(5,3),(5,7),(6,4),(6,7),(7,10),(7,11),(10,8),(10,12),(11,9),(11,12)],13)
=> ? = 2 + 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [[5,5,2],[]]
=> ([(0,5),(0,6),(1,9),(2,8),(3,4),(3,11),(4,2),(4,10),(5,3),(5,7),(6,1),(6,7),(7,9),(7,11),(10,8),(11,10)],12)
=> ? = 1 + 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [[4,4,4,3],[1,1]]
=> ([(0,4),(0,5),(1,3),(1,12),(2,8),(3,9),(4,2),(4,11),(5,11),(5,12),(8,6),(9,7),(10,6),(10,7),(11,8),(11,10),(12,9),(12,10)],13)
=> ? = 1 + 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [[5,5,3],[1]]
=> ([(0,5),(0,11),(1,4),(1,11),(2,3),(2,9),(3,6),(4,8),(5,2),(5,10),(8,7),(9,6),(10,7),(10,9),(11,8),(11,10)],12)
=> ? = 1 + 1
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2,2],[]]
=> ([(0,5),(0,6),(1,9),(2,8),(3,4),(3,11),(4,2),(4,10),(5,3),(5,7),(6,1),(6,7),(7,9),(7,11),(9,12),(10,8),(11,10),(11,12)],13)
=> ? = 1 + 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [[4,4,2,2],[]]
=> ([(0,5),(0,6),(1,9),(2,8),(3,2),(3,10),(4,1),(4,11),(5,3),(5,7),(6,4),(6,7),(7,10),(7,11),(10,8),(11,9)],12)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [[4,4,3,3],[1]]
=> ([(0,4),(0,12),(1,5),(1,12),(2,9),(3,7),(4,2),(4,10),(5,3),(5,11),(8,6),(9,6),(10,8),(10,9),(11,7),(11,8),(12,10),(12,11)],13)
=> ? = 1 + 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [[5,5,4],[2]]
=> ([(0,4),(0,5),(1,2),(1,10),(2,3),(2,11),(3,6),(4,9),(5,9),(5,10),(8,7),(9,8),(10,8),(10,11),(11,6),(11,7)],12)
=> ? = 1 + 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3,2],[1]]
=> ([(0,5),(0,11),(1,4),(1,11),(2,7),(3,8),(4,2),(4,9),(5,3),(5,10),(9,6),(9,7),(10,6),(10,8),(11,9),(11,10)],12)
=> ? = 2 + 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [[4,4,3,3],[1,1]]
=> ([(0,4),(0,5),(1,3),(1,11),(2,7),(3,9),(4,2),(4,10),(5,10),(5,11),(8,6),(9,6),(10,7),(10,8),(11,8),(11,9)],12)
=> ? = 1 + 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [[3,3,2,2,2],[]]
=> ([(0,5),(0,6),(1,9),(2,8),(3,4),(3,11),(4,2),(4,10),(5,3),(5,7),(6,1),(6,7),(7,9),(7,11),(10,8),(11,10)],12)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[4,4,4,4],[2]]
=> ([(0,4),(0,5),(1,3),(1,13),(2,9),(3,10),(4,2),(4,12),(5,12),(5,13),(7,6),(8,6),(9,7),(10,8),(11,7),(11,8),(12,9),(12,11),(13,10),(13,11)],14)
=> ? = 0 + 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [[3,3,3,3,3],[1]]
=> ([(0,5),(0,13),(1,4),(1,13),(2,3),(2,12),(3,10),(4,7),(5,2),(5,11),(7,8),(8,9),(9,6),(10,6),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11)],14)
=> ? = 0 + 1
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [[4,4,4,4],[2,1]]
=> ([(0,4),(0,12),(1,3),(1,11),(2,11),(2,12),(3,8),(4,9),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7),(11,8),(11,10),(12,9),(12,10)],13)
=> ? = 0 + 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [[4,4,4,3],[2]]
=> ([(0,4),(0,5),(1,3),(1,12),(2,8),(3,9),(4,2),(4,11),(5,11),(5,12),(8,6),(9,7),(10,6),(10,7),(11,8),(11,10),(12,9),(12,10)],13)
=> ? = 1 + 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [[5,5,5],[3]]
=> ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,10),(4,5),(4,10),(5,9),(5,11),(7,6),(8,6),(9,7),(10,9),(11,7),(11,8)],12)
=> ? = 0 + 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [[3,3,3,3,2],[1]]
=> ([(0,5),(0,12),(1,4),(1,12),(2,3),(2,11),(3,7),(4,8),(5,2),(5,10),(8,9),(9,6),(10,9),(10,11),(11,6),(11,7),(12,8),(12,10)],13)
=> ? = 1 + 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [[4,4,4,2],[2]]
=> ([(0,4),(0,5),(1,3),(1,11),(2,7),(3,9),(4,2),(4,10),(5,10),(5,11),(8,6),(9,6),(10,7),(10,8),(11,8),(11,9)],12)
=> ? = 1 + 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [[4,4,4,3],[2,1]]
