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Your data matches 220 different statistics following compositions of up to 3 maps.
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Matching statistic: St000167
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
St000167: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> 1
[[],[]]
=> 2
[[[]]]
=> 1
[[],[],[]]
=> 3
[[],[[]]]
=> 2
[[[]],[]]
=> 2
[[[],[]]]
=> 2
[[[[]]]]
=> 1
[[],[],[],[]]
=> 4
[[],[],[[]]]
=> 3
[[],[[]],[]]
=> 3
[[],[[],[]]]
=> 3
[[],[[[]]]]
=> 2
[[[]],[],[]]
=> 3
[[[]],[[]]]
=> 2
[[[],[]],[]]
=> 3
[[[[]]],[]]
=> 2
[[[],[],[]]]
=> 3
[[[],[[]]]]
=> 2
[[[[]],[]]]
=> 2
[[[[],[]]]]
=> 2
[[[[[]]]]]
=> 1
[[],[],[],[],[]]
=> 5
[[],[],[],[[]]]
=> 4
[[],[],[[]],[]]
=> 4
[[],[],[[],[]]]
=> 4
[[],[],[[[]]]]
=> 3
[[],[[]],[],[]]
=> 4
[[],[[]],[[]]]
=> 3
[[],[[],[]],[]]
=> 4
[[],[[[]]],[]]
=> 3
[[],[[],[],[]]]
=> 4
[[],[[],[[]]]]
=> 3
[[],[[[]],[]]]
=> 3
[[],[[[],[]]]]
=> 3
[[],[[[[]]]]]
=> 2
[[[]],[],[],[]]
=> 4
[[[]],[],[[]]]
=> 3
[[[]],[[]],[]]
=> 3
[[[]],[[],[]]]
=> 3
[[[]],[[[]]]]
=> 2
[[[],[]],[],[]]
=> 4
[[[[]]],[],[]]
=> 3
[[[],[]],[[]]]
=> 3
[[[[]]],[[]]]
=> 2
[[[],[],[]],[]]
=> 4
[[[],[[]]],[]]
=> 3
[[[[]],[]],[]]
=> 3
[[[[],[]]],[]]
=> 3
[[[[[]]]],[]]
=> 2
Description
The number of leaves of an ordered tree.
This is the number of nodes which do not have any children.
Matching statistic: St000010
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 94%●distinct values known / distinct values provided: 80%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 94%●distinct values known / distinct values provided: 80%
Values
[[]]
=> [1,0]
=> {{1}}
=> [1]
=> 1
[[],[]]
=> [1,0,1,0]
=> {{1},{2}}
=> [1,1]
=> 2
[[[]]]
=> [1,1,0,0]
=> {{1,2}}
=> [2]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> [1,1,1]
=> 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> [2,1]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> [2,1]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> [2,1]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> [3]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> [2,1,1]
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> [2,1,1]
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> [2,1,1]
=> 3
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> [3,1]
=> 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> [2,1,1]
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> [2,2]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> [2,1,1]
=> 3
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> [3,1]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> [2,1,1]
=> 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> [3,1]
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> [2,2]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> [3,1]
=> 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> [4]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> 4
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> [3,1,1]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> [2,2,1]
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> 4
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> [3,1,1]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> 4
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> [3,1,1]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> [2,2,1]
=> 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> [3,1,1]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> [4,1]
=> 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> [2,2,1]
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> [2,2,1]
=> 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> [3,2]
=> 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 4
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> [3,1,1]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> [2,2,1]
=> 3
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> [3,2]
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> 4
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> [2,2,1]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> [3,1,1]
=> 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> [4,1]
=> 2
[[[[[[[[[]]]]]]]],[]]
=> ?
=> ?
=> ?
=> ? = 2
[[],[[[[[[[[]]]]]]]]]
=> ?
=> ?
=> ?
=> ? = 2
[[[[[[[[[[]]]]]]]]],[]]
=> ?
=> ?
=> ?
=> ? = 2
[[[[[[[[],[]]]]]]],[]]
=> ?
=> ?
=> ?
=> ? = 3
[[],[[[[[[[],[]]]]]]]]
=> ?
=> ?
=> ?
=> ? = 3
[[],[[[[[[[[[]]]]]]]]]]
=> ?
=> ?
=> ?
=> ? = 2
[[[[[[[[[[[]]]]]]]]]],[]]
=> ?
=> ?
=> ?
=> ? = 2
[[[[[[[[[],[]]]]]]]],[]]
=> ?
=> ?
=> ?
=> ? = 3
[[[[[[[[]],[]]]]]],[]]
=> ?
=> ?
=> ?
=> ? = 3
[[[[[[[],[[]]]]]]],[]]
=> ?
=> ?
=> ?
=> ? = 3
[[],[[[[[[[]],[]]]]]]]
=> ?
=> ?
=> ?
=> ? = 3
[[],[[[[[[],[[]]]]]]]]
=> ?
=> ?
=> ?
=> ? = 3
[[],[[[[[[[[],[]]]]]]]]]
=> ?
=> ?
=> ?
=> ? = 3
[[],[[[[[[[[[[]]]]]]]]]]]
=> ?
=> ?
=> ?
=> ? = 2
[[],[],[],[],[],[],[],[],[]]
=> ?
=> ?
=> ?
=> ? = 9
[[],[],[],[],[],[],[],[[]]]
=> ?
=> ?
=> ?
=> ? = 8
[[],[],[],[],[],[],[[],[]]]
=> ?
=> ?
=> ?
=> ? = 8
[[],[],[],[],[],[[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 8
[[],[],[],[],[[],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 8
[[],[],[],[[],[],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 8
[[],[],[[],[],[],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 8
[[],[[],[],[],[],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 8
[[[],[],[],[],[],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 8
[[],[],[],[],[],[],[],[],[],[]]
=> ?
=> ?
=> ?
=> ? = 10
[[],[],[],[],[],[],[],[],[[]]]
=> ?
=> ?
=> ?
=> ? = 9
[[],[],[],[],[],[],[[[]]]]
=> ?
=> ?
=> ?
=> ? = 7
[[],[],[],[],[],[],[],[[],[]]]
=> ?
=> ?
=> ?
=> ? = 9
[[],[],[],[],[],[[[]],[]]]
=> ?
=> ?
=> ?
=> ? = 7
[[],[],[],[],[],[],[[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 9
[[],[],[],[],[[[]],[],[]]]
=> ?
=> ?
=> ?
=> ? = 7
[[],[],[],[],[],[[],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 9
[[],[],[],[[[]],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 7
[[],[],[],[],[[],[],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 9
[[],[],[[[]],[],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 7
[[],[],[],[[],[],[],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 9
[[],[[[]],[],[],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 7
[[],[],[[],[],[],[],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 9
[[[[]],[],[],[],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 7
[[],[[],[],[],[],[],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 9
[[[],[],[],[],[],[],[],[],[]]]
=> ?
=> ?
=> ?
=> ? = 9
Description
The length of the partition.
Matching statistic: St000288
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => => ? = 1 - 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => 1 => 1 = 2 - 1
[[[]]]
=> [[.,.],.]
=> [1,2] => 0 => 0 = 1 - 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 11 => 2 = 3 - 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 10 => 1 = 2 - 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => 10 => 1 = 2 - 1
[[[],[]]]
=> [[.,.],[.,.]]
=> [1,3,2] => 01 => 1 = 2 - 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 00 => 0 = 1 - 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 110 => 2 = 3 - 1
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 110 => 2 = 3 - 1
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 101 => 2 = 3 - 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 100 => 1 = 2 - 1
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 2 = 3 - 1
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 100 => 1 = 2 - 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 101 => 2 = 3 - 1
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 1 = 2 - 1
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 011 => 2 = 3 - 1
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => 010 => 1 = 2 - 1
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => 001 => 1 = 2 - 1
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => 010 => 1 = 2 - 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1111 => 4 = 5 - 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1110 => 3 = 4 - 1
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1110 => 3 = 4 - 1
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 1101 => 3 = 4 - 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 1100 => 2 = 3 - 1
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1110 => 3 = 4 - 1
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1100 => 2 = 3 - 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1101 => 3 = 4 - 1
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1100 => 2 = 3 - 1
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 1011 => 3 = 4 - 1
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 1010 => 2 = 3 - 1
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 1001 => 2 = 3 - 1
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 1010 => 2 = 3 - 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1000 => 1 = 2 - 1
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1110 => 3 = 4 - 1
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 1100 => 2 = 3 - 1
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 1100 => 2 = 3 - 1
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0101 => 2 = 3 - 1
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 1000 => 1 = 2 - 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1011 => 3 = 4 - 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1100 => 2 = 3 - 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1010 => 2 = 3 - 1
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 1000 => 1 = 2 - 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1101 => 3 = 4 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 1001 => 2 = 3 - 1
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 1001 => 2 = 3 - 1
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 1010 => 2 = 3 - 1
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 1000 => 1 = 2 - 1
[[[],[],[],[]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0111 => 3 = 4 - 1
[[],[],[[[],[]],[],[]]]
=> [.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [3,5,4,8,7,6,2,1] => ? => ? = 6 - 1
[[],[[[],[[],[]]],[]]]
=> [.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [3,5,4,2,6,8,7,1] => ? => ? = 5 - 1
[[[],[[],[[],[]]]],[]]
=> [[.,[[.,[[.,[.,.]],[.,.]]],.]],.]
=> [4,3,6,5,2,7,1,8] => ? => ? = 5 - 1
[[],[[],[[],[[],[[],[]]]]]]
=> [.,[[.,[[.,[[.,[[.,.],[.,.]]],.]],.]],.]]
=> [5,7,6,4,8,3,9,2,10,1] => ? => ? = 6 - 1
[[],[[],[[],[[[],[]],[]]]]]
=> [.,[[.,[[.,[[[.,.],[.,.]],[.,.]]],.]],.]]
=> [4,6,5,8,7,3,9,2,10,1] => ? => ? = 6 - 1
[[],[[],[[[],[]],[[],[]]]]]
=> [.,[[.,[[[.,.],[.,.]],[[.,.],[.,.]]]],.]]
=> [3,5,4,7,9,8,6,2,10,1] => ? => ? = 6 - 1
[[],[[],[[[],[[],[]]],[]]]]
=> [.,[[.,[[[.,[[.,.],[.,.]]],.],[.,.]]],.]]
=> [4,6,5,3,7,9,8,2,10,1] => ? => ? = 6 - 1
[[],[[],[[[[],[]],[]],[]]]]
=> [.,[[.,[[[[.,.],[.,.]],[.,.]],[.,.]]],.]]
=> [3,5,4,7,6,9,8,2,10,1] => ? => ? = 6 - 1
[[],[[[],[]],[[],[[],[]]]]]
=> [.,[[[.,.],[.,.]],[[.,[[.,.],[.,.]]],.]]]
=> [2,4,3,7,9,8,6,10,5,1] => ? => ? = 6 - 1
[[],[[[],[]],[[[],[]],[]]]]
=> [.,[[[.,.],[.,.]],[[[.,.],[.,.]],[.,.]]]]
=> [2,4,3,6,8,7,10,9,5,1] => ? => ? = 6 - 1
[[],[[[],[[],[]]],[[],[]]]]
=> [.,[[[.,[[.,.],[.,.]]],.],[[.,.],[.,.]]]]
=> [3,5,4,2,6,8,10,9,7,1] => ? => ? = 6 - 1
[[],[[[[],[]],[]],[[],[]]]]
=> [.,[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [2,4,3,6,5,8,10,9,7,1] => ? => ? = 6 - 1
[[],[[[],[[],[[],[]]]],[]]]
=> [.,[[[.,[[.,[[.,.],[.,.]]],.]],.],[.,.]]]
=> [4,6,5,3,7,2,8,10,9,1] => ? => ? = 6 - 1
[[],[[[],[[[],[]],[]]],[]]]
=> [.,[[[.,[[[.,.],[.,.]],[.,.]]],.],[.,.]]]
=> [3,5,4,7,6,2,8,10,9,1] => ? => ? = 6 - 1
[[],[[[[],[]],[[],[]]],[]]]
=> [.,[[[[.,.],[.,.]],[[.,.],[.,.]]],[.,.]]]
