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Your data matches 22 different statistics following compositions of up to 3 maps.
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Matching statistic: St000381
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => 1
[[],[]]
=> [1,0,1,0]
=> [1,1] => 1
[[[]]]
=> [1,1,0,0]
=> [2] => 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3] => 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
Description
The largest part of an integer composition.
Matching statistic: St000147
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => [1]
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 1
[[[]]]
=> [1,1,0,0]
=> [2] => [2]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,1] => [2,1]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [3,1,1]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [3,1,1]
=> 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> 4
Description
The largest part of an integer partition.
Matching statistic: St000392
(load all 6 compositions to match this statistic)
(load all 6 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
Mp00131: Permutations —descent bottoms⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => => ? = 1 - 1
[[],[]]
=> [[.,.],.]
=> [1,2] => 0 => 0 = 1 - 1
[[[]]]
=> [.,[.,.]]
=> [2,1] => 1 => 1 = 2 - 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => 00 => 0 = 1 - 1
[[],[[]]]
=> [[.,.],[.,.]]
=> [1,3,2] => 01 => 1 = 2 - 1
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => 10 => 1 = 2 - 1
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 10 => 1 = 2 - 1
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 11 => 2 = 3 - 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => 001 => 1 = 2 - 1
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => 010 => 1 = 2 - 1
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => 010 => 1 = 2 - 1
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 011 => 2 = 3 - 1
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 1 = 2 - 1
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 101 => 1 = 2 - 1
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 100 => 1 = 2 - 1
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 2 = 3 - 1
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 100 => 1 = 2 - 1
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 101 => 1 = 2 - 1
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 110 => 2 = 3 - 1
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 110 => 2 = 3 - 1
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0000 => 0 = 1 - 1
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0001 => 1 = 2 - 1
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 0010 => 1 = 2 - 1
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 0010 => 1 = 2 - 1
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 0011 => 2 = 3 - 1
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 0100 => 1 = 2 - 1
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0101 => 1 = 2 - 1
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => 0100 => 1 = 2 - 1
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 0110 => 2 = 3 - 1
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 0100 => 1 = 2 - 1
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0101 => 1 = 2 - 1
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 0110 => 2 = 3 - 1
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 0110 => 2 = 3 - 1
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0111 => 3 = 4 - 1
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 1000 => 1 = 2 - 1
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 1001 => 1 = 2 - 1
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 1010 => 1 = 2 - 1
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1010 => 1 = 2 - 1
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1011 => 2 = 3 - 1
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 1000 => 1 = 2 - 1
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1100 => 2 = 3 - 1
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 1001 => 1 = 2 - 1
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1101 => 2 = 3 - 1
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 1000 => 1 = 2 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => 1010 => 1 = 2 - 1
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 1100 => 2 = 3 - 1
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 1100 => 2 = 3 - 1
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1110 => 3 = 4 - 1
[[[],[],[],[]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1000 => 1 = 2 - 1
[[],[],[[]],[[[]]],[]]
=> [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [1,2,4,3,7,6,5,8] => ? => ? = 3 - 1
[[],[],[[],[]],[],[],[]]
=> [[[[[[.,.],.],[[.,.],.]],.],.],.]
=> [1,2,4,5,3,6,7,8] => ? => ? = 2 - 1
[[],[],[[],[]],[],[[]]]
=> [[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> [1,2,4,5,3,6,8,7] => ? => ? = 2 - 1
[[],[],[[[]]],[[]],[]]
=> [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [1,2,5,4,3,7,6,8] => ? => ? = 3 - 1
[[],[],[[[]],[]],[],[]]
=> [[[[[.,.],.],[[.,[.,.]],.]],.],.]
=> [1,2,5,4,6,3,7,8] => ? => ? = 3 - 1
[[],[],[[],[[]]],[[]]]
=> [[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> [1,2,4,6,5,3,8,7] => ? => ? = 2 - 1
[[],[],[[[]],[]],[[]]]
=> [[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [1,2,5,4,6,3,8,7] => ? => ? = 3 - 1
[[],[],[[[],[]]],[[]]]
=> [[[[.,.],.],[.,[[.,.],.]]],[.,.]]
=> [1,2,5,6,4,3,8,7] => ? => ? = 3 - 1
[[],[],[[[[]]]],[[]]]
=> [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
=> [1,2,6,5,4,3,8,7] => ? => ? = 4 - 1
[[],[],[[],[],[[]]],[]]
=> [[[[.,.],.],[[[.,.],.],[.,.]]],.]
=> [1,2,4,5,7,6,3,8] => ? => ? = 2 - 1
[[],[],[[],[[]],[]],[]]
=> [[[[.,.],.],[[[.,.],[.,.]],.]],.]
=> [1,2,4,6,5,7,3,8] => ? => ? = 2 - 1
[[],[],[[],[[[]]]],[]]
=> [[[[.,.],.],[[.,.],[.,[.,.]]]],.]
=> [1,2,4,7,6,5,3,8] => ? => ? = 3 - 1
[[],[],[[[]],[[]]],[]]
=> [[[[.,.],.],[[.,[.,.]],[.,.]]],.]
=> [1,2,5,4,7,6,3,8] => ? => ? = 3 - 1
[[],[],[[[[]]],[]],[]]
=> [[[[.,.],.],[[.,[.,[.,.]]],.]],.]
=> [1,2,6,5,4,7,3,8] => ? => ? = 4 - 1
[[],[],[[[[]],[]]],[]]
=> [[[[.,.],.],[.,[[.,[.,.]],.]]],.]
=> [1,2,6,5,7,4,3,8] => ? => ? = 4 - 1
[[],[],[[],[],[[]],[]]]
=> [[[.,.],.],[[[[.,.],.],[.,.]],.]]
=> [1,2,4,5,7,6,8,3] => ? => ? = 2 - 1
[[],[],[[],[[],[]],[]]]
=> [[[.,.],.],[[[.,.],[[.,.],.]],.]]