=> ([(0,4),(0,11),(1,3),(1,10),(2,10),(2,11),(3,7),(4,8),(7,5),(8,6),(9,5),(9,6),(10,7),(10,9),(11,8),(11,9)],12)
=> ? = 1 + 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [[3,3,3,2,2],[1]]
=> ([(0,5),(0,11),(1,4),(1,11),(2,3),(2,9),(3,6),(4,8),(5,2),(5,10),(8,7),(9,6),(10,7),(10,9),(11,8),(11,10)],12)
=> ? = 1 + 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [[3,3,3,3,3],[1,1]]
=> ([(0,4),(0,5),(1,2),(1,11),(2,3),(2,12),(3,9),(4,10),(5,10),(5,11),(7,8),(8,6),(9,6),(10,7),(11,7),(11,12),(12,8),(12,9)],13)
=> ? = 0 + 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [[4,4,4,4],[2,2]]
=> ([(0,3),(0,5),(1,2),(1,4),(2,9),(3,10),(4,9),(4,11),(5,10),(5,11),(7,6),(8,6),(9,7),(10,8),(11,7),(11,8)],12)
=> ? = 0 + 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [[3,3,3,3,2],[1,1]]
=> ([(0,4),(0,5),(1,2),(1,10),(2,3),(2,11),(3,6),(4,9),(5,9),(5,10),(8,7),(9,8),(10,8),(10,11),(11,6),(11,7)],12)
=> ? = 1 + 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [[3,3,3,3,3],[1,1,1]]
=> ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,10),(4,5),(4,10),(5,9),(5,11),(7,6),(8,6),(9,7),(10,9),(11,7),(11,8)],12)
=> ? = 0 + 1
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[5,5,5],[1,1]]
=> ([(0,4),(0,5),(1,10),(2,3),(2,12),(3,9),(4,2),(4,11),(5,10),(5,11),(7,8),(8,6),(9,6),(10,7),(11,7),(11,12),(12,8),(12,9)],13)
=> ? = 0 + 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [[4,4,4,4],[1,1,1]]
=> ([(0,3),(0,4),(1,10),(2,7),(3,2),(3,12),(4,5),(4,12),(5,10),(5,11),(7,8),(8,6),(9,6),(10,9),(11,8),(11,9),(12,7),(12,11)],13)
=> ? = 0 + 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [[5,5,5],[2,1]]
=> ([(0,9),(1,4),(1,11),(2,9),(2,11),(3,8),(4,3),(4,10),(6,7),(7,5),(8,5),(9,6),(10,7),(10,8),(11,6),(11,10)],12)
=> ? = 0 + 1
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [[5,5,4],[1,1]]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,6),(4,2),(4,10),(5,9),(5,10),(8,7),(9,8),(10,8),(10,11),(11,6),(11,7)],12)
=> ? = 1 + 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [[6,6],[1]]
=> ([(0,10),(1,5),(1,10),(2,4),(2,7),(3,6),(4,3),(4,9),(5,2),(5,8),(7,9),(8,7),(9,6),(10,8)],11)
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [[4,4,4,3],[1,1,1]]
=> ([(0,3),(0,4),(1,9),(2,8),(3,2),(3,11),(4,5),(4,11),(5,9),(5,10),(8,7),(9,6),(10,6),(10,7),(11,8),(11,10)],12)
=> ? = 1 + 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [[5,5,3],[1,1]]
=> ([(0,4),(0,5),(1,8),(2,3),(2,9),(3,6),(4,2),(4,10),(5,8),(5,10),(8,7),(9,6),(10,7),(10,9)],11)
=> ? = 1 + 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [[5,5,4],[2,1]]
=> ([(0,8),(1,4),(1,10),(2,8),(2,10),(3,6),(4,3),(4,9),(7,5),(8,7),(9,5),(9,6),(10,7),(10,9)],11)
=> ? = 1 + 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [[4,4,3,3],[1,1,1]]
=> ([(0,3),(0,4),(1,9),(2,6),(3,2),(3,10),(4,5),(4,10),(5,8),(5,9),(8,7),(9,7),(10,6),(10,8)],11)
=> ? = 1 + 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [[4,4,4,4],[2,1,1]]
=> ([(0,9),(1,4),(1,11),(2,3),(2,11),(3,8),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7),(11,8),(11,10)],12)
=> ? = 0 + 1
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [[5,5,5],[3,1]]
=> ([(0,9),(1,4),(1,9),(2,3),(2,10),(3,8),(4,7),(4,10),(6,5),(7,6),(8,5),(9,7),(10,6),(10,8)],11)
=> ? = 0 + 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [[4,4,4,3],[2,1,1]]
=> ([(0,8),(1,4),(1,10),(2,3),(2,10),(3,7),(4,8),(4,9),(7,6),(8,5),(9,5),(9,6),(10,7),(10,9)],11)
=> ? = 1 + 1
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [[4,4,4,4],[2,2,1]]
=> ([(0,3),(0,4),(1,8),(2,8),(2,10),(3,9),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0 + 1
Description