=> [2,4,3,6,8,7,5,10,9,1] => ? => ? = 6 - 1
[[],[[[[],[[],[]]],[]],[]]]
=> [.,[[[[.,[[.,.],[.,.]]],.],[.,.]],[.,.]]]
=> [3,5,4,2,6,8,7,10,9,1] => ? => ? = 6 - 1
[[[],[]],[[],[[],[[],[]]]]]
=> [[.,[.,.]],[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [2,1,6,8,7,5,9,4,10,3] => ? => ? = 6 - 1
[[[],[]],[[],[[[],[]],[]]]]
=> [[.,[.,.]],[[.,[[[.,.],[.,.]],[.,.]]],.]]
=> [2,1,5,7,6,9,8,4,10,3] => ? => ? = 6 - 1
[[[],[]],[[[],[]],[[],[]]]]
=> [[.,[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [2,1,4,6,5,8,10,9,7,3] => ? => ? = 6 - 1
[[[],[]],[[[],[[],[]]],[]]]
=> [[.,[.,.]],[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [2,1,5,7,6,4,8,10,9,3] => ? => ? = 6 - 1
[[[],[]],[[[[],[]],[]],[]]]
=> [[.,[.,.]],[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [2,1,4,6,5,8,7,10,9,3] => ? => ? = 6 - 1
[[[],[[],[]]],[[],[[],[]]]]
=> [[.,[[.,[.,.]],[[.,[[.,.],[.,.]]],.]]],.]
=> [3,2,6,8,7,5,9,4,1,10] => ? => ? = 6 - 1
[[[],[[],[]]],[[[],[]],[]]]
=> [[.,[[.,[.,.]],[[[.,.],[.,.]],[.,.]]]],.]
=> [3,2,5,7,6,9,8,4,1,10] => ? => ? = 6 - 1
[[[[],[]],[]],[[],[[],[]]]]
=> [[[.,[.,.]],[[.,[[.,.],[.,.]]],.]],[.,.]]
=> [2,1,5,7,6,4,8,3,10,9] => ? => ? = 6 - 1
[[[[],[]],[]],[[[],[]],[]]]
=> [[[.,[.,.]],[[[.,.],[.,.]],[.,.]]],[.,.]]
=> [2,1,4,6,5,8,7,3,10,9] => ? => ? = 6 - 1
[[[],[[],[[],[]]]],[[],[]]]
=> [[.,[[.,[[.,[.,.]],[[.,.],[.,.]]]],.]],.]
=> [4,3,6,8,7,5,2,9,1,10] => ? => ? = 6 - 1
[[[],[[[],[]],[]]],[[],[]]]
=> [[.,[[[.,[.,.]],[[.,.],[.,.]]],[.,.]]],.]
=> [3,2,5,7,6,4,9,8,1,10] => ? => ? = 6 - 1
[[[[],[[],[]]],[]],[[],[]]]
=> [[[.,[[.,[.,.]],[[.,.],[.,.]]]],.],[.,.]]
=> [3,2,5,7,6,4,1,8,10,9] => ? => ? = 6 - 1
[[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,[[.,[[.,[.,.]],[.,.]]],.]],.]],.]
=> [5,4,7,6,3,8,2,9,1,10] => ? => ? = 6 - 1
[[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,[[[.,[.,.]],[.,.]],[.,.]]],.]],.]
=> [4,3,6,5,8,7,2,9,1,10] => ? => ? = 6 - 1
[[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]],.]
=> [2,4,3,7,9,8,6,5,1,10] => ? => ? = 6 - 1
[[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[[.,[[.,[.,.]],[.,.]]],.],[.,.]]],.]
=> [4,3,6,5,2,7,9,8,1,10] => ? => ? = 6 - 1
[[[[],[]],[[],[[],[]]]],[]]
=> [[[.,.],[.,.]],[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,3,2,7,9,8,6,10,5,4] => ? => ? = 6 - 1
[[[[],[]],[[[],[]],[]]],[]]
=> [[[.,.],[.,.]],[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,3,2,6,8,7,10,9,5,4] => ? => ? = 6 - 1
[[[[],[[],[]]],[[],[]]],[]]
=> [[[.,[[.,.],[.,.]]],.],[.,[[.,.],[.,.]]]]
=> [2,4,3,1,5,8,10,9,7,6] => ? => ? = 6 - 1
[[[[[],[]],[]],[[],[]]],[]]
=> [[[[.,.],[.,.]],[.,.]],[.,[[.,.],[.,.]]]]
=> [1,3,2,5,4,8,10,9,7,6] => ? => ? = 6 - 1
[[[[],[[],[[],[]]]],[]],[]]
=> [[[.,[[.,[[.,[.,.]],[.,.]]],.]],.],[.,.]]
=> [4,3,6,5,2,7,1,8,10,9] => ? => ? = 6 - 1
[[[[],[[[],[]],[]]],[]],[]]
=> [[[.,[[[.,[.,.]],[.,.]],[.,.]]],.],[.,.]]
=> [3,2,5,4,7,6,1,8,10,9] => ? => ? = 6 - 1
[[[[[],[]],[[],[]]],[]],[]]
=> [[[[.,.],[.,.]],[.,[[.,.],[.,.]]]],[.,.]]
=> [1,3,2,6,8,7,5,4,10,9] => ? => ? = 6 - 1
[[[[[],[[],[]]],[]],[]],[]]
=> [[[[.,[[.,[.,.]],[.,.]]],.],[.,.]],[.,.]]
=> [3,2,5,4,1,6,8,7,10,9] => ? => ? = 6 - 1
[[],[[],[[],[[],[[],[[],[]]]]]]]
=> [.,[[.,[[.,[[.,[[.,[[.,.],[.,.]]],.]],.]],.]],.]]
=> [6,8,7,5,9,4,10,3,11,2,12,1] => ? => ? = 7 - 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [.,[[.,[[.,[[.,[[[.,.],[.,.]],[.,.]]],.]],.]],.]]
=> [5,7,6,9,8,4,10,3,11,2,12,1] => ? => ? = 7 - 1
[[],[[],[[],[[[],[]],[[],[]]]]]]
=> [.,[[.,[[.,[[[.,.],[.,.]],[[.,.],[.,.]]]],.]],.]]
=> [4,6,5,8,10,9,7,3,11,2,12,1] => ? => ? = 7 - 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [.,[[.,[[.,[[[.,[[.,.],[.,.]]],.],[.,.]]],.]],.]]
=> [5,7,6,4,8,10,9,3,11,2,12,1] => ? => ? = 7 - 1
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [.,[[.,[[.,[[[[.,.],[.,.]],[.,.]],[.,.]]],.]],.]]
=> [4,6,5,8,7,10,9,3,11,2,12,1] => ? => ? = 7 - 1
[[],[[],[[[],[]],[[],[[],[]]]]]]
=> [.,[[.,[[[.,.],[.,.]],[[.,[[.,.],[.,.]]],.]]],.]]
=> [3,5,4,8,10,9,7,11,6,2,12,1] => ? => ? = 7 - 1
[[],[[],[[[],[]],[[[],[]],[]]]]]
=> [.,[[.,[[[.,.],[.,.]],[[[.,.],[.,.]],[.,.]]]],.]]
=> [3,5,4,7,9,8,11,10,6,2,12,1] => ? => ? = 7 - 1
[[],[[],[[[],[[],[]]],[[],[]]]]]
=> [.,[[.,[[[.,[[.,.],[.,.]]],.],[[.,.],[.,.]]]],.]]
=> [4,6,5,3,7,9,11,10,8,2,12,1] => ? => ? = 7 - 1
[[],[[],[[[[],[]],[]],[[],[]]]]]
=> [.,[[.,[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]]],.]]
=> [3,5,4,7,6,9,11,10,8,2,12,1] => ? => ? = 7 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St001176
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => [1]
=> 0 = 1 - 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => [1,1]
=> 1 = 2 - 1
[[[]]]
=> [[.,.],.]
=> [1,2] => [2]
=> 0 = 1 - 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1]
=> 2 = 3 - 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 1 = 2 - 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 1 = 2 - 1
[[[],[]]]
=> [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 1 = 2 - 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => [3]
=> 0 = 1 - 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1]
=> 3 = 4 - 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1]
=> 2 = 3 - 1
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,1]
=> 2 = 3 - 1
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1]
=> 2 = 3 - 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1]
=> 1 = 2 - 1
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,1]
=> 2 = 3 - 1
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1]
=> 1 = 2 - 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2 = 3 - 1
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,1]
=> 1 = 2 - 1
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1]
=> 2 = 3 - 1
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1]
=> 1 = 2 - 1
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1]
=> 1 = 2 - 1
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1]
=> 1 = 2 - 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [4]
=> 0 = 1 - 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1]
=> 4 = 5 - 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1]
=> 3 = 4 - 1
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,1,1]
=> 3 = 4 - 1
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,1,1,1]
=> 3 = 4 - 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 2 = 3 - 1
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,1,1,1]
=> 3 = 4 - 1
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,1]
=> 2 = 3 - 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,2,1]
=> 3 = 4 - 1
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,1,1]
=> 2 = 3 - 1
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1]
=> 3 = 4 - 1
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1]
=> 2 = 3 - 1
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1]
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,1,1]
=> 2 = 3 - 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,1]
=> 1 = 2 - 1
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [2,1,1,1]
=> 3 = 4 - 1
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,1,1]
=> 2 = 3 - 1
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,1,1]
=> 2 = 3 - 1
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,1,1]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,1]
=> 1 = 2 - 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,2,1]
=> 3 = 4 - 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,1,1]
=> 2 = 3 - 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,2]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [4,1]
=> 1 = 2 - 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,2,1]
=> 3 = 4 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2]
=> 2 = 3 - 1
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [3,2]
=> 2 = 3 - 1
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [3,2]
=> 2 = 3 - 1
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [4,1]
=> 1 = 2 - 1
[[],[],[],[[[],[]],[]]]
=> [.,[.,[.,[[[.,.],[.,.]],[.,.]]]]]
=> [4,6,5,8,7,3,2,1] => ?
=> ? = 6 - 1
[[],[],[[[]],[],[],[]]]
=> [.,[.,[[[.,.],.],[.,[.,[.,.]]]]]]
=> [3,4,8,7,6,5,2,1] => ?
=> ? = 6 - 1
[[],[],[[[],[]],[],[]]]
=> [.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [3,5,4,8,7,6,2,1] => ?
=> ? = 6 - 1
[[],[],[[[[]]],[],[]]]
=> [.,[.,[[[[.,.],.],.],[.,[.,.]]]]]
=> [3,4,5,8,7,6,2,1] => ?
=> ? = 5 - 1
[[],[[[],[[],[]]],[]]]
=> [.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [3,5,4,2,6,8,7,1] => ?
=> ? = 5 - 1
[[],[[[[[],[],[]]]]]]
=> [.,[[[[[.,.],[.,[.,.]]],.],.],.]]
=> [2,5,4,3,6,7,8,1] => ?
=> ? = 4 - 1
[[[],[[],[[],[]]]],[]]
=> [[.,[[.,[[.,[.,.]],[.,.]]],.]],.]
=> [4,3,6,5,2,7,1,8] => ?
=> ? = 5 - 1
[[[[[[],[],[]]]]],[]]
=> [[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> [3,2,1,5,4,6,7,8] => ?
=> ? = 4 - 1
[[[[[[[]],[]]]]],[]]
=> [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 3 - 1
[[],[[],[[],[[],[[],[]]]]]]
=> [.,[[.,[[.,[[.,[[.,.],[.,.]]],.]],.]],.]]
=> [5,7,6,4,8,3,9,2,10,1] => ?
=> ? = 6 - 1
[[],[[],[[],[[[],[]],[]]]]]
=> [.,[[.,[[.,[[[.,.],[.,.]],[.,.]]],.]],.]]
=> [4,6,5,8,7,3,9,2,10,1] => ?
=> ? = 6 - 1
[[],[[],[[[],[]],[[],[]]]]]
=> [.,[[.,[[[.,.],[.,.]],[[.,.],[.,.]]]],.]]
=> [3,5,4,7,9,8,6,2,10,1] => ?
=> ? = 6 - 1
[[],[[],[[[],[[],[]]],[]]]]
=> [.,[[.,[[[.,[[.,.],[.,.]]],.],[.,.]]],.]]