=> [1,2,4,6,7,5,8,3] => ? => ? = 2 - 1
[[],[],[[[],[]],[[]]]]
=> [[[.,.],.],[[.,[[.,.],.]],[.,.]]]
=> [1,2,5,6,4,8,7,3] => ? => ? = 3 - 1
[[],[],[[[[]],[]],[]]]
=> [[[.,.],.],[[.,[[.,[.,.]],.]],.]]
=> [1,2,6,5,7,4,8,3] => ? => ? = 4 - 1
[[],[],[[[[],[]]],[]]]
=> [[[.,.],.],[[.,[.,[[.,.],.]]],.]]
=> [1,2,6,7,5,4,8,3] => ? => ? = 4 - 1
[[],[],[[[],[[],[]]]]]
=> [[[.,.],.],[.,[[.,.],[[.,.],.]]]]
=> [1,2,5,7,8,6,4,3] => ? => ? = 3 - 1
[[],[],[[[[],[]],[]]]]
=> [[[.,.],.],[.,[[.,[[.,.],.]],.]]]
=> [1,2,6,7,5,8,4,3] => ? => ? = 4 - 1
[[],[],[[[[],[[]]]]]]
=> [[[.,.],.],[.,[.,[[.,.],[.,.]]]]]
=> [1,2,6,8,7,5,4,3] => ? => ? = 4 - 1
[[],[[]],[],[[],[]],[]]
=> [[[[[.,.],[.,.]],.],[[.,.],.]],.]
=> [1,3,2,4,6,7,5,8] => ? => ? = 2 - 1
[[],[[]],[],[[[],[]]]]
=> [[[[.,.],[.,.]],.],[.,[[.,.],.]]]
=> [1,3,2,4,7,8,6,5] => ? => ? = 3 - 1
[[],[[]],[[]],[],[[]]]
=> [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [1,3,2,5,4,6,8,7] => ? => ? = 2 - 1
[[],[[]],[[[[],[]]]]]
=> [[[.,.],[.,.]],[.,[.,[[.,.],.]]]]
=> [1,3,2,7,8,6,5,4] => ? => ? = 4 - 1
[[],[[],[]],[[]],[],[]]
=> [[[[[.,.],[[.,.],.]],[.,.]],.],.]
=> [1,3,4,2,6,5,7,8] => ? => ? = 2 - 1
[[],[[[]]],[[]],[],[]]
=> [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
=> [1,4,3,2,6,5,7,8] => ? => ? = 3 - 1
[[],[[[]]],[[]],[[]]]
=> [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
=> [1,4,3,2,6,5,8,7] => ? => ? = 3 - 1
[[],[[],[]],[[[]]],[]]
=> [[[[.,.],[[.,.],.]],[.,[.,.]]],.]
=> [1,3,4,2,7,6,5,8] => ? => ? = 3 - 1
[[],[[[]]],[[[],[]]]]
=> [[[.,.],[.,[.,.]]],[.,[[.,.],.]]]
=> [1,4,3,2,7,8,6,5] => ? => ? = 3 - 1
[[],[[[]],[]],[],[],[]]
=> [[[[[.,.],[[.,[.,.]],.]],.],.],.]
=> [1,4,3,5,2,6,7,8] => ? => ? = 3 - 1
[[],[[[]],[]],[],[[]]]
=> [[[[.,.],[[.,[.,.]],.]],.],[.,.]]
=> [1,4,3,5,2,6,8,7] => ? => ? = 3 - 1
[[],[[[[]]]],[],[[]]]
=> [[[[.,.],[.,[.,[.,.]]]],.],[.,.]]
=> [1,5,4,3,2,6,8,7] => ? => ? = 4 - 1
[[],[[],[[]]],[[[]]]]
=> [[[.,.],[[.,.],[.,.]]],[.,[.,.]]]
=> [1,3,5,4,2,8,7,6] => ? => ? = 3 - 1
[[],[[],[],[[]]],[],[]]
=> [[[[.,.],[[[.,.],.],[.,.]]],.],.]
=> [1,3,4,6,5,2,7,8] => ? => ? = 2 - 1
[[],[[],[[[]]]],[],[]]
=> [[[[.,.],[[.,.],[.,[.,.]]]],.],.]
=> [1,3,6,5,4,2,7,8] => ? => ? = 3 - 1
[[],[[],[],[[]]],[[]]]
=> [[[.,.],[[[.,.],.],[.,.]]],[.,.]]
=> [1,3,4,6,5,2,8,7] => ? => ? = 2 - 1
[[],[[],[[]],[]],[[]]]
=> [[[.,.],[[[.,.],[.,.]],.]],[.,.]]
=> [1,3,5,4,6,2,8,7] => ? => ? = 2 - 1
[[],[[],[[],[]]],[[]]]
=> [[[.,.],[[.,.],[[.,.],.]]],[.,.]]
=> [1,3,5,6,4,2,8,7] => ? => ? = 2 - 1
[[],[[[]],[],[]],[[]]]
=> [[[.,.],[[[.,[.,.]],.],.]],[.,.]]
=> [1,4,3,5,6,2,8,7] => ? => ? = 3 - 1
[[],[[[]],[[]]],[[]]]
=> [[[.,.],[[.,[.,.]],[.,.]]],[.,.]]
=> [1,4,3,6,5,2,8,7] => ? => ? = 3 - 1
[[],[[[],[]],[]],[[]]]
=> [[[.,.],[[.,[[.,.],.]],.]],[.,.]]
=> [1,4,5,3,6,2,8,7] => ? => ? = 3 - 1
[[],[[[[]]],[]],[[]]]
=> [[[.,.],[[.,[.,[.,.]]],.]],[.,.]]
=> [1,5,4,3,6,2,8,7] => ? => ? = 4 - 1
[[],[[[],[],[]]],[[]]]
=> [[[.,.],[.,[[[.,.],.],.]]],[.,.]]
=> [1,4,5,6,3,2,8,7] => ? => ? = 3 - 1
[[],[[[],[[]]]],[[]]]
=> [[[.,.],[.,[[.,.],[.,.]]]],[.,.]]
