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Your data matches 67 different statistics following compositions of up to 3 maps.
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Matching statistic: St000993
Mp00246: Ordered trees —rotate⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[],[],[]]
=> [[],[],[]]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[[],[[]]]
=> [[[]],[]]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[[[],[]]]
=> [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[[],[],[],[]]
=> [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[[],[],[[]]]
=> [[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[[],[[]],[]]
=> [[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[[],[[],[]]]
=> [[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[[],[[[]]]]
=> [[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[[[]],[],[]]
=> [[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[[[]],[[]]]
=> [[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[[[],[]],[]]
=> [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[[[],[],[]]]
=> [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[[[],[[]]]]
=> [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[[[[]],[]]]
=> [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[[[[],[]]]]
=> [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 1
[[],[],[],[[]]]
=> [[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 1
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1
[[],[],[[],[]]]
=> [[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 1
[[],[],[[[]]]]
=> [[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
[[],[[]],[],[]]
=> [[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
[[],[[],[]],[]]
=> [[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[[],[[[]]],[]]
=> [[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[[],[[],[[]]]]
=> [[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 1
[[],[[[]],[]]]
=> [[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[[[]],[],[],[]]
=> [[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[[[]],[],[[]]]
=> [[[],[[]],[]]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 1
[[[]],[[]],[]]
=> [[[[]],[],[]]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[[[]],[[],[]]]
=> [[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[[[]],[[[]]]]
=> [[[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[[[],[]],[],[]]
=> [[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1
[[[[]]],[],[]]
=> [[[[],[],[]]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[[[],[]],[[]]]
=> [[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1
[[[[]]],[[]]]
=> [[[[[]],[]]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[[[],[],[]],[]]
=> [[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1
[[[],[[]]],[]]
=> [[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 1
[[[[]],[]],[]]
=> [[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1
[[[],[],[],[]]]
=> [[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 2
[[[],[],[[]]]]
=> [[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 2
[[[],[[]],[]]]
=> [[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 2
[[[],[[],[]]]]
=> [[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 2
[[[],[[[]]]]]
=> [[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2
[[[[]],[],[]]]
=> [[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2
[[[[]],[[]]]]
=> [[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[[[[],[]],[]]]
=> [[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000678
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[[],[],[]]
=> [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[[],[],[[]]]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[[[],[[]],[]]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[],[[[]]]]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[[]],[],[]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[[[[]],[[]]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[],[[],[[[[[]]]]]]]
=> [.,[[.,[[[[[.,.],.],.],.],.]],.]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[[],[[[],[[[],[]]]]]]
=> [.,[[[.,[[[.,[.,.]],.],.]],.],.]]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[[],[[[],[[[[]]]]]]]
=> [.,[[[.,[[[[.,.],.],.],.]],.],.]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[[],[[[[[],[]]],[]]]]
=> [.,[[[[[.,[.,.]],.],[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[[],[[[[],[],[[]]]]]]
=> [.,[[[[.,[.,[[.,.],.]]],.],.],.]]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[],[[[[],[[],[]]]]]]
=> [.,[[[[.,[[.,[.,.]],.]],.],.],.]]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[[],[[[[[],[],[]]]]]]
=> [.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000326
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,1,3] => 10 => 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [2,3,1] => 10 => 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,3,2] => 01 => 2
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 101 => 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 100 => 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 100 => 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 100 => 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 100 => 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 110 => 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 100 => 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 100 => 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 010 => 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 010 => 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 010 => 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 001 => 3
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 1010 => 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 1000 => 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 1010 => 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 1010 => 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 1000 => 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1010 => 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 1000 => 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => 1001 => 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1000 => 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 1001 => 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 1000 => 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 1000 => 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1000 => 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1000 => 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 1100 => 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 1000 => 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 1100 => 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 1100 => 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1000 => 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => 1100 => 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 1100 => 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 1000 => 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1000 => 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 1001 => 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 1000 => 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => 1010 => 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => 1000 => 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 0101 => 2
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0100 => 2
[[[],[[]],[]]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => 0100 => 2
[[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => 0100 => 2
[[[],[[[]]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 0100 => 2
[[[[]],[],[]]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => 0110 => 2
[[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => 0100 => 2
[[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => 0100 => 2
[[],[[[],[[[],[]]]]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,6,7,1,8,5] => ? => ? = 1
[[[]],[[[],[[]],[]]]]
=> [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,4,5,1,6,8,2,7] => ? => ? = 1
[[[[]]],[[],[],[]],[]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3,8] => ? => ? = 1
[[[[]]],[[[]],[]],[]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> [4,1,5,6,2,7,8,3] => ? => ? = 1
[[[[]],[]],[[[],[]]]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,1,0,0,0]
=> [4,5,7,1,8,2,3,6] => ? => ? = 1
[[[[[]]],[]],[[]],[]]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
=> [5,1,6,7,2,3,4,8] => ? => ? = 1
[[[[[]],[]]],[[]],[]]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0,1,0]
=> [5,1,6,8,2,3,4,7] => ? => ? = 1
[[[[[]]],[],[]],[[]]]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ? => ? = 1
[[[[[]],[[]]]],[[]]]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4,7,8] => ? => ? = 1
[[[[[]]],[[],[]],[]]]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,1,0,0]
=> [1,5,2,6,7,3,8,4] => ? => ? = 2
[[[[[]]],[[],[],[]]]]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> [1,5,6,2,7,3,8,4] => ? => ? = 2
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of {1,…,n,n+1} that contains n+1, this is the minimal element of the set.