=> [4,6,5,3,7,9,8,2,10,1] => ?
=> ? = 6 - 1
[[],[[],[[[[],[]],[]],[]]]]
=> [.,[[.,[[[[.,.],[.,.]],[.,.]],[.,.]]],.]]
=> [3,5,4,7,6,9,8,2,10,1] => ?
=> ? = 6 - 1
[[],[[[],[]],[[],[[],[]]]]]
=> [.,[[[.,.],[.,.]],[[.,[[.,.],[.,.]]],.]]]
=> [2,4,3,7,9,8,6,10,5,1] => ?
=> ? = 6 - 1
[[],[[[],[]],[[[],[]],[]]]]
=> [.,[[[.,.],[.,.]],[[[.,.],[.,.]],[.,.]]]]
=> [2,4,3,6,8,7,10,9,5,1] => ?
=> ? = 6 - 1
[[],[[[],[[],[]]],[[],[]]]]
=> [.,[[[.,[[.,.],[.,.]]],.],[[.,.],[.,.]]]]
=> [3,5,4,2,6,8,10,9,7,1] => ?
=> ? = 6 - 1
[[],[[[[],[]],[]],[[],[]]]]
=> [.,[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [2,4,3,6,5,8,10,9,7,1] => ?
=> ? = 6 - 1
[[],[[[],[[],[[],[]]]],[]]]
=> [.,[[[.,[[.,[[.,.],[.,.]]],.]],.],[.,.]]]
=> [4,6,5,3,7,2,8,10,9,1] => ?
=> ? = 6 - 1
[[],[[[],[[[],[]],[]]],[]]]
=> [.,[[[.,[[[.,.],[.,.]],[.,.]]],.],[.,.]]]
=> [3,5,4,7,6,2,8,10,9,1] => ?
=> ? = 6 - 1
[[],[[[[],[]],[[],[]]],[]]]
=> [.,[[[[.,.],[.,.]],[[.,.],[.,.]]],[.,.]]]
=> [2,4,3,6,8,7,5,10,9,1] => ?
=> ? = 6 - 1
[[],[[[[],[[],[]]],[]],[]]]
=> [.,[[[[.,[[.,.],[.,.]]],.],[.,.]],[.,.]]]
=> [3,5,4,2,6,8,7,10,9,1] => ?
=> ? = 6 - 1
[[[],[]],[[],[[],[[],[]]]]]
=> [[.,[.,.]],[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [2,1,6,8,7,5,9,4,10,3] => ?
=> ? = 6 - 1
[[[],[]],[[],[[[],[]],[]]]]
=> [[.,[.,.]],[[.,[[[.,.],[.,.]],[.,.]]],.]]
=> [2,1,5,7,6,9,8,4,10,3] => ?
=> ? = 6 - 1
[[[],[]],[[[],[]],[[],[]]]]
=> [[.,[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [2,1,4,6,5,8,10,9,7,3] => ?
=> ? = 6 - 1
[[[],[]],[[[],[[],[]]],[]]]
=> [[.,[.,.]],[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [2,1,5,7,6,4,8,10,9,3] => ?
=> ? = 6 - 1
[[[],[]],[[[[],[]],[]],[]]]
=> [[.,[.,.]],[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [2,1,4,6,5,8,7,10,9,3] => ?
=> ? = 6 - 1
[[[],[[],[]]],[[],[[],[]]]]
=> [[.,[[.,[.,.]],[[.,[[.,.],[.,.]]],.]]],.]
=> [3,2,6,8,7,5,9,4,1,10] => ?
=> ? = 6 - 1
[[[],[[],[]]],[[[],[]],[]]]
=> [[.,[[.,[.,.]],[[[.,.],[.,.]],[.,.]]]],.]
=> [3,2,5,7,6,9,8,4,1,10] => ?
=> ? = 6 - 1
[[[[],[]],[]],[[],[[],[]]]]
=> [[[.,[.,.]],[[.,[[.,.],[.,.]]],.]],[.,.]]
=> [2,1,5,7,6,4,8,3,10,9] => ?
=> ? = 6 - 1
[[[[],[]],[]],[[[],[]],[]]]
=> [[[.,[.,.]],[[[.,.],[.,.]],[.,.]]],[.,.]]
=> [2,1,4,6,5,8,7,3,10,9] => ?
=> ? = 6 - 1
[[[],[[],[[],[]]]],[[],[]]]
=> [[.,[[.,[[.,[.,.]],[[.,.],[.,.]]]],.]],.]
=> [4,3,6,8,7,5,2,9,1,10] => ?
=> ? = 6 - 1
[[[],[[[],[]],[]]],[[],[]]]
=> [[.,[[[.,[.,.]],[[.,.],[.,.]]],[.,.]]],.]
=> [3,2,5,7,6,4,9,8,1,10] => ?
=> ? = 6 - 1
[[[[],[[],[]]],[]],[[],[]]]
=> [[[.,[[.,[.,.]],[[.,.],[.,.]]]],.],[.,.]]
=> [3,2,5,7,6,4,1,8,10,9] => ?
=> ? = 6 - 1
[[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,[[.,[[.,[.,.]],[.,.]]],.]],.]],.]
=> [5,4,7,6,3,8,2,9,1,10] => ?
=> ? = 6 - 1
[[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,[[[.,[.,.]],[.,.]],[.,.]]],.]],.]
=> [4,3,6,5,8,7,2,9,1,10] => ?
=> ? = 6 - 1
[[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]],.]
=> [2,4,3,7,9,8,6,5,1,10] => ?
=> ? = 6 - 1
[[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[[.,[[.,[.,.]],[.,.]]],.],[.,.]]],.]
=> [4,3,6,5,2,7,9,8,1,10] => ?
=> ? = 6 - 1
[[[[],[]],[[],[[],[]]]],[]]
=> [[[.,.],[.,.]],[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,3,2,7,9,8,6,10,5,4] => ?
=> ? = 6 - 1
[[[[],[]],[[[],[]],[]]],[]]
=> [[[.,.],[.,.]],[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,3,2,6,8,7,10,9,5,4] => ?
=> ? = 6 - 1
[[[[],[[],[]]],[[],[]]],[]]
=> [[[.,[[.,.],[.,.]]],.],[.,[[.,.],[.,.]]]]
=> [2,4,3,1,5,8,10,9,7,6] => ?
=> ? = 6 - 1
[[[[[],[]],[]],[[],[]]],[]]
=> [[[[.,.],[.,.]],[.,.]],[.,[[.,.],[.,.]]]]
=> [1,3,2,5,4,8,10,9,7,6] => ?
=> ? = 6 - 1
[[[[],[[],[[],[]]]],[]],[]]
=> [[[.,[[.,[[.,[.,.]],[.,.]]],.]],.],[.,.]]
=> [4,3,6,5,2,7,1,8,10,9] => ?
=> ? = 6 - 1
[[[[],[[[],[]],[]]],[]],[]]
=> [[[.,[[[.,[.,.]],[.,.]],[.,.]]],.],[.,.]]
=> [3,2,5,4,7,6,1,8,10,9] => ?
=> ? = 6 - 1
[[[[[],[]],[[],[]]],[]],[]]
=> [[[[.,.],[.,.]],[.,[[.,.],[.,.]]]],[.,.]]
=> [1,3,2,6,8,7,5,4,10,9] => ?
=> ? = 6 - 1
[[[[[],[[],[]]],[]],[]],[]]
=> [[[[.,[[.,[.,.]],[.,.]]],.],[.,.]],[.,.]]
=> [3,2,5,4,1,6,8,7,10,9] => ?
=> ? = 6 - 1
[[],[[],[[],[[],[[],[[],[]]]]]]]
=> [.,[[.,[[.,[[.,[[.,[[.,.],[.,.]]],.]],.]],.]],.]]
=> [6,8,7,5,9,4,10,3,11,2,12,1] => ?
=> ? = 7 - 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [.,[[.,[[.,[[.,[[[.,.],[.,.]],[.,.]]],.]],.]],.]]
=> [5,7,6,9,8,4,10,3,11,2,12,1] => ?
=> ? = 7 - 1
[[],[[],[[],[[[],[]],[[],[]]]]]]
=> [.,[[.,[[.,[[[.,.],[.,.]],[[.,.],[.,.]]]],.]],.]]
=> [4,6,5,8,10,9,7,3,11,2,12,1] => ?
=> ? = 7 - 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [.,[[.,[[.,[[[.,[[.,.],[.,.]]],.],[.,.]]],.]],.]]
=> [5,7,6,4,8,10,9,3,11,2,12,1] => ?
=> ? = 7 - 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000157
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => [[1]]
=> 0 = 1 - 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => [[1],[2]]
=> 1 = 2 - 1
[[[]]]
=> [[.,.],.]
=> [1,2] => [[1,2]]
=> 0 = 1 - 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [[1],[2],[3]]
=> 2 = 3 - 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [[1,2],[3]]
=> 1 = 2 - 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => [[1,3],[2]]
=> 1 = 2 - 1
[[[],[]]]
=> [[.,.],[.,.]]
=> [1,3,2] => [[1,2],[3]]
=> 1 = 2 - 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => [[1,2,3]]
=> 0 = 1 - 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 3 = 4 - 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 2 = 3 - 1
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 2 = 3 - 1
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [[1,2],[3],[4]]
=> 2 = 3 - 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1 = 2 - 1
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 2 = 3 - 1
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2 = 3 - 1
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1 = 2 - 1
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 3 - 1
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [[1,2,3],[4]]
=> 1 = 2 - 1
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 1 = 2 - 1
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 1 = 2 - 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0 = 1 - 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 4 = 5 - 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 3 = 4 - 1
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 2 = 3 - 1
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 3 = 4 - 1
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 2 = 3 - 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [[1,3],[2,4],[5]]
=> 3 = 4 - 1
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 2 = 3 - 1
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [[1,2,3],[4],[5]]
=> 2 = 3 - 1
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> 2 = 3 - 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1 = 2 - 1
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 3 = 4 - 1
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [[1,2,5],[3],[4]]
=> 2 = 3 - 1
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [[1,3,5],[2],[4]]
=> 2 = 3 - 1
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1 = 2 - 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 3 = 4 - 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 2 = 3 - 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 1 = 2 - 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 3 = 4 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 2 = 3 - 1
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 2 = 3 - 1
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> 2 = 3 - 1
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1 = 2 - 1
[[],[],[[[],[]],[],[]]]
=> [.,[.,[[[.,.],[.,.]],[.,[.,.]]]]]
=> [3,5,4,8,7,6,2,1] => ?
=> ? = 6 - 1
[[],[[[],[[],[]]],[]]]
=> [.,[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [3,5,4,2,6,8,7,1] => ?
=> ? = 5 - 1
[[[],[[],[[],[]]]],[]]
=> [[.,[[.,[[.,[.,.]],[.,.]]],.]],.]
=> [4,3,6,5,2,7,1,8] => ?
=> ? = 5 - 1
[[[[[[[]],[]]]]],[]]
=> [[[[[[.,[.,.]],.],[.,.]],.],.],.]
=> [2,1,3,5,4,6,7,8] => ?
=> ? = 3 - 1
[[],[[],[[],[[],[[],[]]]]]]
=> [.,[[.,[[.,[[.,[[.,.],[.,.]]],.]],.]],.]]
=> [5,7,6,4,8,3,9,2,10,1] => ?
=> ? = 6 - 1
[[],[[],[[],[[[],[]],[]]]]]
=> [.,[[.,[[.,[[[.,.],[.,.]],[.,.]]],.]],.]]
=> [4,6,5,8,7,3,9,2,10,1] => ?
=> ? = 6 - 1
[[],[[],[[[],[]],[[],[]]]]]
=> [.,[[.,[[[.,.],[.,.]],[[.,.],[.,.]]]],.]]
=> [3,5,4,7,9,8,6,2,10,1] => ?
=> ? = 6 - 1
[[],[[],[[[],[[],[]]],[]]]]
=> [.,[[.,[[[.,[[.,.],[.,.]]],.],[.,.]]],.]]
=> [4,6,5,3,7,9,8,2,10,1] => ?
=> ? = 6 - 1
[[],[[],[[[[],[]],[]],[]]]]
=> [.,[[.,[[[[.,.],[.,.]],[.,.]],[.,.]]],.]]
