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Your data matches 81 different statistics following compositions of up to 3 maps.
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Matching statistic: St000386
(load all 47 compositions to match this statistic)
(load all 47 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000386: 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]
=> 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[[[.,.],.],.]
=> [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]
=> 0
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[[[[.,.],.],.],.]
=> [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]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [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]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [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]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [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]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000291
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => => ? = 0
[.,[.,.]]
=> [2,1] => 1 => 0
[[.,.],.]
=> [1,2] => 0 => 0
[.,[.,[.,.]]]
=> [3,2,1] => 11 => 0
[.,[[.,.],.]]
=> [2,3,1] => 01 => 0
[[.,.],[.,.]]
=> [1,3,2] => 01 => 0
[[.,[.,.]],.]
=> [2,1,3] => 10 => 1
[[[.,.],.],.]
=> [1,2,3] => 00 => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 011 => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 011 => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 101 => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 001 => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 011 => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 001 => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 101 => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 001 => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 010 => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 010 => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1111 => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0111 => 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 0111 => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1011 => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0011 => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 0111 => 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 0011 => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1011 => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 0011 => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1101 => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0101 => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 0101 => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1001 => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0001 => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0111 => 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 0011 => 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0011 => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 0101 => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 0001 => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1011 => 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1001 => 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 0011 => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 0001 => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1101 => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 0101 => 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0101 => 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 1001 => 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0001 => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1110 => 1
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [5,4,8,7,6,3,2,1] => ? => ? = 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [3,6,5,8,7,4,2,1] => ? => ? = 1
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [3,6,5,7,8,4,2,1] => ? => ? = 1
[.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [3,5,4,8,7,6,2,1] => ? => ? = 1
[.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> [4,5,7,6,3,8,2,1] => ? => ? = 1
[.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> [5,4,3,7,6,8,2,1] => ? => ? = 2
[[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,5,7,8,6,4,3,2] => ? => ? = 0
[[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,5,6,8,7,4,3,2] => ? => ? = 0
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,4,7,6,8,5,3,2] => ? => ? = 1
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,5,4,8,7,6,3,2] => ? => ? = 1
[[.,.],[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,5,6,4,8,7,3,2] => ? => ? = 1
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,5,4,6,8,7,3,2] => ? => ? = 1
[[.,.],[.,[[[[.,.],.],.],[.,.]]]]
=> [1,4,5,6,8,7,3,2] => ? => ? = 0
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,6,7,5,4,8,3,2] => ? => ? = 1
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,5,7,6,4,8,3,2] => ? => ? = 1
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,6,5,7,4,8,3,2] => ? => ? = 2
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> [1,5,6,7,4,8,3,2] => ? => ? = 1
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,5,4,7,6,8,3,2] => ? => ? = 2
[[.,.],[.,[[[[.,.],.],[.,.]],.]]]
=> [1,4,5,7,6,8,3,2] => ? => ? = 1
[[.,.],[.,[[[[.,.],[.,.]],.],.]]]
=> [1,4,6,5,7,8,3,2] => ? => ? = 1
[[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
=> [1,3,7,6,8,5,4,2] => ? => ? = 1
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,3,6,5,8,7,4,2] => ? => ? = 1
[[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,3,7,6,5,8,4,2] => ? => ? = 1
[[.,.],[[[.,.],.],[.,[[.,.],.]]]]
=> [1,3,4,7,8,6,5,2] => ? => ? = 0
[[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,3,4,6,8,7,5,2] => ? => ? = 0
[[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,5,4,3,7,8,6,2] => ? => ? = 1
[[.,.],[[.,[[.,[.,.]],.]],[.,.]]]
=> [1,5,4,6,3,8,7,2] => ? => ? = 2
[[.,.],[[[.,.],[[.,.],.]],[.,.]]]
=> [1,3,5,6,4,8,7,2] => ? => ? = 1
[[.,.],[[[[.,.],.],[.,.]],[.,.]]]
=> [1,3,4,6,5,8,7,2] => ? => ? = 1
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [1,3,5,4,6,8,7,2] => ? => ? = 1
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [1,4,3,5,6,8,7,2] => ? => ? = 1
[[.,.],[[.,[.,[[.,.],[.,.]]]],.]]
=> [1,5,7,6,4,3,8,2] => ? => ? = 1
[[.,.],[[.,[.,[[[.,.],.],.]]],.]]
=> [1,5,6,7,4,3,8,2] => ? => ? = 1
[[.,.],[[.,[[.,.],[.,[.,.]]]],.]]
=> [1,4,7,6,5,3,8,2] => ? => ? = 1
[[.,.],[[.,[[.,[.,.]],[.,.]]],.]]
=> [1,5,4,7,6,3,8,2] => ? => ? = 2
[[.,.],[[.,[[[.,.],.],[.,.]]],.]]
=> [1,4,5,7,6,3,8,2] => ? => ? = 1
[[.,.],[[.,[[.,[.,[.,.]]],.]],.]]
=> [1,6,5,4,7,3,8,2] => ? => ? = 2
[[.,.],[[[.,.],[.,[.,[.,.]]]],.]]
=> [1,3,7,6,5,4,8,2] => ? => ? = 1
[[.,.],[[[.,.],[.,[[.,.],.]]],.]]
=> [1,3,6,7,5,4,8,2] => ? => ? = 1
[[.,.],[[[.,[.,.]],[.,[.,.]]],.]]
=> [1,4,3,7,6,5,8,2] => ? => ? = 2
[[.,.],[[[.,[.,[.,.]]],[.,.]],.]]
=> [1,5,4,3,7,6,8,2] => ? => ? = 2
[[.,.],[[[.,[[.,.],.]],[.,.]],.]]
=> [1,4,5,3,7,6,8,2] => ? => ? = 2
[[.,.],[[[[.,[.,.]],.],[.,.]],.]]
=> [1,4,3,5,7,6,8,2] => ? => ? = 2
[[.,.],[[[.,[.,[[.,.],.]]],.],.]]