=> [1,4,6,5,3,2,8,7] => ? => ? = 3 - 1
[[],[[],[],[[[]]]],[]]
=> [[[.,.],[[[.,.],.],[.,[.,.]]]],.]
=> [1,3,4,7,6,5,2,8] => ? => ? = 3 - 1
[[],[[[]],[],[],[]],[]]
=> [[[.,.],[[[[.,[.,.]],.],.],.]],.]
=> [1,4,3,5,6,7,2,8] => ? => ? = 3 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000013
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1] => [1,0]
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> [2] => [1,1,0,0]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[],[],[],[[]],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[],[],[],[[[],[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[[],[],[],[[],[[]],[]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[[],[],[],[[[],[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[[],[],[[]],[[],[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[[],[],[[]],[[[]]],[]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
[[],[],[[]],[[],[],[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[[],[],[[]],[[[]],[]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
[[],[],[[]],[[[],[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
[[],[],[[[]]],[],[[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 3
[[],[],[[],[]],[[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[[[]]],[[]],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[[],[],[[],[]],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[[],[]],[[[]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[[],[],[[[]]],[[],[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[[],[],[[],[[]]],[],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[[],[],[[[[]]]],[],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
[[],[],[[[]],[]],[[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 3
[[],[],[[[],[]]],[[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 3
[[],[],[[],[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[[],[[]],[]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[[],[],[[],[[],[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[[],[],[[],[[[]]]],[]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
[[],[],[[[]],[[]]],[]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[[],[],[[[[]]],[]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
[[],[],[[[],[[]]]],[]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 3
[[],[],[[[[]],[]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
[[],[],[[[[],[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 4
[[],[],[[],[],[[]],[]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[[],[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[[],[],[[],[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[[],[],[[],[[]],[],[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[[],[],[[],[[],[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000444
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 88%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1] => [1,0]
=> ? = 1
[[],[]]
=> [1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> [2] => [1,1,0,0]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[[]]],[],[],[],[],[]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[[[[]]],[],[],[],[[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3
[[[[]]],[],[],[[]],[]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3
[[[[]]],[],[],[[],[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3
[[[[]]],[],[],[[[]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[[[[]]],[],[[]],[],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[[[[]]],[],[[]],[[]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
[[[[]]],[],[[],[]],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[[[[]]],[],[[[]]],[]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
[[[[]]],[],[[],[],[]]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[[[[]]],[],[[],[[]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3
[[[[]]],[],[[[]],[]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
[[[[]]],[],[[[],[]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3
[[[[]]],[],[[[[]]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[]]],[[]],[],[],[]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3
[[[[]]],[[]],[],[[]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
[[[[]]],[[]],[[]],[]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[[[[]]],[[]],[[],[]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[[[[]]],[[]],[[[]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[[[[]]],[[],[]],[],[]]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3
[[[[]]],[[[]]],[],[]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[[[]]],[[],[]],[[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
[[[[]]],[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[[[[]]],[[],[],[]],[]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3
[[[[]]],[[],[[]]],[]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[[[[]]],[[[]],[]],[]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[[[]]],[[[],[]]],[]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[[[]]],[[[[]]]],[]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[[[[]]],[[],[],[],[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3
[[[[]]],[[],[],[[]]]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3
[[[[]]],[[],[[]],[]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[[[[]]],[[],[[],[]]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3
[[[[]]],[[],[[[]]]]]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[[[[]]],[[[]],[],[]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[[[]]],[[[]],[[]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[[[[]]],[[[],[]],[]]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[[[]]],[[[[]]],[]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[[[[]]],[[[],[],[]]]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[[[[]]],[[[],[[]]]]]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[[[[]]],[[[[]],[]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[[[[]]],[[[[],[]]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4
[[[[]]],[[[[[]]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[[[[]],[]],[],[],[],[]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[[[[],[]]],[],[],[],[]]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3
[[[[[]]]],[],[],[],[]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4
[[[[]],[]],[],[],[[]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3
[[[[],[]]],[],[],[[]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3
[[[[[]]]],[],[],[[]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[[[[]],[]],[],[[]],[]]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000442
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 88%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1] => [1,0]
=> ? = 1 - 1
[[],[]]
=> [1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[[[]]]
=> [1,1,0,0]
=> [2] => [1,1,0,0]
=> 1 = 2 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 0 = 1 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 2 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 2 = 3 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[[[]]],[],[],[],[],[]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[[[]]],[],[],[],[[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 - 1
[[[[]]],[],[],[[]],[]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[[[]]],[],[],[[],[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[[[]]],[],[],[[[]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 1
[[[[]]],[],[[]],[],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[[[]]],[],[[]],[[]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[[]]],[],[[],[]],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[[[]]],[],[[[]]],[]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[[[[]]],[],[[],[],[]]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
=> [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[[[]]],[],[[],[[]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
=> [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[[]]],[],[[[]],[]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[[[[]]],[],[[[],[]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
=> [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 - 1