Matching statistic: St000297
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[[],[],[]]
=> [1,0,1,0,1,0]
=> [2,1,3] => 01 => 0 = 1 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [2,3,1] => 00 => 0 = 1 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,3,2] => 10 => 1 = 2 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 010 => 0 = 1 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 000 => 0 = 1 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 011 => 0 = 1 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 001 => 0 = 1 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 000 => 0 = 1 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 000 => 0 = 1 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 000 => 0 = 1 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 001 => 0 = 1 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 101 => 1 = 2 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 100 => 1 = 2 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 100 => 1 = 2 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 110 => 2 = 3 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 0101 => 0 = 1 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 0001 => 0 = 1 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 0100 => 0 = 1 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 0000 => 0 = 1 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 0000 => 0 = 1 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 0100 => 0 = 1 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0000 => 0 = 1 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => 0110 => 0 = 1 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 0111 => 0 = 1 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 0010 => 0 = 1 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0000 => 0 = 1 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 0011 => 0 = 1 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 0001 => 0 = 1 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0000 => 0 = 1 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 0001 => 0 = 1 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 0001 => 0 = 1 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 0000 => 0 = 1 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 0000 => 0 = 1 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0000 => 0 = 1 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => 0000 => 0 = 1 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 0000 => 0 = 1 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 0000 => 0 = 1 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0000 => 0 = 1 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 0010 => 0 = 1 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 0011 => 0 = 1 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => 0000 => 0 = 1 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => 0001 => 0 = 1 - 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => 1010 => 1 = 2 - 1
[[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1000 => 1 = 2 - 1
[[[],[[]],[]]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => 1011 => 1 = 2 - 1
[[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => 1001 => 1 = 2 - 1
[[[],[[[]]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => 1000 => 1 = 2 - 1
[[[[]],[],[]]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => 1000 => 1 = 2 - 1
[[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => 1000 => 1 = 2 - 1
[[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => 1001 => 1 = 2 - 1
[[],[[[],[[[],[]]]]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,6,7,1,8,5] => ? => ? = 1 - 1
[[[]],[[],[[[],[]]]]]
=> [1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,4,5,7,1,8,2,6] => ? => ? = 1 - 1
[[[]],[[[],[[]],[]]]]
=> [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,4,5,1,6,8,2,7] => ? => ? = 1 - 1
[[[]],[[[[],[]],[]]]]
=> [1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,4,5,1,6,7,2,8] => ? => ? = 1 - 1
[[[[]]],[],[[],[[]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,6,1,2,8,3,7] => ? => ? = 1 - 1
[[[[]]],[],[[[]],[]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
=> [4,5,1,6,8,2,3,7] => ? => ? = 1 - 1
[[[[]]],[[]],[[],[]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> [4,5,1,6,2,3,7,8] => ? => ? = 1 - 1
[[[[]]],[[],[],[]],[]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3,8] => ? => ? = 1 - 1
[[[[]]],[[[]],[]],[]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
=> [4,1,5,6,2,7,8,3] => ? => ? = 1 - 1
[[[[]]],[[],[[[]]]]]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3,8] => ? => ? = 1 - 1
[[[[[]]]],[],[[]],[]]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [5,1,6,7,2,3,8,4] => ? => ? = 1 - 1
[[[[[]]]],[[]],[],[]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,1,6,2,7,8,3,4] => ? => ? = 1 - 1
[[[[]],[]],[[[],[]]]]
=> [1,1,1,0,0,1,0,0,1,1,1,0,1,0,0,0]
=> [4,5,7,1,8,2,3,6] => ? => ? = 1 - 1
[[[[[]]]],[[],[],[]]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
=> [5,6,1,7,2,8,3,4] => ? => ? = 1 - 1
[[[[[]]],[]],[],[[]]]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
=> [5,6,1,2,7,3,4,8] => ? => ? = 1 - 1
[[[[[]]],[]],[[]],[]]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
=> [5,1,6,7,2,3,4,8] => ? => ? = 1 - 1
[[[[[]],[]]],[[]],[]]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0,1,0]
=> [5,1,6,8,2,3,4,7] => ? => ? = 1 - 1
[[[[[]]],[]],[[],[]]]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> [5,6,1,7,2,3,4,8] => ? => ? = 1 - 1
[[[[[],[]]]],[[],[]]]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [5,7,1,8,2,3,4,6] => ? => ? = 1 - 1
[[[[[]]],[],[]],[[]]]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [5,6,1,2,3,7,4,8] => ? => ? = 1 - 1
[[[[[]],[[]]]],[[]]]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4,7,8] => ? => ? = 1 - 1
[[[[[[]],[]],[]]],[]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [6,1,2,3,7,4,5,8] => ? => ? = 1 - 1
[[[[[]]],[],[[],[]]]]
=> [1,1,1,1,0,0,0,1,0,1,1,0,1,0,0,0]
=> [1,5,6,2,7,3,4,8] => ? => ? = 2 - 1
[[[[[]]],[[],[]],[]]]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,1,0,0]
=> [1,5,2,6,7,3,8,4] => ? => ? = 2 - 1
[[[[[]]],[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,5,2,6,7,8,3,4] => ? => ? = 2 - 1
[[[[[]]],[[],[],[]]]]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> [1,5,6,2,7,3,8,4] => ? => ? = 2 - 1
[[[[[]]],[[[]],[]]]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> [1,5,6,2,7,8,3,4] => ? => ? = 2 - 1
[[[[[]],[]],[[[]]]]]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> [1,5,6,8,2,3,4,7] => ? => ? = 2 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000011
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 1
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[[],[],[[]]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2
[[[],[[]],[]]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 2
[[[],[[[]]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 2
[[[[]],[],[]]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[[[]],[[]]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2
[[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[[],[[[],[[[],[]]]]]]
=> [.,[[[.,[[[.,[.,.]],.],.]],.],.]]