=> [1,5,6,4,3,7,8,2] => ? => ? = 1
[[.,.],[[[.,[[.,.],[.,.]]],.],.]]
=> [1,4,6,5,3,7,8,2] => ? => ? = 1
[[.,.],[[[.,[[[.,.],.],.]],.],.]]
=> [1,4,5,6,3,7,8,2] => ? => ? = 1
[[.,.],[[[[.,.],[.,[.,.]]],.],.]]
=> [1,3,6,5,4,7,8,2] => ? => ? = 1
[[.,[.,.]],[[[[.,.],.],[.,.]],.]]
=> [2,1,4,5,7,6,8,3] => ? => ? = 2
[[.,[.,.]],[[[.,[.,[.,.]]],.],.]]
=> [2,1,6,5,4,7,8,3] => ? => ? = 2
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => => => ? = 0
[.,[.,.]]
=> [2,1] => 1 => 1 => 0
[[.,.],.]
=> [1,2] => 0 => 0 => 0
[.,[.,[.,.]]]
=> [3,2,1] => 11 => 11 => 0
[.,[[.,.],.]]
=> [2,3,1] => 01 => 10 => 0
[[.,.],[.,.]]
=> [1,3,2] => 01 => 10 => 0
[[.,[.,.]],.]
=> [2,1,3] => 10 => 01 => 1
[[[.,.],.],.]
=> [1,2,3] => 00 => 00 => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 111 => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 011 => 110 => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 011 => 110 => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 101 => 101 => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 001 => 100 => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 011 => 110 => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 001 => 100 => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 101 => 101 => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 001 => 100 => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 011 => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 010 => 010 => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 010 => 010 => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 001 => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 000 => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1111 => 1111 => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0111 => 1110 => 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 0111 => 1110 => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1011 => 1101 => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0011 => 1100 => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 0111 => 1110 => 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 0011 => 1100 => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1011 => 1101 => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 0011 => 1100 => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1101 => 1011 => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0101 => 1010 => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 0101 => 1010 => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1001 => 1001 => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0001 => 1000 => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0111 => 1110 => 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 0011 => 1100 => 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0011 => 1100 => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 0101 => 1010 => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 0001 => 1000 => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1011 => 1101 => 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1001 => 1001 => 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 0011 => 1100 => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 0001 => 1000 => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1101 => 1011 => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 0101 => 1010 => 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0101 => 1010 => 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 1001 => 1001 => 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0001 => 1000 => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1110 => 0111 => 1
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [5,4,8,7,6,3,2,1] => ? => ? => ? = 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [3,6,5,8,7,4,2,1] => ? => ? => ? = 1
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [3,6,5,7,8,4,2,1] => ? => ? => ? = 1
[.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [3,5,4,8,7,6,2,1] => ? => ? => ? = 1
[.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> [4,5,7,6,3,8,2,1] => ? => ? => ? = 1
[.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> [5,4,3,7,6,8,2,1] => ? => ? => ? = 2
[[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,5,7,8,6,4,3,2] => ? => ? => ? = 0
[[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,5,6,8,7,4,3,2] => ? => ? => ? = 0
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,4,7,6,8,5,3,2] => ? => ? => ? = 1
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,5,4,8,7,6,3,2] => ? => ? => ? = 1
[[.,.],[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,5,6,4,8,7,3,2] => ? => ? => ? = 1
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,5,4,6,8,7,3,2] => ? => ? => ? = 1
[[.,.],[.,[[[[.,.],.],.],[.,.]]]]
=> [1,4,5,6,8,7,3,2] => ? => ? => ? = 0
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,6,7,5,4,8,3,2] => ? => ? => ? = 1
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,5,7,6,4,8,3,2] => ? => ? => ? = 1
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,6,5,7,4,8,3,2] => ? => ? => ? = 2
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> [1,5,6,7,4,8,3,2] => ? => ? => ? = 1
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,5,4,7,6,8,3,2] => ? => ? => ? = 2
[[.,.],[.,[[[[.,.],.],[.,.]],.]]]
=> [1,4,5,7,6,8,3,2] => ? => ? => ? = 1
[[.,.],[.,[[[[.,.],[.,.]],.],.]]]
=> [1,4,6,5,7,8,3,2] => ? => ? => ? = 1
[[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
=> [1,3,7,6,8,5,4,2] => ? => ? => ? = 1
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,3,6,5,8,7,4,2] => ? => ? => ? = 1
[[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,3,7,6,5,8,4,2] => ? => ? => ? = 1
[[.,.],[[[.,.],.],[.,[[.,.],.]]]]
=> [1,3,4,7,8,6,5,2] => ? => ? => ? = 0
[[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,3,4,6,8,7,5,2] => ? => ? => ? = 0
[[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,5,4,3,7,8,6,2] => ? => ? => ? = 1
[[.,.],[[.,[[.,[.,.]],.]],[.,.]]]
=> [1,5,4,6,3,8,7,2] => ? => ? => ? = 2
[[.,.],[[[.,.],[[.,.],.]],[.,.]]]
=> [1,3,5,6,4,8,7,2] => ? => ? => ? = 1
[[.,.],[[[[.,.],.],[.,.]],[.,.]]]
=> [1,3,4,6,5,8,7,2] => ? => ? => ? = 1
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [1,3,5,4,6,8,7,2] => ? => ? => ? = 1
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [1,4,3,5,6,8,7,2] => ? => ? => ? = 1
[[.,.],[[.,[.,[[.,.],[.,.]]]],.]]
=> [1,5,7,6,4,3,8,2] => ? => ? => ? = 1
[[.,.],[[.,[.,[[[.,.],.],.]]],.]]
=> [1,5,6,7,4,3,8,2] => ? => ? => ? = 1
[[.,.],[[.,[[.,.],[.,[.,.]]]],.]]
=> [1,4,7,6,5,3,8,2] => ? => ? => ? = 1
[[.,.],[[.,[[.,[.,.]],[.,.]]],.]]