=> [3,5,4,7,6,9,8,2,10,1] => ?
=> ? = 6 - 1
[[],[[[],[]],[[],[[],[]]]]]
=> [.,[[[.,.],[.,.]],[[.,[[.,.],[.,.]]],.]]]
=> [2,4,3,7,9,8,6,10,5,1] => ?
=> ? = 6 - 1
[[],[[[],[]],[[[],[]],[]]]]
=> [.,[[[.,.],[.,.]],[[[.,.],[.,.]],[.,.]]]]
=> [2,4,3,6,8,7,10,9,5,1] => ?
=> ? = 6 - 1
[[],[[[],[[],[]]],[[],[]]]]
=> [.,[[[.,[[.,.],[.,.]]],.],[[.,.],[.,.]]]]
=> [3,5,4,2,6,8,10,9,7,1] => ?
=> ? = 6 - 1
[[],[[[[],[]],[]],[[],[]]]]
=> [.,[[[[.,.],[.,.]],[.,.]],[[.,.],[.,.]]]]
=> [2,4,3,6,5,8,10,9,7,1] => ?
=> ? = 6 - 1
[[],[[[],[[],[[],[]]]],[]]]
=> [.,[[[.,[[.,[[.,.],[.,.]]],.]],.],[.,.]]]
=> [4,6,5,3,7,2,8,10,9,1] => ?
=> ? = 6 - 1
[[],[[[],[[[],[]],[]]],[]]]
=> [.,[[[.,[[[.,.],[.,.]],[.,.]]],.],[.,.]]]
=> [3,5,4,7,6,2,8,10,9,1] => ?
=> ? = 6 - 1
[[],[[[[],[]],[[],[]]],[]]]
=> [.,[[[[.,.],[.,.]],[[.,.],[.,.]]],[.,.]]]
=> [2,4,3,6,8,7,5,10,9,1] => ?
=> ? = 6 - 1
[[],[[[[],[[],[]]],[]],[]]]
=> [.,[[[[.,[[.,.],[.,.]]],.],[.,.]],[.,.]]]
=> [3,5,4,2,6,8,7,10,9,1] => ?
=> ? = 6 - 1
[[[],[]],[[],[[],[[],[]]]]]
=> [[.,[.,.]],[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [2,1,6,8,7,5,9,4,10,3] => ?
=> ? = 6 - 1
[[[],[]],[[],[[[],[]],[]]]]
=> [[.,[.,.]],[[.,[[[.,.],[.,.]],[.,.]]],.]]
=> [2,1,5,7,6,9,8,4,10,3] => ?
=> ? = 6 - 1
[[[],[]],[[[],[]],[[],[]]]]
=> [[.,[.,.]],[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [2,1,4,6,5,8,10,9,7,3] => ?
=> ? = 6 - 1
[[[],[]],[[[],[[],[]]],[]]]
=> [[.,[.,.]],[[[.,[[.,.],[.,.]]],.],[.,.]]]
=> [2,1,5,7,6,4,8,10,9,3] => ?
=> ? = 6 - 1
[[[],[]],[[[[],[]],[]],[]]]
=> [[.,[.,.]],[[[[.,.],[.,.]],[.,.]],[.,.]]]
=> [2,1,4,6,5,8,7,10,9,3] => ?
=> ? = 6 - 1
[[[],[[],[]]],[[],[[],[]]]]
=> [[.,[[.,[.,.]],[[.,[[.,.],[.,.]]],.]]],.]
=> [3,2,6,8,7,5,9,4,1,10] => ?
=> ? = 6 - 1
[[[],[[],[]]],[[[],[]],[]]]
=> [[.,[[.,[.,.]],[[[.,.],[.,.]],[.,.]]]],.]
=> [3,2,5,7,6,9,8,4,1,10] => ?
=> ? = 6 - 1
[[[[],[]],[]],[[],[[],[]]]]
=> [[[.,[.,.]],[[.,[[.,.],[.,.]]],.]],[.,.]]
=> [2,1,5,7,6,4,8,3,10,9] => ?
=> ? = 6 - 1
[[[[],[]],[]],[[[],[]],[]]]
=> [[[.,[.,.]],[[[.,.],[.,.]],[.,.]]],[.,.]]
=> [2,1,4,6,5,8,7,3,10,9] => ?
=> ? = 6 - 1
[[[],[[],[[],[]]]],[[],[]]]
=> [[.,[[.,[[.,[.,.]],[[.,.],[.,.]]]],.]],.]
=> [4,3,6,8,7,5,2,9,1,10] => ?
=> ? = 6 - 1
[[[],[[[],[]],[]]],[[],[]]]
=> [[.,[[[.,[.,.]],[[.,.],[.,.]]],[.,.]]],.]
=> [3,2,5,7,6,4,9,8,1,10] => ?
=> ? = 6 - 1
[[[[],[]],[[],[]]],[[],[]]]
=> [[[.,[.,.]],[[.,.],[.,.]]],[[.,.],[.,.]]]
=> [2,1,4,6,5,3,8,10,9,7] => [[1,3,4,7,8],[2,5,9],[6,10]]
=> ? = 6 - 1
[[[[],[[],[]]],[]],[[],[]]]
=> [[[.,[[.,[.,.]],[[.,.],[.,.]]]],.],[.,.]]
=> [3,2,5,7,6,4,1,8,10,9] => ?
=> ? = 6 - 1
[[[[[],[]],[]],[]],[[],[]]]
=> [[[[.,[.,.]],[[.,.],[.,.]]],[.,.]],[.,.]]
=> [2,1,4,6,5,3,8,7,10,9] => [[1,3,4,7,9],[2,5,8,10],[6]]
=> ? = 6 - 1
[[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,[[.,[[.,[.,.]],[.,.]]],.]],.]],.]
=> [5,4,7,6,3,8,2,9,1,10] => ?
=> ? = 6 - 1
[[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,[[[.,[.,.]],[.,.]],[.,.]]],.]],.]
=> [4,3,6,5,8,7,2,9,1,10] => ?
=> ? = 6 - 1
[[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[[.,.],[.,.]],[.,[[.,.],[.,.]]]]],.]
=> [2,4,3,7,9,8,6,5,1,10] => ?
=> ? = 6 - 1
[[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[[.,[[.,[.,.]],[.,.]]],.],[.,.]]],.]
=> [4,3,6,5,2,7,9,8,1,10] => ?
=> ? = 6 - 1
[[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]],.]
=> [3,2,5,4,7,6,9,8,1,10] => [[1,3,5,7,10],[2,4,6,8],[9]]
=> ? = 6 - 1
[[[[],[]],[[],[[],[]]]],[]]
=> [[[.,.],[.,.]],[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,3,2,7,9,8,6,10,5,4] => ?
=> ? = 6 - 1
[[[[],[]],[[[],[]],[]]],[]]
=> [[[.,.],[.,.]],[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,3,2,6,8,7,10,9,5,4] => ?
=> ? = 6 - 1
[[[[],[[],[]]],[[],[]]],[]]
=> [[[.,[[.,.],[.,.]]],.],[.,[[.,.],[.,.]]]]
=> [2,4,3,1,5,8,10,9,7,6] => ?
=> ? = 6 - 1
[[[[[],[]],[]],[[],[]]],[]]
=> [[[[.,.],[.,.]],[.,.]],[.,[[.,.],[.,.]]]]
=> [1,3,2,5,4,8,10,9,7,6] => ?
=> ? = 6 - 1
[[[[],[[],[[],[]]]],[]],[]]
=> [[[.,[[.,[[.,[.,.]],[.,.]]],.]],.],[.,.]]
=> [4,3,6,5,2,7,1,8,10,9] => ?
=> ? = 6 - 1
[[[[],[[[],[]],[]]],[]],[]]
=> [[[.,[[[.,[.,.]],[.,.]],[.,.]]],.],[.,.]]
=> [3,2,5,4,7,6,1,8,10,9] => ?
=> ? = 6 - 1
[[[[[],[]],[[],[]]],[]],[]]
=> [[[[.,.],[.,.]],[.,[[.,.],[.,.]]]],[.,.]]
=> [1,3,2,6,8,7,5,4,10,9] => ?
=> ? = 6 - 1
[[[[[],[[],[]]],[]],[]],[]]
=> [[[[.,[[.,[.,.]],[.,.]]],.],[.,.]],[.,.]]
=> [3,2,5,4,1,6,8,7,10,9] => ?
=> ? = 6 - 1
[[],[[],[[],[[],[[],[[],[]]]]]]]
=> [.,[[.,[[.,[[.,[[.,[[.,.],[.,.]]],.]],.]],.]],.]]
=> [6,8,7,5,9,4,10,3,11,2,12,1] => ?
=> ? = 7 - 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [.,[[.,[[.,[[.,[[[.,.],[.,.]],[.,.]]],.]],.]],.]]
=> [5,7,6,9,8,4,10,3,11,2,12,1] => ?
=> ? = 7 - 1
[[],[[],[[],[[[],[]],[[],[]]]]]]
=> [.,[[.,[[.,[[[.,.],[.,.]],[[.,.],[.,.]]]],.]],.]]
=> [4,6,5,8,10,9,7,3,11,2,12,1] => ?
=> ? = 7 - 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [.,[[.,[[.,[[[.,[[.,.],[.,.]]],.],[.,.]]],.]],.]]
=> [5,7,6,4,8,10,9,3,11,2,12,1] => ?
=> ? = 7 - 1
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [.,[[.,[[.,[[[[.,.],[.,.]],[.,.]],[.,.]]],.]],.]]
=> [4,6,5,8,7,10,9,3,11,2,12,1] => ?
=> ? = 7 - 1
[[],[[],[[[],[]],[[],[[],[]]]]]]
=> [.,[[.,[[[.,.],[.,.]],[[.,[[.,.],[.,.]]],.]]],.]]
=> [3,5,4,8,10,9,7,11,6,2,12,1] => ?
=> ? = 7 - 1
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: St000507
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 80%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 80%
Values
[[]]
=> [.,.]
=> [1] => [[1]]
=> 1
[[],[]]
=> [[.,.],.]
=> [1,2] => [[1,2]]
=> 2
[[[]]]
=> [.,[.,.]]
=> [2,1] => [[1],[2]]
=> 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => [[1,2,3]]
=> 3
[[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => [[1,2],[3]]
=> 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => [[1,3],[2]]
=> 2
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [[1,3],[2]]
=> 2
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [[1],[2],[3]]
=> 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [[1,2,3,4]]
=> 4
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 3
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 3
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 3
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[1,2],[3],[4]]
=> 2
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 3
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[1,3],[2],[4]]
=> 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 3
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 3
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[1,4],[2],[3]]
=> 2
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[1,4],[2],[3]]
=> 2
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 5
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[1,2,3,4],[5]]
=> 4
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [[1,2,3,5],[4]]
=> 4
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[1,2,3],[4,5]]
=> 4
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[1,2,3],[4],[5]]
=> 3
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [[1,2,4,5],[3]]
=> 4
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[1,2,4],[3],[5]]
=> 3
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 4
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [[1,2,5],[3],[4]]
=> 3
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[1,2,5],[3,4]]
=> 4
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[1,2],[3,4],[5]]
=> 3
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[1,2],[3,5],[4]]
=> 3
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 3
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[1,2],[3],[4],[5]]
=> 2
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 4
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[1,3,4],[2],[5]]
=> 3
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [[1,3,5],[2],[4]]
=> 3
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[1,3],[2,5],[4]]
=> 3
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[1,3],[2],[4],[5]]
=> 2
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [[1,3,4,5],[2]]
=> 4
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> 3
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[1,3,4],[2],[5]]
=> 3
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[1,4],[2],[3],[5]]
=> 2
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [[1,3,4,5],[2]]
=> 4
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [[1,3,5],[2],[4]]
=> 3
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [[1,4,5],[2],[3]]
=> 3
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [[1,4,5],[2],[3]]
=> 3
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 2
[[],[[],[[],[[[],[]],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]
=> [8,9,7,10,5,6,3,4,1,2] => ?
=> ? = 6
[[],[[],[[[],[]],[[],[]]]]]
=> [[.,.],[[.,.],[[.,[[.,.],.]],[[.,.],.]]]]
=> [9,10,6,7,5,8,3,4,1,2] => ?