=> [4,3,5,6,2,7,8,1] => [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[],[[[],[[[[]]]]]]]
=> [.,[[[.,[[[[.,.],.],.],.]],.],.]]
=> [3,4,5,6,2,7,8,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1
[[],[[[[[],[]]],[]]]]
=> [.,[[[[[.,[.,.]],.],[.,.]],.],.]]
=> [6,3,2,4,5,7,8,1] => ?
=> ? = 1
[[],[[[[],[[],[]]]]]]
=> [.,[[[[.,[[.,[.,.]],.]],.],.],.]]
=> [4,3,5,2,6,7,8,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 1
[[[]],[[[]],[[[]]]]]
=> [[.,.],[[[.,.],[[[.,.],.],.]],.]]
=> [5,6,7,3,4,8,1,2] => ?
=> ? = 1
[[[]],[[[[]]],[[]]]]
=> [[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [6,7,3,4,5,8,1,2] => ?
=> ? = 1
[[[]],[[[[],[]]],[]]]
=> [[.,.],[[[[.,[.,.]],.],[.,.]],.]]
=> [7,4,3,5,6,8,1,2] => ?
=> ? = 1
[[[]],[[[],[[]],[]]]]
=> [[.,.],[[[.,[[.,.],[.,.]]],.],.]]
=> [6,4,5,3,7,8,1,2] => ?
=> ? = 1
[[[]],[[[[]],[],[]]]]
=> [[.,.],[[[[.,.],[.,[.,.]]],.],.]]
=> [6,5,3,4,7,8,1,2] => ?
=> ? = 1
[[[[]]],[[]],[[],[]]]
=> [[[.,.],.],[[.,.],[[.,[.,.]],.]]]
=> [7,6,8,4,5,1,2,3] => ?
=> ? = 1
[[[[[]]],[]],[[]],[]]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [8,6,7,4,1,2,3,5] => ?
=> ? = 1
[[[[[],[]]]],[[],[]]]
=> [[[[.,[.,.]],.],.],[[.,[.,.]],.]]
=> [7,6,8,2,1,3,4,5] => ?
=> ? = 1
[[[[[[]],[]]]],[],[]]
=> [[[[[.,.],[.,.]],.],.],[.,[.,.]]]
=> [8,7,3,1,2,4,5,6] => ?
=> ? = 1
[[[[[]]],[[],[]]],[]]
=> [[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,1,2,3,7] => ?
=> ? = 1
[[[[[[]],[]]],[]],[]]
=> [[[[[.,.],[.,.]],.],[.,.]],[.,.]]
=> [8,6,3,1,2,4,5,7] => ?
=> ? = 1
[[[[[[],[]]]],[]],[]]
=> [[[[[.,[.,.]],.],.],[.,.]],[.,.]]
=> [8,6,2,1,3,4,5,7] => ?
=> ? = 1
[[[[[[]],[],[]]]],[]]
=> [[[[[.,.],[.,[.,.]]],.],.],[.,.]]
=> [8,4,3,1,2,5,6,7] => ?
=> ? = 1
[[[[]],[[[],[[]]]]]]
=> [[[.,.],[[[.,[[.,.],.]],.],.]],.]
=> [4,5,3,6,7,1,2,8] => ?
=> ? = 2
[[[[]],[[[[]],[]]]]]
=> [[[.,.],[[[[.,.],[.,.]],.],.]],.]
=> [5,3,4,6,7,1,2,8] => ?
=> ? = 2
[[[[]],[[[[],[]]]]]]
=> [[[.,.],[[[[.,[.,.]],.],.],.]],.]
=> [4,3,5,6,7,1,2,8] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[[[[[]]],[],[[],[]]]]
=> [[[[.,.],.],[.,[[.,[.,.]],.]]],.]
=> [6,5,7,4,1,2,3,8] => ?
=> ? = 2
[[[[[]]],[[]],[[]]]]
=> [[[[.,.],.],[[.,.],[[.,.],.]]],.]
=> [6,7,4,5,1,2,3,8] => ?
=> ? = 2
[[[[[]]],[[],[],[]]]]
=> [[[[.,.],.],[[.,[.,[.,.]]],.]],.]
=> [6,5,4,7,1,2,3,8] => ?
=> ? = 2
[[[[[]]],[[],[[]]]]]
=> [[[[.,.],.],[[.,[[.,.],.]],.]],.]
=> [5,6,4,7,1,2,3,8] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[[[[[]]],[[[]],[]]]]
=> [[[[.,.],.],[[[.,.],[.,.]],.]],.]
=> [6,4,5,7,1,2,3,8] => ?
=> ? = 2
[[[[[]]],[[[],[]]]]]
=> [[[[.,.],.],[[[.,[.,.]],.],.]],.]
=> [5,4,6,7,1,2,3,8] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> ? = 2
[[[[[]]],[[[[]]]]]]
=> [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [4,5,6,7,1,2,3,8] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 2
[[[[[[]]],[]],[],[]]]
=> [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
=> [7,6,4,1,2,3,5,8] => ?
=> ? = 2
[[[[[[]],[]]],[],[]]]
=> [[[[[.,.],[.,.]],.],[.,[.,.]]],.]
=> [7,6,3,1,2,4,5,8] => ?
=> ? = 2
[[[[[[],[]]]],[],[]]]
=> [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
=> [7,6,2,1,3,4,5,8] => ?
=> ? = 2
[[[[[[]]],[]],[[]]]]
=> [[[[[.,.],.],[.,.]],[[.,.],.]],.]
=> [6,7,4,1,2,3,5,8] => ?