=> [1,5,4,7,6,3,8,2] => ? => ? => ? = 2
[[.,.],[[.,[[[.,.],.],[.,.]]],.]]
=> [1,4,5,7,6,3,8,2] => ? => ? => ? = 1
[[.,.],[[.,[[.,[.,[.,.]]],.]],.]]
=> [1,6,5,4,7,3,8,2] => ? => ? => ? = 2
[[.,.],[[[.,.],[.,[.,[.,.]]]],.]]
=> [1,3,7,6,5,4,8,2] => ? => ? => ? = 1
[[.,.],[[[.,.],[.,[[.,.],.]]],.]]
=> [1,3,6,7,5,4,8,2] => ? => ? => ? = 1
[[.,.],[[[.,[.,.]],[.,[.,.]]],.]]
=> [1,4,3,7,6,5,8,2] => ? => ? => ? = 2
[[.,.],[[[.,[.,[.,.]]],[.,.]],.]]
=> [1,5,4,3,7,6,8,2] => ? => ? => ? = 2
[[.,.],[[[.,[[.,.],.]],[.,.]],.]]
=> [1,4,5,3,7,6,8,2] => ? => ? => ? = 2
[[.,.],[[[[.,[.,.]],.],[.,.]],.]]
=> [1,4,3,5,7,6,8,2] => ? => ? => ? = 2
[[.,.],[[[.,[.,[[.,.],.]]],.],.]]
=> [1,5,6,4,3,7,8,2] => ? => ? => ? = 1
[[.,.],[[[.,[[.,.],[.,.]]],.],.]]
=> [1,4,6,5,3,7,8,2] => ? => ? => ? = 1
[[.,.],[[[.,[[[.,.],.],.]],.],.]]
=> [1,4,5,6,3,7,8,2] => ? => ? => ? = 1
[[.,.],[[[[.,.],[.,[.,.]]],.],.]]
=> [1,3,6,5,4,7,8,2] => ? => ? => ? = 1
[[.,[.,.]],[[[[.,.],.],[.,.]],.]]
=> [2,1,4,5,7,6,8,3] => ? => ? => ? = 2
[[.,[.,.]],[[[.,[.,[.,.]]],.],.]]
=> [2,1,6,5,4,7,8,3] => ? => ? => ? = 2
Description
The number of ascents of a binary word.
Matching statistic: St001712
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [[1]]
=> 0
[.,[.,.]]
=> [2,1] => [[1],[2]]
=> 0
[[.,.],.]
=> [1,2] => [[1,2]]
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [[1],[2],[3]]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [[1,2],[3]]
=> 0
[[.,.],[.,.]]
=> [1,3,2] => [[1,2],[3]]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [[1,3],[2]]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [[1,2,3]]
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [[1,2],[3],[4]]
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [[1,2],[3],[4],[5]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [[1,2,3],[4],[5]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [[1,3],[2,4],[5]]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [[1,2,3],[4],[5]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [[1,2,4],[3],[5]]
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 0
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [5,4,8,7,6,3,2,1] => ?
=> ? = 1
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [6,5,4,8,7,3,2,1] => ?
=> ? = 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [3,6,5,8,7,4,2,1] => ?
=> ? = 1
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [3,6,5,7,8,4,2,1] => ?
=> ? = 1
[.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [3,5,4,8,7,6,2,1] => ?
=> ? = 1
[.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> [4,5,7,6,3,8,2,1] => ?
=> ? = 1
[.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> [5,4,3,7,6,8,2,1] => ?
=> ? = 2
[[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,5,7,8,6,4,3,2] => ?
=> ? = 0
[[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,5,6,8,7,4,3,2] => ?
=> ? = 0
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,4,7,6,8,5,3,2] => ?
=> ? = 1
[[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,5,4,8,7,6,3,2] => ?
=> ? = 1
[[.,.],[.,[[[.,.],.],[[.,.],.]]]]
=> [1,4,5,7,8,6,3,2] => ?
=> ? = 0
[[.,.],[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,6,5,4,8,7,3,2] => ?
=> ? = 1
[[.,.],[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,5,6,4,8,7,3,2] => ?
=> ? = 1
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,5,4,6,8,7,3,2] => ?
=> ? = 1
[[.,.],[.,[[[[.,.],.],.],[.,.]]]]
=> [1,4,5,6,8,7,3,2] => ?
=> ? = 0
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,6,7,5,4,8,3,2] => ?
=> ? = 1
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,5,7,6,4,8,3,2] => ?
=> ? = 1
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,6,5,7,4,8,3,2] => ?
=> ? = 2
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> [1,5,6,7,4,8,3,2] => ?
=> ? = 1
[[.,.],[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,5,4,7,6,8,3,2] => ?
=> ? = 2
[[.,.],[.,[[[[.,.],.],[.,.]],.]]]
=> [1,4,5,7,6,8,3,2] => ?
=> ? = 1
[[.,.],[.,[[[[.,.],[.,.]],.],.]]]
=> [1,4,6,5,7,8,3,2] => ?
=> ? = 1
[[.,.],[[.,.],[.,[[.,[.,.]],.]]]]
=> [1,3,7,6,8,5,4,2] => ?
=> ? = 1
[[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,3,6,5,8,7,4,2] => ?
=> ? = 1
[[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,3,7,6,5,8,4,2] => ?
=> ? = 1
[[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> [1,3,5,7,6,8,4,2] => ?
=> ? = 1
[[.,.],[[[.,.],.],[.,[[.,.],.]]]]
=> [1,3,4,7,8,6,5,2] => ?
=> ? = 0
[[.,.],[[[.,.],.],[[.,.],[.,.]]]]
=> [1,3,4,6,8,7,5,2] => ?
=> ? = 0
[[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,5,4,3,7,8,6,2] => ?
=> ? = 1
[[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> [1,3,5,4,7,8,6,2] => ?
=> ? = 1
[[.,.],[[.,[[.,[.,.]],.]],[.,.]]]
=> [1,5,4,6,3,8,7,2] => ?
=> ? = 2
[[.,.],[[[.,.],[[.,.],.]],[.,.]]]