=> ? = 6
[[],[[],[[[],[[],[]]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]
=> [8,9,6,7,5,10,3,4,1,2] => ?
=> ? = 6
[[],[[],[[[[],[]],[]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]
=> [7,8,6,9,5,10,3,4,1,2] => ?
=> ? = 6
[[],[[[],[]],[[],[[],[]]]]]
=> [[.,.],[[.,[[.,.],.]],[[.,.],[[.,.],.]]]]
=> [9,10,7,8,4,5,3,6,1,2] => ?
=> ? = 6
[[],[[[],[]],[[[],[]],[]]]]
=> [[.,.],[[.,[[.,.],.]],[[.,[[.,.],.]],.]]]
=> [8,9,7,10,4,5,3,6,1,2] => ?
=> ? = 6
[[],[[[],[[],[]]],[[],[]]]]
=> [[.,.],[[.,[[.,.],[[.,.],.]]],[[.,.],.]]]
=> [9,10,6,7,4,5,3,8,1,2] => ?
=> ? = 6
[[],[[[[],[]],[]],[[],[]]]]
=> [[.,.],[[.,[[.,[[.,.],.]],.]],[[.,.],.]]]
=> [9,10,5,6,4,7,3,8,1,2] => ?
=> ? = 6
[[],[[[],[[],[[],[]]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]
=> [8,9,6,7,4,5,3,10,1,2] => ?
=> ? = 6
[[],[[[],[[[],[]],[]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]
=> [7,8,6,9,4,5,3,10,1,2] => ?
=> ? = 6
[[],[[[[],[]],[[],[]]],[]]]
=> [[.,.],[[.,[[.,[[.,.],.]],[[.,.],.]]],.]]
=> [8,9,5,6,4,7,3,10,1,2] => ?
=> ? = 6
[[],[[[[],[[],[]]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,.],[[.,.],.]]],.]],.]]
=> [7,8,5,6,4,9,3,10,1,2] => ?
=> ? = 6
[[],[[[[[],[]],[]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,[[.,.],.]],.]],.]],.]]
=> [6,7,5,8,4,9,3,10,1,2] => ?
=> ? = 6
[[[],[]],[[],[[],[[],[]]]]]
=> [[.,[[.,.],.]],[[.,.],[[.,.],[[.,.],.]]]]
=> [9,10,7,8,5,6,2,3,1,4] => ?
=> ? = 6
[[[],[]],[[],[[[],[]],[]]]]
=> [[.,[[.,.],.]],[[.,.],[[.,[[.,.],.]],.]]]
=> [8,9,7,10,5,6,2,3,1,4] => ?
=> ? = 6
[[[],[]],[[[],[]],[[],[]]]]
=> [[.,[[.,.],.]],[[.,[[.,.],.]],[[.,.],.]]]
=> [9,10,6,7,5,8,2,3,1,4] => ?
=> ? = 6
[[[],[]],[[[],[[],[]]],[]]]
=> [[.,[[.,.],.]],[[.,[[.,.],[[.,.],.]]],.]]
=> [8,9,6,7,5,10,2,3,1,4] => ?
=> ? = 6
[[[],[]],[[[[],[]],[]],[]]]
=> [[.,[[.,.],.]],[[.,[[.,[[.,.],.]],.]],.]]
=> [7,8,6,9,5,10,2,3,1,4] => ?
=> ? = 6
[[[],[[],[]]],[[],[[],[]]]]
=> [[.,[[.,.],[[.,.],.]]],[[.,.],[[.,.],.]]]
=> [9,10,7,8,4,5,2,3,1,6] => ?
=> ? = 6
[[[],[[],[]]],[[[],[]],[]]]
=> [[.,[[.,.],[[.,.],.]]],[[.,[[.,.],.]],.]]
=> [8,9,7,10,4,5,2,3,1,6] => ?
=> ? = 6
[[[[],[]],[]],[[],[[],[]]]]
=> [[.,[[.,[[.,.],.]],.]],[[.,.],[[.,.],.]]]
=> [9,10,7,8,3,4,2,5,1,6] => ?
=> ? = 6
[[[[],[]],[]],[[[],[]],[]]]
=> [[.,[[.,[[.,.],.]],.]],[[.,[[.,.],.]],.]]
=> [8,9,7,10,3,4,2,5,1,6] => ?
=> ? = 6
[[[],[[],[[],[]]]],[[],[]]]
=> [[.,[[.,.],[[.,.],[[.,.],.]]]],[[.,.],.]]
=> [9,10,6,7,4,5,2,3,1,8] => [[1,3,8],[2,5],[4,7],[6,10],[9]]
=> ? = 6
[[[],[[[],[]],[]]],[[],[]]]
=> [[.,[[.,.],[[.,[[.,.],.]],.]]],[[.,.],.]]
=> [9,10,5,6,4,7,2,3,1,8] => ?
=> ? = 6
[[[[],[]],[[],[]]],[[],[]]]
=> [[.,[[.,[[.,.],.]],[[.,.],.]]],[[.,.],.]]
=> [9,10,6,7,3,4,2,5,1,8] => ?
=> ? = 6
[[[[],[[],[]]],[]],[[],[]]]
=> [[.,[[.,[[.,.],[[.,.],.]]],.]],[[.,.],.]]
=> [9,10,5,6,3,4,2,7,1,8] => ?
=> ? = 6
[[[[[],[]],[]],[]],[[],[]]]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],[[.,.],.]]
=> [9,10,4,5,3,6,2,7,1,8] => ?
=> ? = 6
[[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,.],[[.,.],.]]]]],.]
=> [8,9,6,7,4,5,2,3,1,10] => ?
=> ? = 6
[[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,[[.,.],.]],.]]]],.]
=> [7,8,6,9,4,5,2,3,1,10] => ?
=> ? = 6
[[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[.,.],[[.,[[.,.],.]],[[.,.],.]]]],.]
=> [8,9,5,6,4,7,2,3,1,10] => ?
=> ? = 6
[[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,.],[[.,.],.]]],.]]],.]
=> [7,8,5,6,4,9,2,3,1,10] => ?
=> ? = 6
[[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,[[.,.],.]],.]],.]]],.]
=> [6,7,5,8,4,9,2,3,1,10] => ?
=> ? = 6
[[[[],[]],[[],[[],[]]]],[]]
=> [[.,[[.,[[.,.],.]],[[.,.],[[.,.],.]]]],.]
=> [8,9,6,7,3,4,2,5,1,10] => ?
=> ? = 6
[[[[],[]],[[[],[]],[]]],[]]
=> [[.,[[.,[[.,.],.]],[[.,[[.,.],.]],.]]],.]
=> [7,8,6,9,3,4,2,5,1,10] => ?
=> ? = 6
[[[[],[[],[]]],[[],[]]],[]]
=> [[.,[[.,[[.,.],[[.,.],.]]],[[.,.],.]]],.]
=> [8,9,5,6,3,4,2,7,1,10] => ?
=> ? = 6
[[[[[],[]],[]],[[],[]]],[]]
=> [[.,[[.,[[.,[[.,.],.]],.]],[[.,.],.]]],.]
=> [8,9,4,5,3,6,2,7,1,10] => ?
=> ? = 6
[[[[],[[],[[],[]]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,.],[[.,.],.]]]],.]],.]
=> [7,8,5,6,3,4,2,9,1,10] => ?
=> ? = 6
[[[[],[[[],[]],[]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,[[.,.],.]],.]]],.]],.]
=> [6,7,5,8,3,4,2,9,1,10] => ?
=> ? = 6
[[[[[],[]],[[],[]]],[]],[]]
=> [[.,[[.,[[.,[[.,.],.]],[[.,.],.]]],.]],.]
=> [7,8,4,5,3,6,2,9,1,10] => ?
=> ? = 6
[[[[[],[[],[]]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,.],[[.,.],.]]],.]],.]],.]
=> [6,7,4,5,3,8,2,9,1,10] => ?
=> ? = 6
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]]
=> [10,11,9,12,7,8,5,6,3,4,1,2] => ?
=> ? = 7
[[],[[],[[],[[[],[]],[[],[]]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],[[.,.],.]]]]]
=> [11,12,8,9,7,10,5,6,3,4,1,2] => ?
=> ? = 7
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]]
=> [10,11,8,9,7,12,5,6,3,4,1,2] => ?
=> ? = 7
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]]
=> [9,10,8,11,7,12,5,6,3,4,1,2] => ?
=> ? = 7
[[],[[],[[[],[]],[[],[[],[]]]]]]
=> [[.,.],[[.,.],[[.,[[.,.],.]],[[.,.],[[.,.],.]]]]]
=> [11,12,9,10,6,7,5,8,3,4,1,2] => ?
=> ? = 7
[[],[[],[[[],[]],[[[],[]],[]]]]]
=> [[.,.],[[.,.],[[.,[[.,.],.]],[[.,[[.,.],.]],.]]]]
=> [10,11,9,12,6,7,5,8,3,4,1,2] => ?
=> ? = 7
[[],[[],[[[],[[],[]]],[[],[]]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],[[.,.],.]]]]
=> [11,12,8,9,6,7,5,10,3,4,1,2] => ?
=> ? = 7
[[],[[],[[[[],[]],[]],[[],[]]]]]
=> [[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],[[.,.],.]]]]
=> [11,12,7,8,6,9,5,10,3,4,1,2] => ?
=> ? = 7
[[],[[],[[[],[[],[[],[]]]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]]
=> [10,11,8,9,6,7,5,12,3,4,1,2] => ?
=> ? = 7
[[],[[],[[[],[[[],[]],[]]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]]
=> [9,10,8,11,6,7,5,12,3,4,1,2] => ?
=> ? = 7
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000147
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 80%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 80%
Values
[[]]
=> [.,.]
=> [1] => [1]
=> 1
[[],[]]
=> [[.,.],.]
=> [1,2] => [2]
=> 2
[[[]]]
=> [.,[.,.]]
=> [2,1] => [1,1]
=> 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => [3]
=> 3
[[],[[]]]
=> [[.,.],[.,.]]
=> [1,3,2] => [2,1]
=> 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 2
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 2
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1]
=> 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [4]
=> 4
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1]
=> 3
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1]
=> 3
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1]
=> 3
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1]
=> 2
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [3,1]
=> 3
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,2]
=> 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1]
=> 3
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,1,1]
=> 2
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1]
=> 3
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1]
=> 2
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,1]
=> 2
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1]
=> 2
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1]
=> 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [5]
=> 5
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,1]
=> 4
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [4,1]
=> 4
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,1]
=> 4
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,1,1]
=> 3
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [4,1]
=> 4
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [3,2]
=> 3
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [4,1]
=> 4
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [3,1,1]
=> 3
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [4,1]
=> 4
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,1,1]
=> 3
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [3,1,1]
=> 3
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [3,1,1]
=> 3
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,1,1,1]
=> 2
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [4,1]
=> 4
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [3,2]
=> 3
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [3,2]
=> 3
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,2]
=> 3
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,2,1]
=> 2
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [4,1]
=> 4
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,1,1]
=> 3
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2]
=> 3
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [2,2,1]
=> 2
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [4,1]
=> 4
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [3,1,1]
=> 3
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,1,1]
=> 3
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,1,1]
=> 3
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [2,1,1,1]
=> 2
[[],[],[[[],[]],[],[]]]
=> [[[.,.],.],[[[.,[[.,.],.]],.],.]]
=> [1,2,5,6,4,7,8,3] => ?
=> ? = 6
[[],[[],[[],[[[],[]],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]
=> [1,3,5,8,9,7,10,6,4,2] => ?
=> ? = 6
[[],[[],[[[],[]],[[],[]]]]]
=> [[.,.],[[.,.],[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,3,6,7,5,9,10,8,4,2] => ?
=> ? = 6
[[],[[],[[[],[[],[]]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]
=> [1,3,6,8,9,7,5,10,4,2] => ?
=> ? = 6
[[],[[],[[[[],[]],[]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]
=> [1,3,7,8,6,9,5,10,4,2] => ?
=> ? = 6
[[],[[[],[]],[[],[[],[]]]]]
=> [[.,.],[[.,[[.,.],.]],[[.,.],[[.,.],.]]]]
=> [1,4,5,3,7,9,10,8,6,2] => ?