The number of maximal elements of a poset.
Matching statistic: St000353
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 75%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 75%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [1] => ? = 0
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => 0
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 0
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,5,6,3,7,2,1] => ? = 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,4,2,1] => ? = 0
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 0
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,4,5,6,7,3,1] => ? = 0
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,5,6,7,4,3,1] => ? = 0
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [3,5,6,4,7,2,1] => ? = 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,5,6,7,4,1] => ? = 0
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,2,7,1] => ? = 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,3,2,7,1] => ? = 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,4,5,6,3,7,1] => ? = 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,2,6,7,1] => ? = 1
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [4,5,3,6,7,2,1] => ? = 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [3,5,4,6,7,2,1] => ? = 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [4,5,3,6,2,7,1] => ? = 2
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,1] => ? = 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,5,6,4,7,3,1] => ? = 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,4,5,3,6,7,1] => ? = 1
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,6,4,3,7,2,1] => ? = 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,3,2,6,7,1] => ? = 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [3,5,6,4,2,7,1] => ? = 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,5,6,4,7,1] => ? = 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [3,5,4,6,2,7,1] => ? = 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [2,5,6,4,3,7,1] => ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [5,6,4,3,2,7,1] => ? = 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,5,2,1] => ? = 0
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,6,7,5,3,2,1] => ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,7,5,3,1] => ? = 0
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [3,4,6,5,7,2,1] => ? = 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,6,7,5,1] => ? = 0
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,6,5,7,3,2,1] => ? = 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [3,4,2,6,7,5,1] => ? = 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [2,4,6,5,7,3,1] => ? = 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [4,6,5,3,7,2,1] => ? = 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,5,4,2,1] => ? = 0
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,6,7,5,4,1] => ? = 0
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,6,5,7,4,2,1] => ? = 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,6,7,5,4,3,1] => ? = 0
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [6,7,5,4,3,2,1] => ? = 0
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => ? = 0
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => ? = 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => ? = 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 0
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => ? = 1
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 0
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,5,4,6,7,3,1] => ? = 1
Description
The number of inner valleys of a permutation.