=> [1,3,5,6,4,8,7,2] => ?
=> ? = 1
[[.,.],[[[[.,.],.],[.,.]],[.,.]]]
=> [1,3,4,6,5,8,7,2] => ?
=> ? = 1
[[.,.],[[[[.,.],[.,.]],.],[.,.]]]
=> [1,3,5,4,6,8,7,2] => ?
=> ? = 1
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [1,4,3,5,6,8,7,2] => ?
=> ? = 1
[[.,.],[[.,[.,[[.,.],[.,.]]]],.]]
=> [1,5,7,6,4,3,8,2] => ?
=> ? = 1
[[.,.],[[.,[[.,.],[.,[.,.]]]],.]]
=> [1,4,7,6,5,3,8,2] => ?
=> ? = 1
[[.,.],[[.,[[.,[.,.]],[.,.]]],.]]
=> [1,5,4,7,6,3,8,2] => ?
=> ? = 2
[[.,.],[[.,[[[.,.],.],[.,.]]],.]]
=> [1,4,5,7,6,3,8,2] => ?
=> ? = 1
[[.,.],[[.,[[.,[.,[.,.]]],.]],.]]
=> [1,6,5,4,7,3,8,2] => ?
=> ? = 2
[[.,.],[[[.,.],[.,[.,[.,.]]]],.]]
=> [1,3,7,6,5,4,8,2] => ?
=> ? = 1
[[.,.],[[[.,.],[.,[[.,.],.]]],.]]
=> [1,3,6,7,5,4,8,2] => ?
=> ? = 1
[[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> [1,3,5,7,6,4,8,2] => ?
=> ? = 1
[[.,.],[[[.,[.,.]],[.,[.,.]]],.]]
=> [1,4,3,7,6,5,8,2] => ?
=> ? = 2
[[.,.],[[[.,[.,.]],[[.,.],.]],.]]
=> [1,4,3,6,7,5,8,2] => ?
=> ? = 2
[[.,.],[[[.,[.,[.,.]]],[.,.]],.]]
=> [1,5,4,3,7,6,8,2] => ?
=> ? = 2
[[.,.],[[[.,[[.,.],.]],[.,.]],.]]
=> [1,4,5,3,7,6,8,2] => ?
=> ? = 2
[[.,.],[[[[.,[.,.]],.],[.,.]],.]]
=> [1,4,3,5,7,6,8,2] => ?
=> ? = 2
[[.,.],[[[.,[.,[.,[.,.]]]],.],.]]
=> [1,6,5,4,3,7,8,2] => ?
=> ? = 1
Description
The number of natural descents of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Matching statistic: St001037
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1,0]
=> 0
[.,[.,.]]
=> [2,1] => [1,1] => [1,0,1,0]
=> 0
[[.,.],.]
=> [1,2] => [2] => [1,1,0,0]
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[[.,.],[.,.]]
=> [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [5,7,8,6,4,3,2,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [5,6,8,7,4,3,2,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [4,7,8,6,5,3,2,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [4,6,8,7,5,3,2,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> [4,5,8,7,6,3,2,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [4,5,7,8,6,3,2,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [4,6,7,5,8,3,2,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [3,7,8,6,5,4,2,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> [3,4,5,6,8,7,2,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> [4,5,7,6,3,8,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [3,4,5,6,7,8,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [2,7,8,6,5,4,3,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [2,3,4,6,7,8,5,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [2,3,4,5,8,7,6,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [2,3,4,5,7,8,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [2,3,4,5,6,8,7,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[.,[[.,[.,[.,[[[.,.],.],.]]]],.]]
=> [5,6,7,4,3,2,8,1] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [3,4,5,6,7,2,8,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [2,3,4,5,7,6,8,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 1
[.,[[[[[[.,.],.],.],[.,.]],.],.]]
=> [2,3,4,6,5,7,8,1] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 1
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [3,4,5,2,6,7,8,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [2,3,5,4,6,7,8,1] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,7,8,6,5,4,3,2] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,6,8,7,5,4,3,2] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,6,7,8,5,4,3,2] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,5,8,7,6,4,3,2] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,5,7,8,6,4,3,2] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,5,6,8,7,4,3,2] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,6,7,5,8,4,3,2] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[[.,.],[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,5,7,6,8,4,3,2] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> [1,5,6,7,8,4,3,2] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,4,8,7,6,5,3,2] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,4,7,8,6,5,3,2] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,4,6,8,7,5,3,2] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,4,7,6,8,5,3,2] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
=> [1,4,6,7,8,5,3,2] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[[.,.],[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,4,5,8,7,6,3,2] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[[.,.],[.,[[[.,.],.],[[.,.],.]]]]
=> [1,4,5,7,8,6,3,2] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
[[.,.],[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,5,6,4,8,7,3,2] => ? => ?
=> ? = 1
[[.,.],[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,4,6,5,8,7,3,2] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[[.,.],[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,5,4,6,8,7,3,2] => ? => ?
=> ? = 1
[[.,.],[.,[[[[.,.],.],.],[.,.]]]]
=> [1,4,5,6,8,7,3,2] => ? => ?
=> ? = 0
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,6,7,5,4,8,3,2] => ? => ?
=> ? = 1
[[.,.],[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,5,7,6,4,8,3,2] => ? => ?
=> ? = 1
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> [1,5,6,7,4,8,3,2] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[[.,.],[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,4,7,6,5,8,3,2] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[[.,.],[.,[[[.,.],[[.,.],.]],.]]]
=> [1,4,6,7,5,8,3,2] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000568
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> [1] => [.,.]
=> ? = 0 + 1
[.,[.,.]]
=> [.,[.,.]]
=> [2,1] => [[.,.],.]
=> 1 = 0 + 1
[[.,.],.]
=> [[.,.],.]
=> [1,2] => [.,[.,.]]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [2,3,1] => [[.,.],[.,.]]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [1,3,2] => [.,[[.,.],.]]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,.],.],[.,.]]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],.],[.,.]]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 2 = 1 + 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ? = 0 + 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [6,8,7,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 + 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> [6,7,5,8,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [6,5,8,7,4,3,2,1] => [[[[[[.,.],.],.],.],.],[[.,.],.]]