=> ? = 2
[[[[[[],[]]]],[[]]]]
=> [[[[[.,[.,.]],.],.],[[.,.],.]],.]
=> [6,7,2,1,3,4,5,8] => ?
=> ? = 2
[[[[[[]]],[],[]],[]]]
=> [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
=> [7,5,4,1,2,3,6,8] => ?
=> ? = 2
[[[[[[]]],[],[[]]]]]
=> [[[[[.,.],.],[.,[[.,.],.]]],.],.]
=> [5,6,4,1,2,3,7,8] => ?
=> ? = 3
[[[[[[]]],[[]],[]]]]
=> [[[[[.,.],.],[[.,.],[.,.]]],.],.]
=> [6,4,5,1,2,3,7,8] => ?
=> ? = 3
[[[[[[]]],[[],[]]]]]
=> [[[[[.,.],.],[[.,[.,.]],.]],.],.]
=> [5,4,6,1,2,3,7,8] => ?
=> ? = 3
[[[[[[]]],[[[]]]]]]
=> [[[[[.,.],.],[[[.,.],.],.]],.],.]
=> [4,5,6,1,2,3,7,8] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 3
[[[[[[]],[]],[[]]]]]
=> [[[[[.,.],[.,.]],[[.,.],.]],.],.]
=> [5,6,3,1,2,4,7,8] => ?
=> ? = 3
[[[[[[[]]]],[[]]]]]
=> [[[[[[.,.],.],.],[[.,.],.]],.],.]
=> [5,6,1,2,3,4,7,8] => ?
=> ? = 3
[[[[[[[]]],[[]]]]]]
=> [[[[[[.,.],.],[[.,.],.]],.],.],.]
=> [4,5,1,2,3,6,7,8] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000288
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 00 => 0 = 1 - 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 00 => 0 = 1 - 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 01 => 1 = 2 - 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 000 => 0 = 1 - 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 000 => 0 = 1 - 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 000 => 0 = 1 - 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 000 => 0 = 1 - 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 000 => 0 = 1 - 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 000 => 0 = 1 - 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 000 => 0 = 1 - 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 000 => 0 = 1 - 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 001 => 1 = 2 - 1
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 001 => 1 = 2 - 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 001 => 1 = 2 - 1
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 011 => 2 = 3 - 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0000 => 0 = 1 - 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0000 => 0 = 1 - 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 0000 => 0 = 1 - 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0000 => 0 = 1 - 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0000 => 0 = 1 - 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 0000 => 0 = 1 - 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 0000 => 0 = 1 - 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0000 => 0 = 1 - 1
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 0000 => 0 = 1 - 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0000 => 0 = 1 - 1
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0000 => 0 = 1 - 1
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 0000 => 0 = 1 - 1
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0000 => 0 = 1 - 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0000 => 0 = 1 - 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0000 => 0 = 1 - 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 0000 => 0 = 1 - 1
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 0000 => 0 = 1 - 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 0000 => 0 = 1 - 1
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 0000 => 0 = 1 - 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0000 => 0 = 1 - 1
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0000 => 0 = 1 - 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 0000 => 0 = 1 - 1
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 0000 => 0 = 1 - 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0000 => 0 = 1 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 0000 => 0 = 1 - 1
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0000 => 0 = 1 - 1
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 0000 => 0 = 1 - 1
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0001 => 1 = 2 - 1
[[[],[],[[]]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 0001 => 1 = 2 - 1
[[[],[[]],[]]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 0001 => 1 = 2 - 1
[[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 0001 => 1 = 2 - 1
[[[],[[[]]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 0001 => 1 = 2 - 1
[[[[]],[],[]]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 0001 => 1 = 2 - 1
[[[[]],[[]]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 0001 => 1 = 2 - 1
[[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 0001 => 1 = 2 - 1
[[],[[[[[],[]]],[]]]]
=> [.,[[[[[.,[.,.]],.],[.,.]],.],.]]
=> [6,3,2,4,5,7,8,1] => ? => ? = 1 - 1
[[],[[[[],[],[[]]]]]]
=> [.,[[[[.,[.,[[.,.],.]]],.],.],.]]
=> [4,5,3,2,6,7,8,1] => ? => ? = 1 - 1
[[],[[[[],[[],[]]]]]]
=> [.,[[[[.,[[.,[.,.]],.]],.],.],.]]
=> [4,3,5,2,6,7,8,1] => ? => ? = 1 - 1
[[[]],[[[[],[]]]],[]]
=> [[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [8,4,3,5,6,7,1,2] => ? => ? = 1 - 1
[[[]],[[],[[[]],[]]]]
=> [[.,.],[[.,[[[.,.],[.,.]],.]],.]]
=> [6,4,5,7,3,8,1,2] => ? => ? = 1 - 1
[[[]],[[],[[[],[]]]]]
=> [[.,.],[[.,[[[.,[.,.]],.],.]],.]]
=> [5,4,6,7,3,8,1,2] => ? => ? = 1 - 1
[[[]],[[[]],[[[]]]]]
=> [[.,.],[[[.,.],[[[.,.],.],.]],.]]
=> [5,6,7,3,4,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[]]],[[]]]]
=> [[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [6,7,3,4,5,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[]],[]],[]]]
=> [[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> [7,5,3,4,6,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[],[]]],[]]]
=> [[.,.],[[[[.,[.,.]],.],[.,.]],.]]
=> [7,4,3,5,6,8,1,2] => ? => ? = 1 - 1
[[[]],[[[],[[]],[]]]]
=> [[.,.],[[[.,[[.,.],[.,.]]],.],.]]
=> [6,4,5,3,7,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[]],[],[]]]]
=> [[.,.],[[[[.,.],[.,[.,.]]],.],.]]