=> ? = 6
[[],[[[],[]],[[[],[]],[]]]]
=> [[.,.],[[.,[[.,.],.]],[[.,[[.,.],.]],.]]]
=> [1,4,5,3,8,9,7,10,6,2] => ?
=> ? = 6
[[],[[[],[[],[]]],[[],[]]]]
=> [[.,.],[[.,[[.,.],[[.,.],.]]],[[.,.],.]]]
=> [1,4,6,7,5,3,9,10,8,2] => ?
=> ? = 6
[[],[[[[],[]],[]],[[],[]]]]
=> [[.,.],[[.,[[.,[[.,.],.]],.]],[[.,.],.]]]
=> [1,5,6,4,7,3,9,10,8,2] => ?
=> ? = 6
[[],[[[],[[],[[],[]]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]
=> [1,4,6,8,9,7,5,3,10,2] => ?
=> ? = 6
[[],[[[],[[[],[]],[]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]
=> [1,4,7,8,6,9,5,3,10,2] => ?
=> ? = 6
[[],[[[[],[]],[[],[]]],[]]]
=> [[.,.],[[.,[[.,[[.,.],.]],[[.,.],.]]],.]]
=> [1,5,6,4,8,9,7,3,10,2] => ?
=> ? = 6
[[],[[[[],[[],[]]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,.],[[.,.],.]]],.]],.]]
=> [1,5,7,8,6,4,9,3,10,2] => ?
=> ? = 6
[[[],[]],[[],[[],[[],[]]]]]
=> [[.,[[.,.],.]],[[.,.],[[.,.],[[.,.],.]]]]
=> [2,3,1,5,7,9,10,8,6,4] => ?
=> ? = 6
[[[],[]],[[],[[[],[]],[]]]]
=> [[.,[[.,.],.]],[[.,.],[[.,[[.,.],.]],.]]]
=> [2,3,1,5,8,9,7,10,6,4] => ?
=> ? = 6
[[[],[]],[[[],[]],[[],[]]]]
=> [[.,[[.,.],.]],[[.,[[.,.],.]],[[.,.],.]]]
=> [2,3,1,6,7,5,9,10,8,4] => ?
=> ? = 6
[[[],[]],[[[],[[],[]]],[]]]
=> [[.,[[.,.],.]],[[.,[[.,.],[[.,.],.]]],.]]
=> [2,3,1,6,8,9,7,5,10,4] => ?
=> ? = 6
[[[],[]],[[[[],[]],[]],[]]]
=> [[.,[[.,.],.]],[[.,[[.,[[.,.],.]],.]],.]]
=> [2,3,1,7,8,6,9,5,10,4] => ?
=> ? = 6
[[[],[[],[]]],[[],[[],[]]]]
=> [[.,[[.,.],[[.,.],.]]],[[.,.],[[.,.],.]]]
=> [2,4,5,3,1,7,9,10,8,6] => ?
=> ? = 6
[[[],[[],[]]],[[[],[]],[]]]
=> [[.,[[.,.],[[.,.],.]]],[[.,[[.,.],.]],.]]
=> [2,4,5,3,1,8,9,7,10,6] => ?
=> ? = 6
[[[[],[]],[]],[[],[[],[]]]]
=> [[.,[[.,[[.,.],.]],.]],[[.,.],[[.,.],.]]]
=> [3,4,2,5,1,7,9,10,8,6] => ?
=> ? = 6
[[[[],[]],[]],[[[],[]],[]]]
=> [[.,[[.,[[.,.],.]],.]],[[.,[[.,.],.]],.]]
=> [3,4,2,5,1,8,9,7,10,6] => ?
=> ? = 6
[[[],[[],[[],[]]]],[[],[]]]
=> [[.,[[.,.],[[.,.],[[.,.],.]]]],[[.,.],.]]
=> [2,4,6,7,5,3,1,9,10,8] => ?
=> ? = 6
[[[],[[[],[]],[]]],[[],[]]]
=> [[.,[[.,.],[[.,[[.,.],.]],.]]],[[.,.],.]]
=> [2,5,6,4,7,3,1,9,10,8] => ?
=> ? = 6
[[[[],[]],[[],[]]],[[],[]]]
=> [[.,[[.,[[.,.],.]],[[.,.],.]]],[[.,.],.]]
=> [3,4,2,6,7,5,1,9,10,8] => ?
=> ? = 6
[[[[],[[],[]]],[]],[[],[]]]
=> [[.,[[.,[[.,.],[[.,.],.]]],.]],[[.,.],.]]
=> [3,5,6,4,2,7,1,9,10,8] => ?
=> ? = 6
[[[[[],[]],[]],[]],[[],[]]]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],[[.,.],.]]
=> [4,5,3,6,2,7,1,9,10,8] => ?
=> ? = 6
[[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,.],[[.,.],.]]]]],.]
=> [2,4,6,8,9,7,5,3,1,10] => ?
=> ? = 6
[[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,[[.,.],.]],.]]]],.]
=> [2,4,7,8,6,9,5,3,1,10] => ?
=> ? = 6
[[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[.,.],[[.,[[.,.],.]],[[.,.],.]]]],.]
=> [2,5,6,4,8,9,7,3,1,10] => ?
=> ? = 6
[[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,.],[[.,.],.]]],.]]],.]
=> [2,5,7,8,6,4,9,3,1,10] => ?
=> ? = 6
[[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,[[.,.],.]],.]],.]]],.]
=> [2,6,7,5,8,4,9,3,1,10] => ?
=> ? = 6
[[[[],[]],[[],[[],[]]]],[]]
=> [[.,[[.,[[.,.],.]],[[.,.],[[.,.],.]]]],.]
=> [3,4,2,6,8,9,7,5,1,10] => ?
=> ? = 6
[[[[],[]],[[[],[]],[]]],[]]
=> [[.,[[.,[[.,.],.]],[[.,[[.,.],.]],.]]],.]
=> [3,4,2,7,8,6,9,5,1,10] => ?
=> ? = 6
[[[[],[[],[]]],[[],[]]],[]]
=> [[.,[[.,[[.,.],[[.,.],.]]],[[.,.],.]]],.]
=> [3,5,6,4,2,8,9,7,1,10] => ?
=> ? = 6
[[[[[],[]],[]],[[],[]]],[]]
=> [[.,[[.,[[.,[[.,.],.]],.]],[[.,.],.]]],.]
=> [4,5,3,6,2,8,9,7,1,10] => ?
=> ? = 6
[[[[],[[],[[],[]]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,.],[[.,.],.]]]],.]],.]
=> [3,5,7,8,6,4,2,9,1,10] => ?
=> ? = 6
[[[[],[[[],[]],[]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,[[.,.],.]],.]]],.]],.]
=> [3,6,7,5,8,4,2,9,1,10] => ?
=> ? = 6
[[[[[],[]],[[],[]]],[]],[]]
=> [[.,[[.,[[.,[[.,.],.]],[[.,.],.]]],.]],.]
=> [4,5,3,7,8,6,2,9,1,10] => ?
=> ? = 6
[[[[[],[[],[]]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,.],[[.,.],.]]],.]],.]],.]
=> [4,6,7,5,3,8,2,9,1,10] => ?
=> ? = 6
[[],[[],[[],[[],[[],[[],[]]]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]]
=> [1,3,5,7,9,11,12,10,8,6,4,2] => ?
=> ? = 7
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]]
=> [1,3,5,7,10,11,9,12,8,6,4,2] => ?
=> ? = 7
[[],[[],[[],[[[],[]],[[],[]]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],[[.,.],.]]]]]
=> [1,3,5,8,9,7,11,12,10,6,4,2] => ?
=> ? = 7
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]]
=> [1,3,5,8,10,11,9,7,12,6,4,2] => ?
=> ? = 7
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]]
=> [1,3,5,9,10,8,11,7,12,6,4,2] => ?
=> ? = 7
[[],[[],[[[],[]],[[],[[],[]]]]]]
=> [[.,.],[[.,.],[[.,[[.,.],.]],[[.,.],[[.,.],.]]]]]
=> [1,3,6,7,5,9,11,12,10,8,4,2] => ?
=> ? = 7
[[],[[],[[[],[]],[[[],[]],[]]]]]
=> [[.,.],[[.,.],[[.,[[.,.],.]],[[.,[[.,.],.]],.]]]]
=> [1,3,6,7,5,10,11,9,12,8,4,2] => ?
=> ? = 7
[[],[[],[[[],[[],[]]],[[],[]]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],[[.,.],.]]]]
=> [1,3,6,8,9,7,5,11,12,10,4,2] => ?
=> ? = 7
[[],[[],[[[[],[]],[]],[[],[]]]]]
=> [[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],[[.,.],.]]]]
=> [1,3,7,8,6,9,5,11,12,10,4,2] => ?
=> ? = 7
[[],[[],[[[],[[],[[],[]]]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]]
=> [1,3,6,8,10,11,9,7,5,12,4,2] => ?
=> ? = 7
Description
The largest part of an integer partition.
Matching statistic: St000097
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 80%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 80%
Values
[[]]
=> [1,0]
=> [1] => ([],1)
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2
[[[]]]
=> [1,1,0,0]
=> [2] => ([],2)
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,2,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[],[[],[[],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,2,2,3,1,1] => ([(0,8),(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[],[[],[[[],[]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,2,3,1,2,1] => ([(0,9),(1,7),(1,8),(1,9),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[],[[],[[[],[[],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,2,3,2,1,1] => ([(0,8),(0,9),(1,7),(1,8),(1,9),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[],[[],[[[[],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,2,4,1,1,1] => ([(0,7),(0,8),(0,9),(1,7),(1,8),(1,9),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[],[[[],[]],[[],[[],[]]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,1,2,2,1] => ([(0,9),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[],[[[],[]],[[[],[]],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,1,3,1,1] => ([(0,8),(0,9),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[],[[[],[[],[]]],[[],[]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,3,2,1,2,1] => ([(0,9),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[],[[[[],[]],[]],[[],[]]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,4,1,1,2,1] => ([(0,9),(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[],[[[],[[],[[],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,3,2,2,1,1] => ([(0,8),(0,9),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[],[[[],[[[],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,3,3,1,1,1] => ([(0,7),(0,8),(0,9),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[],[[[[],[]],[[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,1,2,1,1] => ([(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[],[[[[],[[],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,4,2,1,1,1] => ([(0,7),(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[],[[[[[],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,5,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[],[]],[[],[[],[[],[]]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[],[]],[[],[[[],[]],[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,1,2,3,1,1] => ([(0,8),(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[],[]],[[[],[]],[[],[]]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [2,1,3,1,2,1] => ([(0,9),(1,7),(1,8),(1,9),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[],[]],[[[],[[],[]]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,1,3,2,1,1] => ([(0,8),(0,9),(1,7),(1,8),(1,9),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[],[]],[[[[],[]],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,1,4,1,1,1] => ([(0,7),(0,8),(0,9),(1,7),(1,8),(1,9),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[],[[],[]]],[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,2,1,2,2,1] => ([(0,9),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[],[[],[]]],[[[],[]],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,2,1,3,1,1] => ([(0,8),(0,9),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[[],[]],[]],[[],[[],[]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [3,1,1,2,2,1] => ([(0,9),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[[],[]],[]],[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [3,1,1,3,1,1] => ([(0,8),(0,9),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[],[[],[[],[]]]],[[],[]]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [2,2,2,1,2,1] => ([(0,9),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[],[[[],[]],[]]],[[],[]]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> [2,3,1,1,2,1] => ([(0,9),(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[[],[]],[[],[]]],[[],[]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,1,2,1,2,1] => ([(0,9),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[[],[[],[]]],[]],[[],[]]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> [3,2,1,1,2,1] => ([(0,9),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[[[],[]],[]],[]],[[],[]]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [4,1,1,1,2,1] => ([(0,9),(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[],[[],[[],[[],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [2,2,2,2,1,1] => ([(0,8),(0,9),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[],[[],[[[],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [2,2,3,1,1,1] => ([(0,7),(0,8),(0,9),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[],[[[],[]],[[],[]]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [2,3,1,2,1,1] => ([(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[],[[[],[[],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [2,3,2,1,1,1] => ([(0,7),(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[],[[[[],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [2,4,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[[],[]],[[],[[],[]]]],[]]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [3,1,2,2,1,1] => ([(0,8),(0,9),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[[],[]],[[[],[]],[]]],[]]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [3,1,3,1,1,1] => ([(0,7),(0,8),(0,9),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[[],[[],[]]],[[],[]]],[]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,1,0]
=> [3,2,1,2,1,1] => ([(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[[[],[]],[]],[[],[]]],[]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [4,1,1,2,1,1] => ([(0,8),(0,9),(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[[],[[],[[],[]]]],[]],[]]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [3,2,2,1,1,1] => ([(0,7),(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[[],[[[],[]],[]]],[]],[]]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0]
=> [3,3,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[[[],[]],[[],[]]],[]],[]]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,1,2,1,1,1] => ([(0,7),(0,8),(0,9),(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[[[],[[],[]]],[]],[]],[]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,0]
=> [4,2,1,1,1,1] => ([(0,6),(0,7),(0,8),(0,9),(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[[[[[],[]],[]],[]],[]],[]]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [5,1,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 6
[[],[[],[[],[[],[[],[[],[]]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,2,2,2,2,2,1] => ([(0,11),(1,10),(1,11),(2,9),(2,10),(2,11),(3,8),(3,9),(3,10),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,2,2,2,3,1,1] => ([(0,10),(0,11),(1,10),(1,11),(2,9),(2,10),(2,11),(3,8),(3,9),(3,10),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[],[[],[[],[[[],[]],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,2,2,3,1,2,1] => ([(0,11),(1,9),(1,10),(1,11),(2,9),(2,10),(2,11),(3,8),(3,9),(3,10),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,0]
=> [1,2,2,3,2,1,1] => ([(0,10),(0,11),(1,9),(1,10),(1,11),(2,9),(2,10),(2,11),(3,8),(3,9),(3,10),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,0]
=> [1,2,2,4,1,1,1] => ([(0,9),(0,10),(0,11),(1,9),(1,10),(1,11),(2,9),(2,10),(2,11),(3,8),(3,9),(3,10),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[],[[],[[[],[]],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,2,3,1,2,2,1] => ([(0,11),(1,10),(1,11),(2,8),(2,9),(2,10),(2,11),(3,8),(3,9),(3,10),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[],[[],[[[],[]],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,2,3,1,3,1,1] => ([(0,10),(0,11),(1,10),(1,11),(2,8),(2,9),(2,10),(2,11),(3,8),(3,9),(3,10),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
[[],[[],[[[],[[],[]]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,0]
=> [1,2,3,2,1,2,1] => ([(0,11),(1,9),(1,10),(1,11),(2,8),(2,9),(2,10),(2,11),(3,8),(3,9),(3,10),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 7
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St001389
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 80%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 80%
Values
[[]]
=> [.,.]