[[[[]]],[],[[[[]]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[[[[]]],[[]],[],[],[]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[[[]]],[[]],[],[[]]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[[[[]]],[[]],[[]],[]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[[[]]],[[]],[[],[]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[[[]]],[[]],[[[]]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[[[[]]],[[],[]],[],[]]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[[[]]],[[[]]],[],[]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[[[[]]],[[],[]],[[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[[[[]]],[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[[[[]]],[[],[],[]],[]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[[[]]],[[],[[]]],[]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[[[]]],[[[]],[]],[]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[[[[]]],[[[],[]]],[]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[[[[]]],[[[[]]]],[]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4 - 1
[[[[]]],[[],[],[],[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[[[]]],[[],[],[[]]]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
=> [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 - 1
[[[[]]],[[],[[]],[]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[[[]]],[[],[[],[]]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[[[]]],[[],[[[]]]]]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[[[[]]],[[[]],[],[]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[[[[]]],[[[]],[[]]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[[[[]]],[[[],[]],[]]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[[[[]]],[[[[]]],[]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4 - 1
[[[[]]],[[[],[],[]]]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 - 1
[[[[]]],[[[],[[]]]]]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 - 1
[[[[]]],[[[[]],[]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4 - 1
[[[[]]],[[[[],[]]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4 - 1
[[[[]]],[[[[[]]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[[[[]],[]],[],[],[],[]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[[[],[]]],[],[],[],[]]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[[[[]]]],[],[],[],[]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[[[[]],[]],[],[],[[]]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 - 1
[[[[],[]]],[],[],[[]]]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3 - 1
[[[[[]]]],[],[],[[]]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 4 - 1
[[[[]],[]],[],[[]],[]]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St001062
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [[]]
=> [1,0]
=> {{1}}
=> ? = 1
[[],[]]
=> [[],[]]
=> [1,0,1,0]
=> {{1},{2}}
=> 1
[[[]]]
=> [[[]]]
=> [1,1,0,0]
=> {{1,2}}
=> 2
[[],[],[]]
=> [[],[],[]]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[[],[[]]]
=> [[[]],[]]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[[[]],[]]
=> [[],[[]]]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2
[[[],[]]]
=> [[[],[]]]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[[[[]]]]
=> [[[[]]]]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[[],[],[],[]]
=> [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[[],[],[[]]]
=> [[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
[[],[[]],[]]
=> [[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[[],[[],[]]]
=> [[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 2
[[],[[[]]]]
=> [[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[[[]],[],[]]
=> [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
[[[]],[[]]]
=> [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[[[],[]],[]]
=> [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[[[[]]],[]]
=> [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 3
[[[],[],[]]]
=> [[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 2
[[[],[[]]]]
=> [[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[[[[]],[]]]
=> [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 3
[[[[],[]]]]
=> [[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[[[[[]]]]]
=> [[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1
[[],[],[],[[]]]
=> [[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 2
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 2
[[],[],[[],[]]]
=> [[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 2
[[],[],[[[]]]]
=> [[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[[],[[]],[],[]]
=> [[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 2
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2
[[],[[],[]],[]]
=> [[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 2
[[],[[[]]],[]]
=> [[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 3
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 2
[[],[[],[[]]]]
=> [[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 2
[[],[[[]],[]]]
=> [[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 3
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 4
[[[]],[],[],[]]
=> [[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 2
[[[]],[],[[]]]
=> [[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[[[]],[[]],[]]
=> [[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 2
[[[]],[[],[]]]
=> [[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 2
[[[]],[[[]]]]
=> [[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3
[[[],[]],[],[]]
=> [[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 2
[[[[]]],[],[]]
=> [[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 3
[[[],[]],[[]]]
=> [[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2
[[[[]]],[[]]]
=> [[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 3
[[[],[],[]],[]]
=> [[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 2
[[[],[[]]],[]]
=> [[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 2
[[[[]],[]],[]]
=> [[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 3
[[[[[]]]],[]]
=> [[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 4
[[[],[],[],[]]]
=> [[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 2
[[],[],[],[],[],[],[],[]]
=> [[],[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[[],[],[],[],[],[],[[]]]
=> [[[]],[],[],[],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 2
[[],[],[],[],[],[[]],[]]
=> [[],[[]],[],[],[],[],[]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2,3},{4},{5},{6},{7},{8}}
=> ? = 2
[[],[],[],[],[],[[],[]]]
=> [[[],[]],[],[],[],[],[]]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 2
[[],[],[],[],[],[[[]]]]
=> [[[[]]],[],[],[],[],[]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 3
[[],[],[],[],[[]],[],[]]
=> [[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3,4},{5},{6},{7},{8}}
=> ? = 2
[[],[],[],[],[[]],[[]]]
=> [[[]],[[]],[],[],[],[]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 2
[[],[],[],[],[[],[]],[]]
=> [[],[[],[]],[],[],[],[]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> {{1},{2,4},{3},{5},{6},{7},{8}}
=> ? = 2
[[],[],[],[],[[[]]],[]]
=> [[],[[[]]],[],[],[],[]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6},{7},{8}}
=> ? = 3
[[],[],[],[],[[],[],[]]]
=> [[[],[],[]],[],[],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,4},{2},{3},{5},{6},{7},{8}}
=> ? = 2
[[],[],[],[],[[],[[]]]]
=> [[[[]],[]],[],[],[],[]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,4},{2,3},{5},{6},{7},{8}}
=> ? = 2
[[],[],[],[],[[[]],[]]]
=> [[[],[[]]],[],[],[],[]]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,3,4},{2},{5},{6},{7},{8}}
=> ? = 3
[[],[],[],[],[[[],[]]]]
=> [[[[],[]]],[],[],[],[]]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,4},{3},{5},{6},{7},{8}}
=> ? = 3
[[],[],[],[],[[[[]]]]]
=> [[[[[]]]],[],[],[],[]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 4
[[],[],[],[[]],[],[],[]]
=> [[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 2
[[],[],[],[[]],[],[[]]]
=> [[[]],[],[[]],[],[],[]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4,5},{6},{7},{8}}
=> ? = 2
[[],[],[],[[]],[[]],[]]
=> [[],[[]],[[]],[],[],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> {{1},{2,3},{4,5},{6},{7},{8}}
=> ? = 2
[[],[],[],[[]],[[],[]]]
=> [[[],[]],[[]],[],[],[]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> {{1,3},{2},{4,5},{6},{7},{8}}
=> ? = 2
[[],[],[],[[]],[[[]]]]
=> [[[[]]],[[]],[],[],[]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 3