=> [6,5,3,4,7,8,1,2] => ? => ? = 1 - 1
[[[[]]],[],[[[]],[]]]
=> [[[.,.],.],[.,[[[.,.],[.,.]],.]]]
=> [7,5,6,8,4,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[]],[[],[]]]
=> [[[.,.],.],[[.,.],[[.,[.,.]],.]]]
=> [7,6,8,4,5,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[]],[[]]]
=> [[[.,.],.],[[.,[.,.]],[[.,.],.]]]
=> [7,8,5,4,6,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[[]]],[]]
=> [[[.,.],.],[[.,[[.,.],.]],[.,.]]]
=> [8,5,6,4,7,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[[]],[]]]
=> [[[.,.],.],[[.,[[.,.],[.,.]]],.]]
=> [7,5,6,4,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[[[]]]]]
=> [[[.,.],.],[[.,[[[.,.],.],.]],.]]
=> [5,6,7,4,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[]],[],[]]]
=> [[[.,.],.],[[[.,.],[.,[.,.]]],.]]
=> [7,6,4,5,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[],[]],[]]]
=> [[[.,.],.],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[[]]],[]]]
=> [[[.,.],.],[[[[.,.],.],[.,.]],.]]
=> [7,4,5,6,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[],[[]]]]]
=> [[[.,.],.],[[[.,[[.,.],.]],.],.]]
=> [5,6,4,7,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[[]],[]]]]
=> [[[.,.],.],[[[[.,.],[.,.]],.],.]]
=> [6,4,5,7,8,1,2,3] => ? => ? = 1 - 1
[[[[[]]],[]],[],[[]]]
=> [[[[.,.],.],[.,.]],[.,[[.,.],.]]]
=> [7,8,6,4,1,2,3,5] => ? => ? = 1 - 1
[[[[[]]],[]],[[]],[]]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [8,6,7,4,1,2,3,5] => ? => ? = 1 - 1
[[[[[],[]]]],[[]],[]]
=> [[[[.,[.,.]],.],.],[[.,.],[.,.]]]
=> [8,6,7,2,1,3,4,5] => ? => ? = 1 - 1
[[[[[]],[]]],[[],[]]]
=> [[[[.,.],[.,.]],.],[[.,[.,.]],.]]
=> [7,6,8,3,1,2,4,5] => ? => ? = 1 - 1
[[[[[],[]]]],[[],[]]]
=> [[[[.,[.,.]],.],.],[[.,[.,.]],.]]
=> [7,6,8,2,1,3,4,5] => ? => ? = 1 - 1
[[[[[]]],[[]]],[],[]]
=> [[[[.,.],.],[[.,.],.]],[.,[.,.]]]
=> [8,7,4,5,1,2,3,6] => ? => ? = 1 - 1
[[[[[[]],[]]]],[],[]]
=> [[[[[.,.],[.,.]],.],.],[.,[.,.]]]
=> [8,7,3,1,2,4,5,6] => ? => ? = 1 - 1
[[[[[]],[]],[]],[[]]]
=> [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [7,8,5,3,1,2,4,6] => ? => ? = 1 - 1
[[[[[]]],[],[[]]],[]]
=> [[[[.,.],.],[.,[[.,.],.]]],[.,.]]
=> [8,5,6,4,1,2,3,7] => ? => ? = 1 - 1
[[[[[]]],[[]],[]],[]]
=> [[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> [8,6,4,5,1,2,3,7] => ? => ? = 1 - 1
[[[[[]]],[[],[]]],[]]
=> [[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,1,2,3,7] => ? => ? = 1 - 1
[[[[[[]]],[]],[]],[]]
=> [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [8,6,4,1,2,3,5,7] => ? => ? = 1 - 1
[[[[[[]],[]]],[]],[]]
=> [[[[[.,.],[.,.]],.],[.,.]],[.,.]]
=> [8,6,3,1,2,4,5,7] => ? => ? = 1 - 1
[[[[[[],[]]]],[]],[]]
=> [[[[[.,[.,.]],.],.],[.,.]],[.,.]]
=> [8,6,2,1,3,4,5,7] => ? => ? = 1 - 1
[[[[[[]]],[],[]]],[]]
=> [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
=> [8,5,4,1,2,3,6,7] => ? => ? = 1 - 1
[[[[[[]]],[[]]]],[]]
=> [[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> [8,4,5,1,2,3,6,7] => ? => ? = 1 - 1
[[[[[[]],[]],[]]],[]]
=> [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [8,5,3,1,2,4,6,7] => ? => ? = 1 - 1
[[[[[[]],[],[]]]],[]]
=> [[[[[.,.],[.,[.,.]]],.],.],[.,.]]
=> [8,4,3,1,2,5,6,7] => ? => ? = 1 - 1
[[[[[[]],[[]]]]],[]]
=> [[[[[.,.],[[.,.],.]],.],.],[.,.]]
=> [8,3,4,1,2,5,6,7] => ? => ? = 1 - 1
[[[[]],[[[],[[]]]]]]
=> [[[.,.],[[[.,[[.,.],.]],.],.]],.]
=> [4,5,3,6,7,1,2,8] => ? => ? = 2 - 1
[[[[]],[[[[]],[]]]]]
=> [[[.,.],[[[[.,.],[.,.]],.],.]],.]
=> [5,3,4,6,7,1,2,8] => ? => ? = 2 - 1
[[[[[]]],[],[[],[]]]]
=> [[[[.,.],.],[.,[[.,[.,.]],.]]],.]
=> [6,5,7,4,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[]],[[]]]]
=> [[[[.,.],.],[[.,.],[[.,.],.]]],.]