=> ? = 1 + 1
[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [5,8,7,6,4,3,2,1] => [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 + 1
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [5,7,8,6,4,3,2,1] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1 + 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [5,7,6,8,4,3,2,1] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1 + 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [5,6,8,7,4,3,2,1] => [[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> ? = 1 + 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> [6,7,5,4,8,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [6,5,4,8,7,3,2,1] => ?
=> ? = 1 + 1
[.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> [.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [6,5,4,7,8,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
[.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
=> [5,6,4,7,8,3,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> ? = 0 + 1
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [5,4,8,7,6,3,2,1] => ?
=> ? = 1 + 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [4,8,7,6,5,3,2,1] => [[[[.,.],.],.],[[[[.,.],.],.],.]]
=> ? = 1 + 1
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [4,7,8,6,5,3,2,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 1 + 1
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> [.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> [4,7,6,8,5,3,2,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 1 + 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [4,6,8,7,5,3,2,1] => [[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> ? = 2 + 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [4,7,6,5,8,3,2,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 1 + 1
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [4,6,7,5,8,3,2,1] => [[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> ? = 1 + 1
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> [4,5,8,7,6,3,2,1] => [[[[.,.],.],.],[.,[[[.,.],.],.]]]
=> ? = 1 + 1
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [7,6,5,4,3,8,2,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [.,[.,[[.,[.,[.,[[.,.],.]]]],.]]]
=> [6,7,5,4,3,8,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [5,7,6,4,3,8,2,1] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1 + 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
=> [5,4,7,6,3,8,2,1] => [[[[[.,.],.],.],.],[[.,.],[.,.]]]
=> ? = 1 + 1
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> [.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [4,7,6,5,3,8,2,1] => [[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> ? = 1 + 1
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> [4,5,7,6,3,8,2,1] => ?
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [5,4,3,8,7,6,2,1] => [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 + 1
[.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> [5,4,3,7,6,8,2,1] => ?
=> ? = 1 + 1
[.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> [.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> ? = 0 + 1
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [3,8,7,6,5,4,2,1] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 + 1
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [3,7,6,8,5,4,2,1] => [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ? = 1 + 1
[.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> [3,7,6,5,8,4,2,1] => [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ? = 1 + 1
[.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
=> [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [3,6,5,8,7,4,2,1] => ?
=> ? = 2 + 1
[.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> [.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [3,6,5,7,8,4,2,1] => ?
=> ? = 1 + 1
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [3,7,6,5,4,8,2,1] => [[[.,.],.],[[[[.,.],.],.],[.,.]]]
=> ? = 1 + 1
[.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [3,5,4,8,7,6,2,1] => ?
=> ? = 2 + 1
[.,[.,[[[[.,.],[.,.]],[.,.]],.]]]
=> [.,[.,[[[[.,.],[.,.]],[.,.]],.]]]
=> [3,5,4,7,6,8,2,1] => [[[.,.],.],[[.,.],[[.,.],[.,.]]]]
=> ? = 2 + 1
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> [3,4,5,6,8,7,2,1] => [[[.,.],.],[.,[.,[.,[[.,.],.]]]]]
=> ? = 1 + 1
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [3,4,5,6,7,8,2,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 0 + 1
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> ? = 0 + 1
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [6,7,5,4,3,2,8,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> ? = 0 + 1
Description
The hook number of a binary tree.
A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.
Matching statistic: St000157
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [[1]]
=> 0
[.,[.,.]]
=> [2,1] => [1,2] => [[1,2]]
=> 0
[[.,.],.]
=> [1,2] => [1,2] => [[1,2]]
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [[1,2,3]]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => [[1,2,3]]
=> 0
[[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => [[1,2,3]]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => [[1,2],[3]]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => [[1,2,4,5],[3]]
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [8,6,7,5,4,3,2,1] => ? => ?
=> ? = 0
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [8,7,5,6,4,3,2,1] => ? => ?
=> ? = 0
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [8,6,5,7,4,3,2,1] => ? => ?
=> ? = 1
[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [8,5,6,7,4,3,2,1] => ? => ?
=> ? = 0
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> [6,7,5,8,4,3,2,1] => ? => ?
=> ? = 1
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [8,7,5,4,6,3,2,1] => ? => ?
=> ? = 1
[.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> [8,7,4,5,6,3,2,1] => ? => ?
=> ? = 0
[.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [7,8,4,5,6,3,2,1] => ? => ?
=> ? = 0
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [8,6,5,4,7,3,2,1] => ? => ?
=> ? = 1
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> [6,7,5,4,8,3,2,1] => ? => ?
=> ? = 1
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> [7,5,6,4,8,3,2,1] => ? => ?
=> ? = 1
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [6,7,4,5,8,3,2,1] => ? => ?
=> ? = 1
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [6,5,4,7,8,3,2,1] => ? => ?
=> ? = 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,3,4,2,1] => ? => ?
=> ? = 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [8,6,5,7,3,4,2,1] => ? => ?
=> ? = 1
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> [7,6,5,8,3,4,2,1] => ? => ?
=> ? = 1
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [6,5,7,8,3,4,2,1] => ? => ?
=> ? = 1
[.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [8,7,5,3,4,6,2,1] => ? => ?
=> ? = 1
[.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> [8,3,4,5,6,7,2,1] => ? => ?
=> ? = 0
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [7,5,6,4,3,8,2,1] => ? => ?
=> ? = 1
[.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [7,6,4,5,3,8,2,1] => ? => ?
=> ? = 1
[.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
=> [7,5,4,6,3,8,2,1] => ? => ?
=> ? = 2
[.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> [7,4,5,6,3,8,2,1] => ? => ?
=> ? = 1
[.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> [7,5,4,3,6,8,2,1] => ? => ?
=> ? = 2
[.,[.,[[[[.,.],[.,.]],[.,.]],.]]]
=> [7,5,3,4,6,8,2,1] => ? => ?
=> ? = 2
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [3,4,5,6,7,8,2,1] => ? => ?