[[],[],[],[[],[]],[],[]]
=> [[],[],[[],[]],[],[],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> {{1},{2},{3,5},{4},{6},{7},{8}}
=> ? = 2
[[],[],[],[[[]]],[],[]]
=> [[],[],[[[]]],[],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1},{2},{3,4,5},{6},{7},{8}}
=> ? = 3
[[],[],[],[[],[]],[[]]]
=> [[[]],[[],[]],[],[],[]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3,5},{4},{6},{7},{8}}
=> ? = 2
[[],[],[],[[[]]],[[]]]
=> [[[]],[[[]]],[],[],[]]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2},{3,4,5},{6},{7},{8}}
=> ? = 3
[[],[],[],[[],[],[]],[]]
=> [[],[[],[],[]],[],[],[]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> {{1},{2,5},{3},{4},{6},{7},{8}}
=> ? = 2
[[],[],[],[[],[[]]],[]]
=> [[],[[[]],[]],[],[],[]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> {{1},{2,5},{3,4},{6},{7},{8}}
=> ? = 2
[[],[],[],[[[]],[]],[]]
=> [[],[[],[[]]],[],[],[]]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> {{1},{2,4,5},{3},{6},{7},{8}}
=> ? = 3
[[],[],[],[[[],[]]],[]]
=> [[],[[[],[]]],[],[],[]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> {{1},{2,3,5},{4},{6},{7},{8}}
=> ? = 3
[[],[],[],[[[[]]]],[]]
=> [[],[[[[]]]],[],[],[]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> {{1},{2,3,4,5},{6},{7},{8}}
=> ? = 4
[[],[],[],[[],[],[],[]]]
=> [[[],[],[],[]],[],[],[]]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 2
[[],[],[],[[],[],[[]]]]
=> [[[[]],[],[]],[],[],[]]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> {{1,5},{2,3},{4},{6},{7},{8}}
=> ? = 2
[[],[],[],[[],[[]],[]]]
=> [[[],[[]],[]],[],[],[]]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> {{1,5},{2},{3,4},{6},{7},{8}}
=> ? = 2
[[],[],[],[[],[[],[]]]]
=> [[[[],[]],[]],[],[],[]]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> {{1,5},{2,4},{3},{6},{7},{8}}
=> ? = 2
[[],[],[],[[],[[[]]]]]
=> [[[[[]]],[]],[],[],[]]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> {{1,5},{2,3,4},{6},{7},{8}}
=> ? = 3
[[],[],[],[[[]],[],[]]]
=> [[[],[],[[]]],[],[],[]]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,4,5},{2},{3},{6},{7},{8}}
=> ? = 3
[[],[],[],[[[]],[[]]]]
=> [[[[]],[[]]],[],[],[]]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,4,5},{2,3},{6},{7},{8}}
=> ? = 3
[[],[],[],[[[],[]],[]]]
=> [[[],[[],[]]],[],[],[]]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> {{1,3,5},{2},{4},{6},{7},{8}}
=> ? = 3
[[],[],[],[[[[]]],[]]]
=> [[[],[[[]]]],[],[],[]]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> {{1,3,4,5},{2},{6},{7},{8}}
=> ? = 4
[[],[],[],[[[],[],[]]]]
=> [[[[],[],[]]],[],[],[]]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,5},{3},{4},{6},{7},{8}}
=> ? = 3
[[],[],[],[[[],[[]]]]]
=> [[[[[]],[]]],[],[],[]]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,5},{3,4},{6},{7},{8}}
=> ? = 3
[[],[],[],[[[[]],[]]]]
=> [[[[],[[]]]],[],[],[]]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,4,5},{3},{6},{7},{8}}
=> ? = 4
[[],[],[],[[[[],[]]]]]
=> [[[[[],[]]]],[],[],[]]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,5},{4},{6},{7},{8}}
=> ? = 4
[[],[],[],[[[[[]]]]]]
=> [[[[[[]]]]],[],[],[]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 5
[[],[],[[]],[],[],[],[]]
=> [[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> {{1},{2},{3},{4},{5,6},{7},{8}}
=> ? = 2
[[],[],[[]],[],[],[[]]]
=> [[[]],[],[],[[]],[],[]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4},{5,6},{7},{8}}
=> ? = 2
[[],[],[[]],[],[[]],[]]
=> [[],[[]],[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5,6},{7},{8}}
=> ? = 2
[[],[],[[]],[],[[],[]]]
=> [[[],[]],[],[[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5,6},{7},{8}}
=> ? = 2
[[],[],[[]],[],[[[]]]]
=> [[[[]]],[],[[]],[],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> {{1,2,3},{4},{5,6},{7},{8}}
=> ? = 3
[[],[],[[]],[[]],[],[]]
=> [[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 2
[[],[],[[]],[[]],[[]]]
=> [[[]],[[]],[[]],[],[]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 2
Description
The maximal size of a block of a set partition.
Matching statistic: St000308
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 88%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [.,.]
=> [1] => 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => 1
[[[]]]
=> [[.,.],.]
=> [1,2] => 2
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => 2
[[[],[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 3
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 3
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 3
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 4
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 2
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 2
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 2
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 2
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 2
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 3
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 2
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 3
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 3
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 4
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 2
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 2
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 2
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 3
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 2
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 3
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 2
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 3
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 2
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 2
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 3
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 3
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 4
[[],[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [7,6,5,4,3,2,1] => ? = 1
[[],[],[],[],[],[[]]]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => ? = 2
[[],[],[],[],[[]],[]]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [6,5,7,4,3,2,1] => ? = 2
[[],[],[],[],[[],[]]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [7,5,6,4,3,2,1] => ? = 2
[[],[],[],[[]],[],[]]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [6,5,4,7,3,2,1] => ? = 2
[[],[],[],[[]],[[]]]
=> [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => ? = 2
[[],[],[],[[],[]],[]]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [7,5,4,6,3,2,1] => ? = 2
[[],[],[],[[],[],[]]]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [7,6,4,5,3,2,1] => ? = 2
[[],[],[],[[],[[]]]]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => ? = 2
[[],[],[[]],[],[],[]]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [6,5,4,3,7,2,1] => ? = 2
[[],[],[[]],[],[[]]]
=> [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => ? = 2
[[],[],[[]],[[]],[]]
=> [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [5,4,6,3,7,2,1] => ? = 2
[[],[],[[]],[[],[]]]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [7,5,6,3,4,2,1] => ? = 2
[[],[],[[],[]],[],[]]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [7,6,4,3,5,2,1] => ? = 2
[[],[],[[],[]],[[]]]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => ? = 2
[[],[],[[],[],[]],[]]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [7,5,4,3,6,2,1] => ? = 2
[[],[],[[],[[]]],[]]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [7,4,5,3,6,2,1] => ? = 2
[[],[],[[],[],[],[]]]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [7,6,5,3,4,2,1] => ? = 2
[[],[],[[],[],[[]]]]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => ? = 2
[[],[],[[],[[]],[]]]
=> [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [6,5,7,3,4,2,1] => ? = 2
[[],[],[[],[[],[]]]]
=> [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [6,4,5,3,7,2,1] => ? = 2
[[],[],[[],[[[]]]]]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => ? = 3
[[],[],[[[]],[[]]]]
=> [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => ? = 3
[[],[],[[[],[[]]]]]
=> [.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => ? = 3
[[],[[]],[],[],[],[]]
=> [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [6,5,4,3,2,7,1] => ? = 2
[[],[[]],[],[],[[]]]
=> [.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [5,6,4,3,2,7,1] => ? = 2
[[],[[]],[],[[]],[]]
=> [.,[[.,[.,[[.,[.,.]],.]]],.]]