=> [6,7,4,5,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[],[]],[]]]
=> [[[[.,.],.],[[.,[.,.]],[.,.]]],.]
=> [7,5,4,6,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[],[],[]]]]
=> [[[[.,.],.],[[.,[.,[.,.]]],.]],.]
=> [6,5,4,7,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[[]],[]]]]
=> [[[[.,.],.],[[[.,.],[.,.]],.]],.]
=> [6,4,5,7,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[[[]]]]]]
=> [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [4,5,6,7,1,2,3,8] => ? => ? = 2 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000392
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 00 => 0 = 1 - 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 00 => 0 = 1 - 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 01 => 1 = 2 - 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 000 => 0 = 1 - 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 000 => 0 = 1 - 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 000 => 0 = 1 - 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 000 => 0 = 1 - 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 000 => 0 = 1 - 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 000 => 0 = 1 - 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 000 => 0 = 1 - 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 000 => 0 = 1 - 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 001 => 1 = 2 - 1
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 001 => 1 = 2 - 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 001 => 1 = 2 - 1
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 011 => 2 = 3 - 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0000 => 0 = 1 - 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0000 => 0 = 1 - 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 0000 => 0 = 1 - 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0000 => 0 = 1 - 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0000 => 0 = 1 - 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 0000 => 0 = 1 - 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 0000 => 0 = 1 - 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0000 => 0 = 1 - 1
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 0000 => 0 = 1 - 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0000 => 0 = 1 - 1
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0000 => 0 = 1 - 1
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 0000 => 0 = 1 - 1
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0000 => 0 = 1 - 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0000 => 0 = 1 - 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0000 => 0 = 1 - 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 0000 => 0 = 1 - 1
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 0000 => 0 = 1 - 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 0000 => 0 = 1 - 1
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 0000 => 0 = 1 - 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0000 => 0 = 1 - 1
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0000 => 0 = 1 - 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 0000 => 0 = 1 - 1
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 0000 => 0 = 1 - 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0000 => 0 = 1 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 0000 => 0 = 1 - 1
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0000 => 0 = 1 - 1
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 0000 => 0 = 1 - 1
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0001 => 1 = 2 - 1
[[[],[],[[]]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 0001 => 1 = 2 - 1
[[[],[[]],[]]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 0001 => 1 = 2 - 1
[[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 0001 => 1 = 2 - 1
[[[],[[[]]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 0001 => 1 = 2 - 1
[[[[]],[],[]]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 0001 => 1 = 2 - 1
[[[[]],[[]]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 0001 => 1 = 2 - 1
[[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 0001 => 1 = 2 - 1
[[],[[[[[],[]]],[]]]]
=> [.,[[[[[.,[.,.]],.],[.,.]],.],.]]
=> [6,3,2,4,5,7,8,1] => ? => ? = 1 - 1
[[],[[[[],[],[[]]]]]]
=> [.,[[[[.,[.,[[.,.],.]]],.],.],.]]
=> [4,5,3,2,6,7,8,1] => ? => ? = 1 - 1
[[],[[[[],[[],[]]]]]]
=> [.,[[[[.,[[.,[.,.]],.]],.],.],.]]
=> [4,3,5,2,6,7,8,1] => ? => ? = 1 - 1
[[[]],[[[[],[]]]],[]]
=> [[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [8,4,3,5,6,7,1,2] => ? => ? = 1 - 1
[[[]],[[],[[[]],[]]]]
=> [[.,.],[[.,[[[.,.],[.,.]],.]],.]]
=> [6,4,5,7,3,8,1,2] => ? => ? = 1 - 1
[[[]],[[],[[[],[]]]]]
=> [[.,.],[[.,[[[.,[.,.]],.],.]],.]]
=> [5,4,6,7,3,8,1,2] => ? => ? = 1 - 1
[[[]],[[[]],[[[]]]]]
=> [[.,.],[[[.,.],[[[.,.],.],.]],.]]
=> [5,6,7,3,4,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[]]],[[]]]]
=> [[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [6,7,3,4,5,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[]],[]],[]]]
=> [[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> [7,5,3,4,6,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[],[]]],[]]]
=> [[.,.],[[[[.,[.,.]],.],[.,.]],.]]
=> [7,4,3,5,6,8,1,2] => ? => ? = 1 - 1
[[[]],[[[],[[]],[]]]]
=> [[.,.],[[[.,[[.,.],[.,.]]],.],.]]
=> [6,4,5,3,7,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[]],[],[]]]]
=> [[.,.],[[[[.,.],[.,[.,.]]],.],.]]
=> [6,5,3,4,7,8,1,2] => ? => ? = 1 - 1
[[[[]]],[],[[[]],[]]]
=> [[[.,.],.],[.,[[[.,.],[.,.]],.]]]
=> [7,5,6,8,4,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[]],[[],[]]]
=> [[[.,.],.],[[.,.],[[.,[.,.]],.]]]
=> [7,6,8,4,5,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[]],[[]]]
=> [[[.,.],.],[[.,[.,.]],[[.,.],.]]]
=> [7,8,5,4,6,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[[]]],[]]
=> [[[.,.],.],[[.,[[.,.],.]],[.,.]]]
=> [8,5,6,4,7,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[[]],[]]]
=> [[[.,.],.],[[.,[[.,.],[.,.]]],.]]
=> [7,5,6,4,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[[[]]]]]
=> [[[.,.],.],[[.,[[[.,.],.],.]],.]]
=> [5,6,7,4,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[]],[],[]]]
=> [[[.,.],.],[[[.,.],[.,[.,.]]],.]]