=> ? = 0
[.,[[.,.],[.,[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,4,2,3,1] => ? => ?
=> ? = 1
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [6,7,8,2,3,4,5,1] => ? => ?
=> ? = 0
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [8,7,2,3,4,5,6,1] => ? => ?
=> ? = 0
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [7,8,2,3,4,5,6,1] => ? => ?
=> ? = 0
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [8,2,3,4,5,6,7,1] => ? => ?
=> ? = 0
[.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
=> [7,5,6,4,3,2,8,1] => ? => ?
=> ? = 1
[.,[[.,[.,[.,[[[.,.],.],.]]]],.]]
=> [5,6,7,4,3,2,8,1] => ? => ?
=> ? = 1
[.,[[.,[.,[[.,.],[.,[.,.]]]]],.]]
=> [7,6,4,5,3,2,8,1] => ? => ?
=> ? = 1
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [6,5,4,7,3,2,8,1] => ? => ?
=> ? = 2
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [3,4,5,6,7,2,8,1] => ? => ?
=> ? = 1
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [7,2,3,4,5,6,8,1] => ? => ?
=> ? = 1
[.,[[[[[[.,.],.],.],[.,.]],.],.]]
=> [6,2,3,4,5,7,8,1] => ? => ?
=> ? = 1
[.,[[[[[.,[.,.]],[.,.]],.],.],.]]
=> [5,3,2,4,6,7,8,1] => ? => ?
=> ? = 2
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,4,3,1,2] => ? => ?
=> ? = 1
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,4,3,1,2] => ? => ?
=> ? = 0
[[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> [7,8,5,6,4,3,1,2] => ? => ?
=> ? = 0
[[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
=> [8,6,5,7,4,3,1,2] => ? => ?
=> ? = 1
[[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
=> [8,5,6,7,4,3,1,2] => ? => ?
=> ? = 0
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [7,6,5,8,4,3,1,2] => ? => ?
=> ? = 1
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> [6,7,5,8,4,3,1,2] => ? => ?
=> ? = 1
[[.,.],[.,[.,[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,4,3,1,2] => ? => ?
=> ? = 1
[[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
=> [8,6,7,4,5,3,1,2] => ? => ?
=> ? = 0
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
=> [7,6,8,4,5,3,1,2] => ? => ?
=> ? = 1
[[.,.],[.,[[.,.],[[[.,.],.],.]]]]
=> [6,7,8,4,5,3,1,2] => ? => ?
=> ? = 0
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000919
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [.,.]
=> ? = 0
[.,[.,.]]
=> [2,1] => [1,2] => [.,[.,.]]
=> 0
[[.,.],.]
=> [1,2] => [1,2] => [.,[.,.]]
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 1
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [8,6,7,5,4,3,2,1] => ? => ?
=> ? = 0
[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [8,7,5,6,4,3,2,1] => ? => ?
=> ? = 0
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [8,6,5,7,4,3,2,1] => ? => ?
=> ? = 1
[.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> [8,5,6,7,4,3,2,1] => ? => ?
=> ? = 0
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> [6,7,5,8,4,3,2,1] => ? => ?
=> ? = 1
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [8,7,5,4,6,3,2,1] => ? => ?
=> ? = 1
[.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> [8,7,4,5,6,3,2,1] => ? => ?
=> ? = 0
[.,[.,[.,[[[.,.],.],[[.,.],.]]]]]
=> [7,8,4,5,6,3,2,1] => ? => ?
=> ? = 0
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [8,6,5,4,7,3,2,1] => ? => ?
=> ? = 1
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> [6,7,5,4,8,3,2,1] => ? => ?
=> ? = 1
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> [7,5,6,4,8,3,2,1] => ? => ?
=> ? = 1
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [6,7,4,5,8,3,2,1] => ? => ?
=> ? = 1
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [6,5,4,7,8,3,2,1] => ? => ?
=> ? = 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,3,4,2,1] => ? => ?
=> ? = 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [8,6,5,7,3,4,2,1] => ? => ?
=> ? = 1
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> [7,6,5,8,3,4,2,1] => ? => ?
=> ? = 1
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [6,5,7,8,3,4,2,1] => ? => ?
=> ? = 1
[.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [8,7,5,3,4,6,2,1] => ? => ?
=> ? = 1
[.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> [8,3,4,5,6,7,2,1] => ? => ?
=> ? = 0
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [7,5,6,4,3,8,2,1] => ? => ?
=> ? = 1
[.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [7,6,4,5,3,8,2,1] => ? => ?
=> ? = 1
[.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
=> [7,5,4,6,3,8,2,1] => ? => ?
=> ? = 2
[.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> [7,4,5,6,3,8,2,1] => ? => ?
=> ? = 1
[.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> [7,5,4,3,6,8,2,1] => ? => ?
=> ? = 2
[.,[.,[[[[.,.],[.,.]],[.,.]],.]]]
=> [7,5,3,4,6,8,2,1] => ? => ?
=> ? = 2
[.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [3,4,5,6,7,8,2,1] => ? => ?
=> ? = 0
[.,[[.,.],[.,[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,4,2,3,1] => ? => ?
=> ? = 1
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [6,7,8,2,3,4,5,1] => ? => ?
=> ? = 0
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [8,7,2,3,4,5,6,1] => ? => ?
=> ? = 0
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [7,8,2,3,4,5,6,1] => ? => ?
=> ? = 0
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [8,2,3,4,5,6,7,1] => ? => ?
=> ? = 0
[.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
=> [7,5,6,4,3,2,8,1] => ? => ?
=> ? = 1
[.,[[.,[.,[.,[[[.,.],.],.]]]],.]]
=> [5,6,7,4,3,2,8,1] => ? => ?
=> ? = 1
[.,[[.,[.,[[.,.],[.,[.,.]]]]],.]]
=> [7,6,4,5,3,2,8,1] => ? => ?
=> ? = 1
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [6,5,4,7,3,2,8,1] => ? => ?
=> ? = 2
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [3,4,5,6,7,2,8,1] => ? => ?