The number of valleys including the boundary is [[St000099]].
Matching statistic: St000256
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 75%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 75%
Values
[.,.]
=> [1,0]
=> [1,0]
=> []
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 0
[[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[.,[[.,.],[.,[[.,.],.]]]]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> ? = 0
[.,[[[.,[.,.]],.],[.,.]]]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1]
=> ? = 1
[.,[[[[.,.],.],.],[.,.]]]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> ? = 0
[.,[[.,[.,[[.,.],.]]],.]]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> ? = 1
[.,[[[.,.],[.,[.,.]]],.]]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> ? = 1
[.,[[[[.,.],.],[.,.]],.]]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> ? = 1
[.,[[[.,[.,[.,.]]],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> ? = 1
[.,[[[[.,.],[.,.]],.],.]]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> ? = 1
[.,[[[[.,[.,.]],.],.],.]]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? = 1
[.,[[[[[.,.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? = 0
[[.,.],[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2]
=> ? = 1
[[.,.],[[[.,.],.],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> ? = 1
[[.,.],[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2]
=> ? = 2
[[.,.],[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> ? = 2
[[.,.],[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 1
[[.,[.,.]],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> ? = 1
[[.,[.,.]],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> ? = 2
[[.,[.,.]],[[[.,.],.],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> ? = 1
[[[.,.],.],[[[.,.],.],.]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? = 1
[[.,[.,[.,.]]],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1]
=> ? = 1
[[.,[[.,.],.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> ? = 1
[[.,[[.,.],.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> ? = 1
[[[.,.],[.,.]],[[.,.],.]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> ? = 2
[[[.,[.,.]],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> ? = 1
[[.,[[.,.],[.,.]]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> ? = 1
[[.,[[.,[.,.]],.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1]
=> ? = 2
[[.,[[[.,.],.],.]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> ? = 1
[[[.,.],[[.,.],.]],[.,.]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> ? = 2
[[[.,[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> ? = 2
[[[[.,.],.],[.,.]],[.,.]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> ? = 2
[[[.,[[.,.],.]],.],[.,.]]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> ? = 1
[[[[.,.],[.,.]],.],[.,.]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> ? = 2
[[.,[[[.,.],.],[.,.]]],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> ? = 0
[[.,[[.,[.,[.,.]]],.]],.]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1]
=> ? = 1
[[.,[[[.,.],[.,.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> ? = 1
[[.,[[[.,[.,.]],.],.]],.]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1]
=> ? = 1
[[.,[[[[.,.],.],.],.]],.]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> ? = 0
[[[.,.],[[[.,.],.],.]],.]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> ? = 1
[[[.,[.,.]],[[.,.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> ? = 1
[[[[.,.],.],[[.,.],.]],.]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> ? = 1
[[[.,[[.,.],.]],[.,.]],.]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> ? = 1
[[[[.,.],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> ? = 2
[[[[.,[.,.]],.],[.,.]],.]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> ? = 1
[[[.,[[[.,.],.],.]],.],.]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> ? = 0
[[[[.,.],[[.,.],.]],.],.]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> ? = 1
[[[[.,[.,.]],[.,.]],.],.]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> ? = 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [4,3,2,1,1]
=> ? = 0
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1]
=> ? = 0
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [5,3,2,1]
=> ? = 1
Description
The number of parts from which one can substract 2 and still get an integer partition.
The following 4 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001960The number of descents of a permutation minus one if its first entry is not one. St001487The number of inner corners of a skew partition.
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