=> [5,4,6,3,2,7,1] => ? = 2
[[],[[]],[],[[],[]]]
=> [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [7,5,6,4,2,3,1] => ? = 2
[[],[[]],[[]],[],[]]
=> [.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [5,4,3,6,2,7,1] => ? = 2
[[],[[]],[[]],[[]]]
=> [.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => ? = 2
[[],[[]],[[],[]],[]]
=> [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [7,5,4,6,2,3,1] => ? = 2
[[],[[]],[[],[],[]]]
=> [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [7,6,4,5,2,3,1] => ? = 2
[[],[[]],[[],[[]]]]
=> [.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => ? = 2
[[],[[]],[[[]],[]]]
=> [.,[[.,.],[[[.,.],.],[.,.]]]]
=> [7,4,5,6,2,3,1] => ? = 3
[[],[[]],[[[],[]]]]
=> [.,[[.,.],[[[.,.],[.,.]],.]]]
=> [6,4,5,7,2,3,1] => ? = 3
[[],[[],[]],[],[],[]]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [7,6,5,3,2,4,1] => ? = 2
[[],[[],[]],[],[[]]]
=> [.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [6,7,5,3,2,4,1] => ? = 2
[[],[[],[]],[[]],[]]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [6,5,7,3,2,4,1] => ? = 2
[[],[[],[]],[[],[]]]
=> [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [7,5,6,3,2,4,1] => ? = 2
[[],[[],[]],[[[]]]]
=> [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => ? = 3
[[],[[[]]],[[],[]]]
=> [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [7,5,6,2,3,4,1] => ? = 3
[[],[[],[],[]],[],[]]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [7,6,4,3,2,5,1] => ? = 2
[[],[[],[[]]],[],[]]
=> [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [7,6,3,4,2,5,1] => ? = 2
[[],[[],[],[]],[[]]]
=> [.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [6,7,4,3,2,5,1] => ? = 2
[[],[[],[[]]],[[]]]
=> [.,[[.,[[.,.],.]],[[.,.],.]]]
=> [6,7,3,4,2,5,1] => ? = 2
[[],[[[]],[]],[[]]]
=> [.,[[[.,[.,.]],.],[[.,.],.]]]
=> [6,7,3,2,4,5,1] => ? = 3
[[],[[[],[]]],[[]]]
=> [.,[[[.,[.,.]],[[.,.],.]],.]]
=> [5,6,3,2,4,7,1] => ? = 3
[[],[[],[],[],[]],[]]
=> [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [7,5,4,3,2,6,1] => ? = 2
[[],[[],[],[[]]],[]]
=> [.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [7,4,5,3,2,6,1] => ? = 2
[[],[[],[[]],[]],[]]
=> [.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [7,4,3,5,2,6,1] => ? = 2
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St001372
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 88%
Mp00093: Dyck paths —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> 10 => 1
[[],[]]
=> [1,0,1,0]
=> 1010 => 1
[[[]]]
=> [1,1,0,0]
=> 1100 => 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> 101010 => 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> 101100 => 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> 110010 => 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> 110100 => 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> 111000 => 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 4
[[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 10101010101010 => ? = 1
[[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 10101010101100 => ? = 2
[[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> 10101010110010 => ? = 2
[[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 10101010110100 => ? = 2
[[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 10101010111000 => ? = 3
[[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> 10101011001010 => ? = 2
[[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 10101011001100 => ? = 2
[[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> 10101011010010 => ? = 2
[[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> 10101011100010 => ? = 3
[[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 10101011010100 => ? = 2
[[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> 10101011011000 => ? = 2
[[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 10101011100100 => ? = 3
[[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 10101011101000 => ? = 3
[[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 10101011110000 => ? = 4
[[],[],[[]],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> 10101100101010 => ? = 2
[[],[],[[]],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 10101100101100 => ? = 2
[[],[],[[]],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> 10101100110010 => ? = 2
[[],[],[[]],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> 10101100110100 => ? = 2
[[],[],[[]],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> 10101100111000 => ? = 3
[[],[],[[],[]],[],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> 10101101001010 => ? = 2
[[],[],[[[]]],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> 10101110001010 => ? = 3
[[],[],[[],[]],[[]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> 10101101001100 => ? = 2
[[],[],[[[]]],[[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> 10101110001100 => ? = 3
[[],[],[[],[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> 10101101010010 => ? = 2
[[],[],[[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> 10101101100010 => ? = 2
[[],[],[[[]],[]],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> 10101110010010 => ? = 3
[[],[],[[[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> 10101110100010 => ? = 3
[[],[],[[[[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> 10101111000010 => ? = 4
[[],[],[[],[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 10101101010100 => ? = 2
[[],[],[[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> 10101101011000 => ? = 2
[[],[],[[],[[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> 10101101100100 => ? = 2