=> ? = 1
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [7,2,3,4,5,6,8,1] => ? => ?
=> ? = 1
[.,[[[[[[.,.],.],.],[.,.]],.],.]]
=> [6,2,3,4,5,7,8,1] => ? => ?
=> ? = 1
[.,[[[[[.,[.,.]],[.,.]],.],.],.]]
=> [5,3,2,4,6,7,8,1] => ? => ?
=> ? = 2
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> [7,6,8,5,4,3,1,2] => ? => ?
=> ? = 1
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> [6,7,8,5,4,3,1,2] => ? => ?
=> ? = 0
[[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> [7,8,5,6,4,3,1,2] => ? => ?
=> ? = 0
[[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
=> [8,6,5,7,4,3,1,2] => ? => ?
=> ? = 1
[[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
=> [8,5,6,7,4,3,1,2] => ? => ?
=> ? = 0
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [7,6,5,8,4,3,1,2] => ? => ?
=> ? = 1
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> [6,7,5,8,4,3,1,2] => ? => ?
=> ? = 1
[[.,.],[.,[.,[[[.,.],[.,.]],.]]]]
=> [7,5,6,8,4,3,1,2] => ? => ?
=> ? = 1
[[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
=> [8,6,7,4,5,3,1,2] => ? => ?
=> ? = 0
[[.,.],[.,[[.,.],[[.,[.,.]],.]]]]
=> [7,6,8,4,5,3,1,2] => ? => ?
=> ? = 1
Description
The number of maximal left branches of a binary tree.
A maximal left branch of a binary tree is an inclusion wise maximal path which consists of left edges only. This statistic records the number of distinct maximal left branches in the tree.
Matching statistic: St000069
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 1 + 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 1 + 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ([(0,6),(3,7),(4,2),(5,1),(6,3),(7,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> [.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(5,4),(6,7),(7,3)],8)
=> ([(0,5),(4,7),(5,4),(6,2),(6,3),(7,1),(7,6)],8)
=> ? = 2 + 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ([(0,6),(3,7),(4,2),(5,1),(6,3),(7,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 1 + 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> [.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> ([(0,6),(1,4),(3,7),(4,7),(5,2),(6,3),(7,5)],8)
=> ([(0,6),(3,4),(4,1),(5,2),(6,7),(7,3),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
=> [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ([(0,5),(4,3),(5,6),(6,2),(6,7),(7,1),(7,4)],8)
=> ? = 2 + 1
[.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> [.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> ([(0,6),(1,6),(2,3),(3,7),(4,5),(6,7),(7,4)],8)
=> ([(0,5),(4,3),(5,7),(6,1),(6,2),(7,4),(7,6)],8)
=> ? = 2 + 1
[.,[.,[[[[.,.],[.,.]],[.,.]],.]]]
=> [.,[.,[[[[.,.],[.,.]],[.,.]],.]]]
=> ([(0,7),(1,6),(2,6),(3,5),(5,4),(6,7),(7,3)],8)
=> ([(0,5),(4,7),(5,4),(6,2),(6,3),(7,1),(7,6)],8)
=> ? = 2 + 1
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[[.,.],[.,[.,[[.,[.,.]],.]]]]]
=> [.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 1 + 1
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
=> [.,[[.,.],[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(2,7),(4,5),(5,7),(6,4),(7,3)],8)
=> ([(0,7),(4,5),(5,6),(6,2),(6,3),(7,1),(7,4)],8)
=> ? = 2 + 1
[.,[[.,[.,[.,[[[.,.],.],.]]]],.]]
=> [.,[[.,.],[.,[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,[.,[[.,.],[.,[.,.]]]]],.]]
=> [.,[[.,.],[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,7),(2,3),(3,7),(5,6),(6,4),(7,5)],8)
=> ([(0,7),(4,6),(5,3),(6,2),(6,5),(7,1),(7,4)],8)
=> ? = 2 + 1
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [.,[[.,.],[[[[[.,.],.],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 1 + 1
[.,[[[[[[.,.],.],.],[.,.]],.],.]]
=> [.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> ([(0,6),(1,5),(2,7),(3,7),(5,2),(6,3),(7,4)],8)
=> ([(0,7),(3,5),(4,6),(5,2),(6,1),(7,3),(7,4)],8)
=> ? = 1 + 1
[.,[[[[[.,[.,.]],[.,.]],.],.],.]]
=> [.,[[[[[.,.],.],.],[.,.]],[.,.]]]
=> ([(0,7),(1,6),(2,4),(4,5),(5,6),(6,7),(7,3)],8)
=> ([(0,7),(4,5),(5,3),(6,2),(6,4),(7,1),(7,6)],8)
=> ? = 2 + 1
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [.,[[[[[[.,.],.],.],[.,.]],.],.]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ([(0,7),(3,5),(4,3),(5,2),(6,1),(7,4),(7,6)],8)
=> ? = 1 + 1
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
=> ? = 1 + 1
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> [[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 1 + 1
[[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
=> [[.,[.,[.,[[.,[.,.]],[.,.]]]]],.]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
Description
The number of maximal elements of a poset.
Matching statistic: St000071
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2 = 1 + 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 1 + 1
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> [.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[.,[[.,[[.,.],.]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 1 + 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [.,[.,[.,[.,[[[.,.],.],[.,.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> [.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[.,[.,.]]],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],[.,[.,.]]]]]]
=> ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ([(0,6),(3,7),(4,2),(5,1),(6,3),(7,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[.,[[.,.],.]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[[.,.],[.,.]]],.]]]]
=> [.,[.,[.,[[.,.],[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(5,4),(6,7),(7,3)],8)
=> ([(0,5),(4,7),(5,4),(6,2),(6,3),(7,1),(7,6)],8)
=> ? = 2 + 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> [.,[.,[.,[[[.,.],[[.,.],.]],.]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [.,[.,[.,[[[.,.],.],[.,[.,.]]]]]]
=> ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ([(0,6),(3,7),(4,2),(5,1),(6,3),(7,4),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> [.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 1 + 1
[.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> [.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> [.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> [.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> ([(0,6),(1,4),(3,7),(4,7),(5,2),(6,3),(7,5)],8)
=> ([(0,6),(3,4),(4,1),(5,2),(6,7),(7,3),(7,5)],8)
=> ? = 1 + 1
[.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[.,[.,[[.,.],[.,.]]]],.]]]
=> [.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[.,[[.,.],[.,[.,.]]]],.]]]
=> [.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[.,[[.,[.,.]],[.,.]]],.]]]
=> [.,[.,[[.,.],[[.,[.,.]],[.,.]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ([(0,5),(4,3),(5,6),(6,2),(6,7),(7,1),(7,4)],8)
=> ? = 2 + 1
[.,[.,[[.,[[[.,.],.],[.,.]]],.]]]
=> [.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> ([(0,6),(1,6),(2,3),(3,7),(4,5),(6,7),(7,4)],8)
=> ([(0,5),(4,3),(5,7),(6,1),(6,2),(7,4),(7,6)],8)
=> ? = 2 + 1
[.,[.,[[[[.,.],[.,.]],[.,.]],.]]]
=> [.,[.,[[[[.,.],[.,.]],[.,.]],.]]]
=> ([(0,7),(1,6),(2,6),(3,5),(5,4),(6,7),(7,3)],8)
=> ([(0,5),(4,7),(5,4),(6,2),(6,3),(7,1),(7,6)],8)
=> ? = 2 + 1
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[[.,.],[.,[.,[[.,[.,.]],.]]]]]
=> [.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 1 + 1
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,[.,[.,[[.,.],[.,.]]]]],.]]
=> [.,[[.,.],[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(2,7),(4,5),(5,7),(6,4),(7,3)],8)
=> ([(0,7),(4,5),(5,6),(6,2),(6,3),(7,1),(7,4)],8)
=> ? = 2 + 1
[.,[[.,[.,[.,[[[.,.],.],.]]]],.]]
=> [.,[[.,.],[.,[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,[.,[[.,.],[.,[.,.]]]]],.]]
=> [.,[[.,.],[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[[.,[.,[[.,[.,[.,.]]],.]]],.]]
=> [.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,7),(2,3),(3,7),(5,6),(6,4),(7,5)],8)
=> ([(0,7),(4,6),(5,3),(6,2),(6,5),(7,1),(7,4)],8)
=> ? = 2 + 1
[.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [.,[[.,.],[[[[[.,.],.],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 1 + 1
[.,[[[[[[.,.],.],.],[.,.]],.],.]]
=> [.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[.,[[[[.,[[[.,.],.],.]],.],.],.]]
=> [.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> ([(0,6),(1,5),(2,7),(3,7),(5,2),(6,3),(7,4)],8)
=> ([(0,7),(3,5),(4,6),(5,2),(6,1),(7,3),(7,4)],8)
=> ? = 1 + 1
[.,[[[[[.,[.,.]],[.,.]],.],.],.]]
=> [.,[[[[[.,.],.],.],[.,.]],[.,.]]]
=> ([(0,7),(1,6),(2,4),(4,5),(5,6),(6,7),(7,3)],8)
=> ([(0,7),(4,5),(5,3),(6,2),(6,4),(7,1),(7,6)],8)
=> ? = 2 + 1
[.,[[[[[[.,.],.],[.,.]],.],.],.]]
=> [.,[[[[[[.,.],.],.],[.,.]],.],.]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ([(0,6),(3,7),(4,5),(5,1),(6,3),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[[[[[.,[[.,.],.]],.],.],.],.]]
=> [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> ([(0,6),(1,4),(2,7),(4,7),(5,2),(6,5),(7,3)],8)
=> ([(0,7),(3,5),(4,3),(5,2),(6,1),(7,4),(7,6)],8)
=> ? = 1 + 1
[.,[[[[[[.,.],[.,.]],.],.],.],.]]
=> [.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,1),(6,7),(7,2),(7,4)],8)
=> ? = 1 + 1
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ([(0,7),(3,4),(4,6),(5,3),(6,1),(7,2),(7,5)],8)
=> ? = 1 + 1
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> [[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 1 + 1
[[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
=> [[.,[.,[.,[[.,[.,.]],[.,.]]]]],.]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 1 + 1
Description
The number of maximal chains in a poset.
The following 71 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000527The width of the poset. St000068The number of minimal elements in a poset. St000201The number of leaf nodes in a binary tree. St001083The number of boxed occurrences of 132 in a permutation. St000010The length of the partition. St000257The number of distinct parts of a partition that occur at least twice. St000097The order of the largest clique of the graph. St001581The achromatic number of a graph. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000632The jump number of the poset. St001840The number of descents of a set partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001354The number of series nodes in the modular decomposition of a graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001304The number of maximally independent sets of vertices of a graph. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St000354The number of recoils of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000353The number of inner valleys of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000647The number of big descents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000672The number of minimal elements in Bruhat order not less than the permutation. St000455The second largest eigenvalue of a graph if it is integral. St000779The tier of a permutation. St000710The number of big deficiencies of a permutation. St001728The number of invisible descents of a permutation. St000711The number of big exceedences of a permutation. St000646The number of big ascents of a permutation. St000251The number of nonsingleton blocks of a set partition. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000360The number of occurrences of the pattern 32-1. St000092The number of outer peaks of a permutation. St000023The number of inner peaks of a permutation. St000523The number of 2-protected nodes of a rooted tree. St000099The number of valleys of a permutation, including the boundary. St000021The number of descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001812The biclique partition number of a graph. St000325The width of the tree associated to a permutation. St000409The number of pitchforks in a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St001597The Frobenius rank of a skew partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000298The order dimension or Dushnik-Miller dimension of a poset. St000035The number of left outer peaks of a permutation. St000307The number of rowmotion orbits of a poset. St000884The number of isolated descents of a permutation. St001330The hat guessing number of a graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000640The rank of the largest boolean interval in a poset. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000252The number of nodes of degree 3 of a binary tree. St001896The number of right descents of a signed permutations. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001823The Stasinski-Voll length of a signed permutation. St000805The number of peaks of the associated bargraph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
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