=> [7,6,4,5,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[],[]],[]]]
=> [[[.,.],.],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[[]]],[]]]
=> [[[.,.],.],[[[[.,.],.],[.,.]],.]]
=> [7,4,5,6,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[],[[]]]]]
=> [[[.,.],.],[[[.,[[.,.],.]],.],.]]
=> [5,6,4,7,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[[]],[]]]]
=> [[[.,.],.],[[[[.,.],[.,.]],.],.]]
=> [6,4,5,7,8,1,2,3] => ? => ? = 1 - 1
[[[[[]]],[]],[],[[]]]
=> [[[[.,.],.],[.,.]],[.,[[.,.],.]]]
=> [7,8,6,4,1,2,3,5] => ? => ? = 1 - 1
[[[[[]]],[]],[[]],[]]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [8,6,7,4,1,2,3,5] => ? => ? = 1 - 1
[[[[[],[]]]],[[]],[]]
=> [[[[.,[.,.]],.],.],[[.,.],[.,.]]]
=> [8,6,7,2,1,3,4,5] => ? => ? = 1 - 1
[[[[[]],[]]],[[],[]]]
=> [[[[.,.],[.,.]],.],[[.,[.,.]],.]]
=> [7,6,8,3,1,2,4,5] => ? => ? = 1 - 1
[[[[[],[]]]],[[],[]]]
=> [[[[.,[.,.]],.],.],[[.,[.,.]],.]]
=> [7,6,8,2,1,3,4,5] => ? => ? = 1 - 1
[[[[[]]],[[]]],[],[]]
=> [[[[.,.],.],[[.,.],.]],[.,[.,.]]]
=> [8,7,4,5,1,2,3,6] => ? => ? = 1 - 1
[[[[[[]],[]]]],[],[]]
=> [[[[[.,.],[.,.]],.],.],[.,[.,.]]]
=> [8,7,3,1,2,4,5,6] => ? => ? = 1 - 1
[[[[[]],[]],[]],[[]]]
=> [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [7,8,5,3,1,2,4,6] => ? => ? = 1 - 1
[[[[[]]],[],[[]]],[]]
=> [[[[.,.],.],[.,[[.,.],.]]],[.,.]]
=> [8,5,6,4,1,2,3,7] => ? => ? = 1 - 1
[[[[[]]],[[]],[]],[]]
=> [[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> [8,6,4,5,1,2,3,7] => ? => ? = 1 - 1
[[[[[]]],[[],[]]],[]]
=> [[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,1,2,3,7] => ? => ? = 1 - 1
[[[[[[]]],[]],[]],[]]
=> [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [8,6,4,1,2,3,5,7] => ? => ? = 1 - 1
[[[[[[]],[]]],[]],[]]
=> [[[[[.,.],[.,.]],.],[.,.]],[.,.]]
=> [8,6,3,1,2,4,5,7] => ? => ? = 1 - 1
[[[[[[],[]]]],[]],[]]
=> [[[[[.,[.,.]],.],.],[.,.]],[.,.]]
=> [8,6,2,1,3,4,5,7] => ? => ? = 1 - 1
[[[[[[]]],[],[]]],[]]
=> [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
=> [8,5,4,1,2,3,6,7] => ? => ? = 1 - 1
[[[[[[]]],[[]]]],[]]
=> [[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> [8,4,5,1,2,3,6,7] => ? => ? = 1 - 1
[[[[[[]],[]],[]]],[]]
=> [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [8,5,3,1,2,4,6,7] => ? => ? = 1 - 1
[[[[[[]],[],[]]]],[]]
=> [[[[[.,.],[.,[.,.]]],.],.],[.,.]]
=> [8,4,3,1,2,5,6,7] => ? => ? = 1 - 1
[[[[[[]],[[]]]]],[]]
=> [[[[[.,.],[[.,.],.]],.],.],[.,.]]
=> [8,3,4,1,2,5,6,7] => ? => ? = 1 - 1
[[[[]],[[[],[[]]]]]]
=> [[[.,.],[[[.,[[.,.],.]],.],.]],.]
=> [4,5,3,6,7,1,2,8] => ? => ? = 2 - 1
[[[[]],[[[[]],[]]]]]
=> [[[.,.],[[[[.,.],[.,.]],.],.]],.]
=> [5,3,4,6,7,1,2,8] => ? => ? = 2 - 1
[[[[[]]],[],[[],[]]]]
=> [[[[.,.],.],[.,[[.,[.,.]],.]]],.]
=> [6,5,7,4,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[]],[[]]]]
=> [[[[.,.],.],[[.,.],[[.,.],.]]],.]
=> [6,7,4,5,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[],[]],[]]]
=> [[[[.,.],.],[[.,[.,.]],[.,.]]],.]
=> [7,5,4,6,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[],[],[]]]]
=> [[[[.,.],.],[[.,[.,[.,.]]],.]],.]
=> [6,5,4,7,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[[]],[]]]]
=> [[[[.,.],.],[[[.,.],[.,.]],.]],.]