[[],[],[[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> 10101101101000 => ? = 2
[[],[],[[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> 10101101110000 => ? = 3
[[],[],[[[]],[],[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> 10101110010100 => ? = 3
[[],[],[[[]],[[]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> 10101110011000 => ? = 3
[[],[],[[[],[]],[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> 10101110100100 => ? = 3
[[],[],[[[[]]],[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 10101111000100 => ? = 4
[[],[],[[[],[],[]]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> 10101110101000 => ? = 3
[[],[],[[[],[[]]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> 10101110110000 => ? = 3
[[],[],[[[[]],[]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> 10101111001000 => ? = 4
[[],[],[[[[],[]]]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 10101111010000 => ? = 4
[[],[],[[[[[]]]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 10101111100000 => ? = 5
[[],[[]],[],[],[],[]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> 10110010101010 => ? = 2
[[],[[]],[],[],[[]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 10110010101100 => ? = 2
[[],[[]],[],[[]],[]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> 10110010110010 => ? = 2
[[],[[]],[],[[],[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> 10110010110100 => ? = 2
[[],[[]],[],[[[]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> 10110010111000 => ? = 3
[[],[[]],[[]],[],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> 10110011001010 => ? = 2
[[],[[]],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 10110011001100 => ? = 2
[[],[[]],[[],[]],[]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> 10110011010010 => ? = 2
Description
The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St000846
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 75%
Mp00047: Ordered trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 75%
Values
[[]]
=> [[]]
=> ([(0,1)],2)
=> 1
[[],[]]
=> [[[]]]
=> ([(0,2),(2,1)],3)
=> 1
[[[]]]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> 2
[[],[],[]]
=> [[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[[]]]
=> [[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[[[]],[]]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[[],[]]]
=> [[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[[[]]]]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[[],[],[],[]]
=> [[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[[]]]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[[],[[]],[]]
=> [[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[[],[[],[]]]
=> [[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[[],[[[]]]]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3
[[[]],[],[]]
=> [[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[[[]],[[]]]
=> [[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[[[],[]],[]]
=> [[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[[[[]]],[]]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[[[],[],[]]]
=> [[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[[[],[[]]]]
=> [[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[[[[]],[]]]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[[[[],[]]]]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[[[[[]]]]]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[[],[],[],[],[]]
=> [[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[],[],[],[[]]]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[[],[],[[]],[]]
=> [[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[[],[],[[],[]]]
=> [[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 2
[[],[],[[[]]]]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> 3
[[],[[]],[],[]]
=> [[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[[],[[]],[[]]]
=> [[[],[[],[]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
[[],[[],[]],[]]
=> [[[[]],[[]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> 2
[[],[[[]]],[]]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 3
[[],[[],[],[]]]
=> [[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 2
[[],[[],[[]]]]
=> [[[[],[]],[]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> 2
[[],[[[]],[]]]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 3
[[],[[[],[]]]]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 3
[[],[[[[]]]]]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> 4
[[[]],[],[],[]]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[[[]],[],[[]]]
=> [[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 2
[[[]],[[]],[]]
=> [[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 2
[[[]],[[],[]]]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 2
[[[]],[[[]]]]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> 3
[[[],[]],[],[]]
=> [[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[[[[]]],[],[]]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[[[],[]],[[]]]
=> [[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[[[[]]],[[]]]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[[[],[],[]],[]]
=> [[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[[[],[[]]],[]]
=> [[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[[[[]],[]],[]]
=> [[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[[[[],[]]],[]]
=> [[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[[[[[]]]],[]]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[[],[],[],[],[],[],[]]
=> [[[[[[[[]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1
[[],[],[],[],[],[[]]]