=> [1] => [1]
=> 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => [2]
=> 2
[[[]]]
=> [[.,.],.]
=> [1,2] => [1,1]
=> 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3]
=> 3
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 2
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 2
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4]
=> 4
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1]
=> 3
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [3,1]
=> 3
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> 3
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,1]
=> 2
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1]
=> 3
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,1]
=> 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1]
=> 3
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,1,1]
=> 2
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1]
=> 3
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,1,1]
=> 2
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,1,1]
=> 2
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5]
=> 5
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1]
=> 4
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,1]
=> 4
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,1]
=> 4
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 3
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,1]
=> 4
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,1]
=> 3
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [4,1]
=> 4
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [3,1,1]
=> 3
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,1]
=> 4
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,1]
=> 3
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [3,1,1]
=> 3
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,1,1]
=> 3
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,1,1,1]
=> 2
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,1]
=> 4
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [3,1,1]
=> 3
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,1,1]
=> 3
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [3,1,1]
=> 3
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,1,1]
=> 2
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,1]
=> 4
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,1,1]
=> 3
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,1]
=> 3
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [2,1,1,1]
=> 2
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1]
=> 4
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [3,1,1]
=> 3
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,1]
=> 3
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [3,1,1]
=> 3
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [2,1,1,1]
=> 2
[[[],[]],[[[],[]],[]]]
=> [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,2,1,3] => ?
=> ? = 5
[[[],[[],[]]],[[],[]]]
=> [[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [7,6,8,3,2,4,1,5] => ?
=> ? = 5
[[[[],[]],[]],[[],[]]]
=> [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [7,6,8,4,2,1,3,5] => ?
=> ? = 5
[[[[],[]],[[],[]]],[]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,2,1,3,7] => ?
=> ? = 5
[[],[[],[[],[[[],[]],[]]]]]
=> [.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]]
=> [7,5,4,6,8,3,9,2,10,1] => ?
=> ? = 6
[[],[[],[[[],[]],[[],[]]]]]
=> [.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]]
=> [7,6,8,4,3,5,9,2,10,1] => ?
=> ? = 6
[[],[[],[[[],[[],[]]],[]]]]
=> [.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]]
=> [8,5,4,6,3,7,9,2,10,1] => ?
=> ? = 6
[[],[[],[[[[],[]],[]],[]]]]
=> [.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]]
=> [8,6,4,3,5,7,9,2,10,1] => ?
=> ? = 6
[[],[[[],[]],[[],[[],[]]]]]
=> [.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]]
=> [7,6,8,5,9,3,2,4,10,1] => ?
=> ? = 6
[[],[[[],[]],[[[],[]],[]]]]
=> [.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]]
=> [8,6,5,7,9,3,2,4,10,1] => ?
=> ? = 6
[[],[[[],[[],[]]],[[],[]]]]
=> [.,[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]]
=> [8,7,9,4,3,5,2,6,10,1] => ?
=> ? = 6
[[],[[[[],[]],[]],[[],[]]]]
=> [.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]]
=> [8,7,9,5,3,2,4,6,10,1] => ?
=> ? = 6
[[],[[[],[[],[[],[]]]],[]]]
=> [.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]]
=> [9,5,4,6,3,7,2,8,10,1] => ?
=> ? = 6
[[],[[[],[[[],[]],[]]],[]]]
=> [.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]]
=> [9,6,4,3,5,7,2,8,10,1] => ?
=> ? = 6
[[],[[[[],[]],[[],[]]],[]]]
=> [.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]]
=> [9,6,5,7,3,2,4,8,10,1] => ?
=> ? = 6
[[],[[[[],[[],[]]],[]],[]]]
=> [.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]]
=> [9,7,4,3,5,2,6,8,10,1] => ?
=> ? = 6
[[],[[[[[],[]],[]],[]],[]]]
=> [.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]]
=> [9,7,5,3,2,4,6,8,10,1] => ?
=> ? = 6
[[[],[]],[[],[[],[[],[]]]]]
=> [[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [7,6,8,5,9,4,10,2,1,3] => ?
=> ? = 6
[[[],[]],[[],[[[],[]],[]]]]
=> [[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [8,6,5,7,9,4,10,2,1,3] => ?
=> ? = 6
[[[],[]],[[[],[]],[[],[]]]]
=> [[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [8,7,9,5,4,6,10,2,1,3] => ?
=> ? = 6
[[[],[]],[[[],[[],[]]],[]]]
=> [[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [9,6,5,7,4,8,10,2,1,3] => ?
=> ? = 6
[[[],[]],[[[[],[]],[]],[]]]
=> [[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [9,7,5,4,6,8,10,2,1,3] => ?
=> ? = 6
[[[],[[],[]]],[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]]
=> [8,7,9,6,10,3,2,4,1,5] => ?
=> ? = 6
[[[],[[],[]]],[[[],[]],[]]]
=> [[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]]
=> [9,7,6,8,10,3,2,4,1,5] => ?
=> ? = 6
[[[[],[]],[]],[[],[[],[]]]]
=> [[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [8,7,9,6,10,4,2,1,3,5] => ?
=> ? = 6
[[[[],[]],[]],[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [9,7,6,8,10,4,2,1,3,5] => ?
=> ? = 6
[[[],[[],[[],[]]]],[[],[]]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]]
=> [9,8,10,4,3,5,2,6,1,7] => ?
=> ? = 6
[[[],[[[],[]],[]]],[[],[]]]
=> [[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]]
=> [9,8,10,5,3,2,4,6,1,7] => ?
=> ? = 6
[[[[],[]],[[],[]]],[[],[]]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [9,8,10,5,4,6,2,1,3,7] => ?
=> ? = 6
[[[[],[[],[]]],[]],[[],[]]]
=> [[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]]
=> [9,8,10,6,3,2,4,1,5,7] => ?
=> ? = 6
[[[[[],[]],[]],[]],[[],[]]]
=> [[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]]
=> [9,8,10,6,4,2,1,3,5,7] => ?
=> ? = 6
[[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]]
=> [10,5,4,6,3,7,2,8,1,9] => ?
=> ? = 6
[[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]]
=> [10,6,4,3,5,7,2,8,1,9] => ?
=> ? = 6
[[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]]
=> [10,6,5,7,3,2,4,8,1,9] => ?
=> ? = 6
[[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]]
=> [10,7,4,3,5,2,6,8,1,9] => ?
=> ? = 6
[[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]]
=> [10,7,5,3,2,4,6,8,1,9] => ?
=> ? = 6
[[[[],[]],[[],[[],[]]]],[]]
=> [[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [10,6,5,7,4,8,2,1,3,9] => ?
=> ? = 6
[[[[],[]],[[[],[]],[]]],[]]
=> [[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [10,7,5,4,6,8,2,1,3,9] => ?
=> ? = 6
[[[[],[[],[]]],[[],[]]],[]]
=> [[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]]
=> [10,7,6,8,3,2,4,1,5,9] => ?
=> ? = 6
[[[[[],[]],[]],[[],[]]],[]]
=> [[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]]
=> [10,7,6,8,4,2,1,3,5,9] => ?
=> ? = 6
[[[[],[[],[[],[]]]],[]],[]]
=> [[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]]
=> [10,8,4,3,5,2,6,1,7,9] => ?
=> ? = 6
[[[[],[[[],[]],[]]],[]],[]]
=> [[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]]
=> [10,8,5,3,2,4,6,1,7,9] => ?
=> ? = 6
[[[[[],[]],[[],[]]],[]],[]]
=> [[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]]
=> [10,8,5,4,6,2,1,3,7,9] => ?
=> ? = 6
[[[[[],[[],[]]],[]],[]],[]]
=> [[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]]
=> [10,8,6,3,2,4,1,5,7,9] => ?
=> ? = 6
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [.,[[.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]],.]]
=> [8,6,5,7,9,4,10,3,11,2,12,1] => ?
=> ? = 7
[[],[[],[[],[[[],[]],[[],[]]]]]]
=> [.,[[.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]],.]]
=> [8,7,9,5,4,6,10,3,11,2,12,1] => ?
=> ? = 7
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [.,[[.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]],.]]
=> [9,6,5,7,4,8,10,3,11,2,12,1] => ?
=> ? = 7
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [.,[[.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]],.]]
=> [9,7,5,4,6,8,10,3,11,2,12,1] => ?
=> ? = 7
[[],[[],[[[],[]],[[],[[],[]]]]]]
=> [.,[[.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]],.]]
=> [8,7,9,6,10,4,3,5,11,2,12,1] => ?
=> ? = 7
[[],[[],[[[],[]],[[[],[]],[]]]]]
=> [.,[[.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]],.]]
=> [9,7,6,8,10,4,3,5,11,2,12,1] => ?
=> ? = 7
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St001918
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 80%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 80%
Values
[[]]
=> [.,.]
=> [1] => [1]
=> 0 = 1 - 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => [2]
=> 1 = 2 - 1
[[[]]]
=> [[.,.],.]
=> [1,2] => [1,1]
=> 0 = 1 - 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3]
=> 2 = 3 - 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,1]
=> 1 = 2 - 1
[[[]],[]]
=> [[.,.],[.,.]]