=> [[[[[[[],[]]]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 2
[[],[],[],[],[[]],[]]
=> [[[[[[],[[]]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[[],[],[],[],[[],[]]]
=> [[[[[[[]],[]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 2
[[],[],[],[],[[[]]]]
=> [[[[[[],[],[]]]]]]
=> ([(0,7),(1,7),(2,7),(4,5),(5,3),(6,4),(7,6)],8)
=> ? = 3
[[],[],[],[[]],[],[]]
=> [[[[[],[[[]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ? = 2
[[],[],[],[[]],[[]]]
=> [[[[[],[[],[]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(5,4),(6,7),(7,3)],8)
=> ? = 2
[[],[],[],[[],[]],[]]
=> [[[[[[]],[[]]]]]]
=> ([(0,4),(1,3),(3,7),(4,7),(5,2),(6,5),(7,6)],8)
=> ? = 2
[[],[],[],[[[]]],[]]
=> [[[[[],[],[[]]]]]]
=> ([(0,7),(1,7),(2,4),(4,7),(5,3),(6,5),(7,6)],8)
=> ? = 3
[[],[],[],[[],[],[]]]
=> [[[[[[[]]],[]]]]]
=> ([(0,7),(1,5),(3,7),(4,2),(5,3),(6,4),(7,6)],8)
=> ? = 2
[[],[],[],[[],[[]]]]
=> [[[[[[],[]],[]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(5,4),(6,7),(7,3)],8)
=> ? = 2
[[],[],[],[[[]],[]]]
=> [[[[[],[[]],[]]]]]
=> ([(0,7),(1,7),(2,4),(4,7),(5,3),(6,5),(7,6)],8)
=> ? = 3
[[],[],[],[[[],[]]]]
=> [[[[[[]],[],[]]]]]
=> ([(0,7),(1,7),(2,4),(4,7),(5,3),(6,5),(7,6)],8)
=> ? = 3
[[],[],[],[[[[]]]]]
=> [[[[[],[],[],[]]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(6,5),(7,4)],8)
=> ? = 4
[[],[],[[]],[],[],[]]
=> [[[[],[[[[]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ? = 2
[[],[],[[]],[],[[]]]
=> [[[[],[[[],[]]]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(6,4),(7,3)],8)
=> ? = 2
[[],[],[[]],[[]],[]]
=> [[[[],[[],[[]]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ? = 2
[[],[],[[]],[[],[]]]
=> [[[[],[[[]],[]]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ? = 2
[[],[],[[]],[[[]]]]
=> [[[[],[[],[],[]]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(6,4),(7,6)],8)
=> ? = 3
[[],[],[[],[]],[],[]]
=> [[[[[]],[[[]]]]]]
=> ([(0,6),(1,4),(3,7),(4,7),(5,2),(6,3),(7,5)],8)
=> ? = 2
[[],[],[[[]]],[],[]]
=> [[[[],[],[[[]]]]]]
=> ([(0,7),(1,7),(2,5),(4,7),(5,4),(6,3),(7,6)],8)
=> ? = 3
[[],[],[[],[]],[[]]]
=> [[[[[]],[[],[]]]]]
=> ([(0,6),(1,6),(2,3),(3,7),(4,5),(6,7),(7,4)],8)
=> ? = 2
[[],[],[[[]]],[[]]]
=> [[[[],[],[[],[]]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(6,7),(7,4)],8)
=> ? = 3
[[],[],[[],[],[]],[]]
=> [[[[[[]]],[[]]]]]
=> ([(0,6),(1,4),(3,7),(4,7),(5,2),(6,3),(7,5)],8)
=> ? = 2
[[],[],[[],[[]]],[]]
=> [[[[[],[]],[[]]]]]
=> ([(0,6),(1,6),(2,3),(3,7),(4,5),(6,7),(7,4)],8)
=> ? = 2
[[],[],[[[]],[]],[]]
=> [[[[],[[]],[[]]]]]
=> ([(0,7),(1,5),(2,4),(4,7),(5,7),(6,3),(7,6)],8)
=> ? = 3
[[],[],[[[],[]]],[]]
=> [[[[[]],[],[[]]]]]
=> ([(0,7),(1,5),(2,4),(4,7),(5,7),(6,3),(7,6)],8)
=> ? = 3
[[],[],[[[[]]]],[]]
=> [[[[],[],[],[[]]]]]
=> ([(0,7),(1,7),(2,7),(3,4),(4,7),(5,6),(7,5)],8)
=> ? = 4
[[],[],[[],[],[],[]]]
=> [[[[[[[]]]],[]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ? = 2
[[],[],[[],[],[[]]]]
=> [[[[[[],[]]],[]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,7),(6,4),(7,3)],8)
=> ? = 2
[[],[],[[],[[]],[]]]
=> [[[[[],[[]]],[]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ? = 2
[[],[],[[],[[],[]]]]
=> [[[[[[]],[]],[]]]]
=> ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
=> ? = 2
[[],[],[[],[[[]]]]]
=> [[[[[],[],[]],[]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,5),(6,4),(7,6)],8)
=> ? = 3
[[],[],[[[]],[],[]]]
=> [[[[],[[[]]],[]]]]
=> ([(0,7),(1,7),(2,5),(4,7),(5,4),(6,3),(7,6)],8)
=> ? = 3
[[],[],[[[]],[[]]]]
=> [[[[],[[],[]],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(6,7),(7,4)],8)
=> ? = 3
[[],[],[[[],[]],[]]]
=> [[[[[]],[[]],[]]]]
=> ([(0,7),(1,5),(2,4),(4,7),(5,7),(6,3),(7,6)],8)
=> ? = 3
[[],[],[[[[]]],[]]]
=> [[[[],[],[[]],[]]]]
=> ([(0,7),(1,7),(2,7),(3,4),(4,7),(5,6),(7,5)],8)
=> ? = 4
[[],[],[[[],[],[]]]]
=> [[[[[[]]],[],[]]]]
=> ([(0,7),(1,7),(2,5),(4,7),(5,4),(6,3),(7,6)],8)
=> ? = 3
[[],[],[[[],[[]]]]]
=> [[[[[],[]],[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(6,7),(7,4)],8)
=> ? = 3
[[],[],[[[[]],[]]]]
=> [[[[],[[]],[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,4),(4,7),(5,6),(7,5)],8)
=> ? = 4
[[],[],[[[[],[]]]]]
=> [[[[[]],[],[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,4),(4,7),(5,6),(7,5)],8)
=> ? = 4
[[],[],[[[[[]]]]]]
=> [[[[],[],[],[],[]]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(7,5)],8)
=> ? = 5
[[],[[]],[],[],[],[]]
=> [[[],[[[[[]]]]]]]
=> ([(0,7),(1,6),(2,7),(4,5),(5,2),(6,4),(7,3)],8)
=> ? = 2
[[],[[]],[],[],[[]]]
=> [[[],[[[[],[]]]]]]
=> ([(0,6),(1,6),(2,7),(4,5),(5,7),(6,4),(7,3)],8)
=> ? = 2
[[],[[]],[],[[]],[]]
=> [[[],[[[],[[]]]]]]
=> ([(0,6),(1,7),(2,3),(3,7),(5,6),(6,4),(7,5)],8)
=> ? = 2
[[],[[]],[],[[],[]]]
=> [[[],[[[[]],[]]]]]
=> ([(0,6),(1,7),(2,3),(3,7),(5,6),(6,4),(7,5)],8)
=> ? = 2
[[],[[]],[],[[[]]]]
=> [[[],[[[],[],[]]]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(6,5),(7,4)],8)
=> ? = 3
[[],[[]],[[]],[],[]]
=> [[[],[[],[[[]]]]]]
=> ([(0,7),(1,6),(2,4),(4,5),(5,6),(6,7),(7,3)],8)
=> ? = 2
[[],[[]],[[]],[[]]]
=> [[[],[[],[[],[]]]]]
=> ([(0,6),(1,7),(2,5),(3,5),(5,6),(6,7),(7,4)],8)
=> ? = 2
[[],[[]],[[],[]],[]]
=> [[[],[[[]],[[]]]]]
=> ([(0,6),(1,4),(2,3),(3,7),(4,7),(6,5),(7,6)],8)
=> ? = 2
Description
The maximal number of elements covering an element of a poset.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000328The maximum number of child nodes in a tree. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St000845The maximal number of elements covered by an element in a poset. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2.
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