=> [3,1,2] => [2,1]
=> 1 = 2 - 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1]
=> 1 = 2 - 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,1,1]
=> 0 = 1 - 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4]
=> 3 = 4 - 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1]
=> 2 = 3 - 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [3,1]
=> 2 = 3 - 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,1]
=> 2 = 3 - 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,1]
=> 1 = 2 - 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1]
=> 2 = 3 - 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,1]
=> 1 = 2 - 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1]
=> 2 = 3 - 1
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,1,1]
=> 1 = 2 - 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1]
=> 2 = 3 - 1
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,1,1]
=> 1 = 2 - 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,1,1]
=> 1 = 2 - 1
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,1]
=> 1 = 2 - 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,1,1,1]
=> 0 = 1 - 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5]
=> 4 = 5 - 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1]
=> 3 = 4 - 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,1]
=> 3 = 4 - 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,1]
=> 3 = 4 - 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1]
=> 2 = 3 - 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,1]
=> 3 = 4 - 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [3,1,1]
=> 2 = 3 - 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [4,1]
=> 3 = 4 - 1
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [3,1,1]
=> 2 = 3 - 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,1]
=> 3 = 4 - 1
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,1,1]
=> 2 = 3 - 1
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [3,1,1]
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,1,1]
=> 2 = 3 - 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1 = 2 - 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,1]
=> 3 = 4 - 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [3,1,1]
=> 2 = 3 - 1
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,1,1]
=> 2 = 3 - 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [3,1,1]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1 = 2 - 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,1]
=> 3 = 4 - 1
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,1,1]
=> 2 = 3 - 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,1]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1 = 2 - 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1]
=> 3 = 4 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [3,1,1]
=> 2 = 3 - 1
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,1]
=> 2 = 3 - 1
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [3,1,1]
=> 2 = 3 - 1
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1 = 2 - 1
[[[],[]],[[[],[]],[]]]
=> [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,2,1,3] => ?
=> ? = 5 - 1
[[[],[[],[]]],[[],[]]]
=> [[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [7,6,8,3,2,4,1,5] => ?
=> ? = 5 - 1
[[[[],[]],[]],[[],[]]]
=> [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [7,6,8,4,2,1,3,5] => ?
=> ? = 5 - 1
[[[[],[]],[[],[]]],[]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,2,1,3,7] => ?
=> ? = 5 - 1
[[],[[],[[],[[[],[]],[]]]]]
=> [.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]]
=> [7,5,4,6,8,3,9,2,10,1] => ?
=> ? = 6 - 1
[[],[[],[[[],[]],[[],[]]]]]
=> [.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]]
=> [7,6,8,4,3,5,9,2,10,1] => ?
=> ? = 6 - 1
[[],[[],[[[],[[],[]]],[]]]]
=> [.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]]
=> [8,5,4,6,3,7,9,2,10,1] => ?
=> ? = 6 - 1
[[],[[],[[[[],[]],[]],[]]]]
=> [.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]]
=> [8,6,4,3,5,7,9,2,10,1] => ?
=> ? = 6 - 1
[[],[[[],[]],[[],[[],[]]]]]
=> [.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]]
=> [7,6,8,5,9,3,2,4,10,1] => ?
=> ? = 6 - 1
[[],[[[],[]],[[[],[]],[]]]]
=> [.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]]
=> [8,6,5,7,9,3,2,4,10,1] => ?
=> ? = 6 - 1
[[],[[[],[[],[]]],[[],[]]]]
=> [.,[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]]
=> [8,7,9,4,3,5,2,6,10,1] => ?
=> ? = 6 - 1
[[],[[[[],[]],[]],[[],[]]]]
=> [.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]]
=> [8,7,9,5,3,2,4,6,10,1] => ?
=> ? = 6 - 1
[[],[[[],[[],[[],[]]]],[]]]
=> [.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]]
=> [9,5,4,6,3,7,2,8,10,1] => ?
=> ? = 6 - 1
[[],[[[],[[[],[]],[]]],[]]]
=> [.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]]
=> [9,6,4,3,5,7,2,8,10,1] => ?
=> ? = 6 - 1
[[],[[[[],[]],[[],[]]],[]]]
=> [.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]]
=> [9,6,5,7,3,2,4,8,10,1] => ?
=> ? = 6 - 1
[[],[[[[],[[],[]]],[]],[]]]
=> [.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]]
=> [9,7,4,3,5,2,6,8,10,1] => ?
=> ? = 6 - 1
[[],[[[[[],[]],[]],[]],[]]]
=> [.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]]
=> [9,7,5,3,2,4,6,8,10,1] => ?
=> ? = 6 - 1
[[[],[]],[[],[[],[[],[]]]]]
=> [[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [7,6,8,5,9,4,10,2,1,3] => ?
=> ? = 6 - 1
[[[],[]],[[],[[[],[]],[]]]]
=> [[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [8,6,5,7,9,4,10,2,1,3] => ?
=> ? = 6 - 1
[[[],[]],[[[],[]],[[],[]]]]
=> [[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [8,7,9,5,4,6,10,2,1,3] => ?
=> ? = 6 - 1
[[[],[]],[[[],[[],[]]],[]]]
=> [[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [9,6,5,7,4,8,10,2,1,3] => ?
=> ? = 6 - 1
[[[],[]],[[[[],[]],[]],[]]]
=> [[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [9,7,5,4,6,8,10,2,1,3] => ?
=> ? = 6 - 1
[[[],[[],[]]],[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]]
=> [8,7,9,6,10,3,2,4,1,5] => ?
=> ? = 6 - 1
[[[],[[],[]]],[[[],[]],[]]]
=> [[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]]
=> [9,7,6,8,10,3,2,4,1,5] => ?
=> ? = 6 - 1
[[[[],[]],[]],[[],[[],[]]]]
=> [[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [8,7,9,6,10,4,2,1,3,5] => ?
=> ? = 6 - 1
[[[[],[]],[]],[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [9,7,6,8,10,4,2,1,3,5] => ?
=> ? = 6 - 1
[[[],[[],[[],[]]]],[[],[]]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]]
=> [9,8,10,4,3,5,2,6,1,7] => ?
=> ? = 6 - 1
[[[],[[[],[]],[]]],[[],[]]]
=> [[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]]
=> [9,8,10,5,3,2,4,6,1,7] => ?
=> ? = 6 - 1
[[[[],[]],[[],[]]],[[],[]]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [9,8,10,5,4,6,2,1,3,7] => ?
=> ? = 6 - 1
[[[[],[[],[]]],[]],[[],[]]]
=> [[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]]
=> [9,8,10,6,3,2,4,1,5,7] => ?
=> ? = 6 - 1
[[[[[],[]],[]],[]],[[],[]]]
=> [[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]]
=> [9,8,10,6,4,2,1,3,5,7] => ?
=> ? = 6 - 1
[[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]]
=> [10,5,4,6,3,7,2,8,1,9] => ?
=> ? = 6 - 1
[[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]]
=> [10,6,4,3,5,7,2,8,1,9] => ?
=> ? = 6 - 1
[[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]]
=> [10,6,5,7,3,2,4,8,1,9] => ?
=> ? = 6 - 1
[[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]]
=> [10,7,4,3,5,2,6,8,1,9] => ?
=> ? = 6 - 1
[[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]]
=> [10,7,5,3,2,4,6,8,1,9] => ?
=> ? = 6 - 1
[[[[],[]],[[],[[],[]]]],[]]
=> [[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [10,6,5,7,4,8,2,1,3,9] => ?
=> ? = 6 - 1
[[[[],[]],[[[],[]],[]]],[]]
=> [[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [10,7,5,4,6,8,2,1,3,9] => ?
=> ? = 6 - 1
[[[[],[[],[]]],[[],[]]],[]]
=> [[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]]
=> [10,7,6,8,3,2,4,1,5,9] => ?
=> ? = 6 - 1
[[[[[],[]],[]],[[],[]]],[]]
=> [[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]]
=> [10,7,6,8,4,2,1,3,5,9] => ?
=> ? = 6 - 1
[[[[],[[],[[],[]]]],[]],[]]
=> [[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]]
=> [10,8,4,3,5,2,6,1,7,9] => ?
=> ? = 6 - 1
[[[[],[[[],[]],[]]],[]],[]]
=> [[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]]
=> [10,8,5,3,2,4,6,1,7,9] => ?
=> ? = 6 - 1
[[[[[],[]],[[],[]]],[]],[]]
=> [[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]]
=> [10,8,5,4,6,2,1,3,7,9] => ?
=> ? = 6 - 1
[[[[[],[[],[]]],[]],[]],[]]
=> [[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],[.,.]]
=> [10,8,6,3,2,4,1,5,7,9] => ?
=> ? = 6 - 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [.,[[.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]],.]]
=> [8,6,5,7,9,4,10,3,11,2,12,1] => ?
=> ? = 7 - 1
[[],[[],[[],[[[],[]],[[],[]]]]]]
=> [.,[[.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]],.]]
=> [8,7,9,5,4,6,10,3,11,2,12,1] => ?
=> ? = 7 - 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [.,[[.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]],.]]
=> [9,6,5,7,4,8,10,3,11,2,12,1] => ?
=> ? = 7 - 1
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [.,[[.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]],.]]
=> [9,7,5,4,6,8,10,3,11,2,12,1] => ?
=> ? = 7 - 1
[[],[[],[[[],[]],[[],[[],[]]]]]]
=> [.,[[.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]],.]]
=> [8,7,9,6,10,4,3,5,11,2,12,1] => ?
=> ? = 7 - 1
[[],[[],[[[],[]],[[[],[]],[]]]]]
=> [.,[[.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]],.]]
=> [9,7,6,8,10,4,3,5,11,2,12,1] => ?
=> ? = 7 - 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$
\sum_{p\in\lambda} [p]_{q^{N/p}},
$$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$
\left(1 - \frac{1}{\lambda_1}\right) N,
$$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.
The following 210 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000098The chromatic number of a graph. St000676The number of odd rises of a Dyck path. St000011The number of touch points (or returns) of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000703The number of deficiencies of a permutation. St001581The achromatic number of a graph. St000925The number of topologically connected components of a set partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001494The Alon-Tarsi number of a graph. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000172The Grundy number of a graph. St001029The size of the core of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000527The width of the poset. St000071The number of maximal chains in a poset. St000068The number of minimal elements in a poset. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St000632The jump number of the poset. St000374The number of exclusive right-to-left minima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000245The number of ascents of a permutation. St000528The height of a poset. St000912The number of maximal antichains in a poset. St000306The bounce count of a Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000996The number of exclusive left-to-right maxima of a permutation. St000159The number of distinct parts of the integer partition. St001461The number of topologically connected components of the chord diagram of a permutation. St000015The number of peaks of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000378The diagonal inversion number of an integer partition. St000809The reduced reflection length of the permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000031The number of cycles in the cycle decomposition of a permutation. St000702The number of weak deficiencies of a permutation. St000141The maximum drop size of a permutation. St000308The height of the tree associated to a permutation. St000007The number of saliances of the permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000164The number of short pairs. St000153The number of adjacent cycles of a permutation. St000332The positive inversions of an alternating sign matrix. St000069The number of maximal elements of a poset. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000733The row containing the largest entry of a standard tableau. St000225Difference between largest and smallest parts in a partition. St001427The number of descents of a signed permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000542The number of left-to-right-minima of a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000168The number of internal nodes of an ordered tree. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001180Number of indecomposable injective modules with projective dimension at most 1. St000035The number of left outer peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000201The number of leaf nodes in a binary tree. St000237The number of small exceedances. St000389The number of runs of ones of odd length in a binary word. St000742The number of big ascents of a permutation after prepending zero. St000820The number of compositions obtained by rotating the composition. St000822The Hadwiger number of the graph. St000991The number of right-to-left minima of a permutation. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000238The number of indices that are not small weak excedances. St000386The number of factors DDU in a Dyck path. St000778The metric dimension of a graph. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001777The number of weak descents in an integer composition. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000061The number of nodes on the left branch of a binary tree. St000216The absolute length of a permutation. St001812The biclique partition number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001323The independence gap of a graph. St001642The Prague dimension of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001644The dimension of a graph. St000741The Colin de Verdière graph invariant. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000711The number of big exceedences of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000746The number of pairs with odd minimum in a perfect matching. St001621The number of atoms of a lattice. St000619The number of cyclic descents of a permutation. St000087The number of induced subgraphs. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St001110The 3-dynamic chromatic number of a graph. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000171The degree of the graph. St000236The number of cyclical small weak excedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001152The number of pairs with even minimum in a perfect matching. St001270The bandwidth of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001668The number of points of the poset minus the width of the poset. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001712The number of natural descents of a standard Young tableau. St000710The number of big deficiencies of a permutation. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one.
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