=> [6,4,5,7,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[[[]]]]]]
=> [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [4,5,6,7,1,2,3,8] => ? => ? = 2 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 00 => 0 = 1 - 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 00 => 0 = 1 - 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 01 => 1 = 2 - 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 000 => 0 = 1 - 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 000 => 0 = 1 - 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 000 => 0 = 1 - 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 000 => 0 = 1 - 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 000 => 0 = 1 - 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 000 => 0 = 1 - 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 000 => 0 = 1 - 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 000 => 0 = 1 - 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 001 => 1 = 2 - 1
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 001 => 1 = 2 - 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 001 => 1 = 2 - 1
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 011 => 2 = 3 - 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0000 => 0 = 1 - 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0000 => 0 = 1 - 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 0000 => 0 = 1 - 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0000 => 0 = 1 - 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0000 => 0 = 1 - 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 0000 => 0 = 1 - 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 0000 => 0 = 1 - 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0000 => 0 = 1 - 1
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 0000 => 0 = 1 - 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0000 => 0 = 1 - 1
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0000 => 0 = 1 - 1
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 0000 => 0 = 1 - 1
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0000 => 0 = 1 - 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0000 => 0 = 1 - 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0000 => 0 = 1 - 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 0000 => 0 = 1 - 1
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 0000 => 0 = 1 - 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 0000 => 0 = 1 - 1
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 0000 => 0 = 1 - 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0000 => 0 = 1 - 1
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0000 => 0 = 1 - 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 0000 => 0 = 1 - 1
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 0000 => 0 = 1 - 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0000 => 0 = 1 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 0000 => 0 = 1 - 1
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0000 => 0 = 1 - 1
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 0000 => 0 = 1 - 1
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0001 => 1 = 2 - 1
[[[],[],[[]]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 0001 => 1 = 2 - 1
[[[],[[]],[]]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 0001 => 1 = 2 - 1
[[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 0001 => 1 = 2 - 1
[[[],[[[]]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 0001 => 1 = 2 - 1
[[[[]],[],[]]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 0001 => 1 = 2 - 1
[[[[]],[[]]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 0001 => 1 = 2 - 1
[[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 0001 => 1 = 2 - 1
[[],[[[[[],[]]],[]]]]
=> [.,[[[[[.,[.,.]],.],[.,.]],.],.]]
=> [6,3,2,4,5,7,8,1] => ? => ? = 1 - 1
[[],[[[[],[],[[]]]]]]
=> [.,[[[[.,[.,[[.,.],.]]],.],.],.]]
=> [4,5,3,2,6,7,8,1] => ? => ? = 1 - 1
[[],[[[[],[[],[]]]]]]
=> [.,[[[[.,[[.,[.,.]],.]],.],.],.]]
=> [4,3,5,2,6,7,8,1] => ? => ? = 1 - 1
[[[]],[[[[],[]]]],[]]
=> [[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [8,4,3,5,6,7,1,2] => ? => ? = 1 - 1
[[[]],[[],[[[]],[]]]]
=> [[.,.],[[.,[[[.,.],[.,.]],.]],.]]
=> [6,4,5,7,3,8,1,2] => ? => ? = 1 - 1
[[[]],[[],[[[],[]]]]]
=> [[.,.],[[.,[[[.,[.,.]],.],.]],.]]
=> [5,4,6,7,3,8,1,2] => ? => ? = 1 - 1
[[[]],[[[]],[[[]]]]]
=> [[.,.],[[[.,.],[[[.,.],.],.]],.]]
=> [5,6,7,3,4,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[]]],[[]]]]
=> [[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [6,7,3,4,5,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[]],[]],[]]]
=> [[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> [7,5,3,4,6,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[],[]]],[]]]
=> [[.,.],[[[[.,[.,.]],.],[.,.]],.]]
=> [7,4,3,5,6,8,1,2] => ? => ? = 1 - 1
[[[]],[[[],[[]],[]]]]
=> [[.,.],[[[.,[[.,.],[.,.]]],.],.]]
=> [6,4,5,3,7,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[]],[],[]]]]
=> [[.,.],[[[[.,.],[.,[.,.]]],.],.]]
=> [6,5,3,4,7,8,1,2] => ? => ? = 1 - 1
[[[[]]],[],[[[]],[]]]
=> [[[.,.],.],[.,[[[.,.],[.,.]],.]]]
=> [7,5,6,8,4,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[]],[[],[]]]
=> [[[.,.],.],[[.,.],[[.,[.,.]],.]]]
=> [7,6,8,4,5,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[]],[[]]]
=> [[[.,.],.],[[.,[.,.]],[[.,.],.]]]
=> [7,8,5,4,6,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[[]]],[]]
=> [[[.,.],.],[[.,[[.,.],.]],[.,.]]]
=> [8,5,6,4,7,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[[]],[]]]
=> [[[.,.],.],[[.,[[.,.],[.,.]]],.]]
=> [7,5,6,4,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[[[]]]]]
=> [[[.,.],.],[[.,[[[.,.],.],.]],.]]
=> [5,6,7,4,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[]],[],[]]]
=> [[[.,.],.],[[[.,.],[.,[.,.]]],.]]
=> [7,6,4,5,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[],[]],[]]]
=> [[[.,.],.],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[[]]],[]]]
=> [[[.,.],.],[[[[.,.],.],[.,.]],.]]
=> [7,4,5,6,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[],[[]]]]]
=> [[[.,.],.],[[[.,[[.,.],.]],.],.]]
=> [5,6,4,7,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[[]],[]]]]
=> [[[.,.],.],[[[[.,.],[.,.]],.],.]]
=> [6,4,5,7,8,1,2,3] => ? => ? = 1 - 1
[[[[[]]],[]],[],[[]]]
=> [[[[.,.],.],[.,.]],[.,[[.,.],.]]]
=> [7,8,6,4,1,2,3,5] => ? => ? = 1 - 1
[[[[[]]],[]],[[]],[]]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [8,6,7,4,1,2,3,5] => ? => ? = 1 - 1
[[[[[],[]]]],[[]],[]]
=> [[[[.,[.,.]],.],.],[[.,.],[.,.]]]
=> [8,6,7,2,1,3,4,5] => ? => ? = 1 - 1
[[[[[]],[]]],[[],[]]]
=> [[[[.,.],[.,.]],.],[[.,[.,.]],.]]
=> [7,6,8,3,1,2,4,5] => ? => ? = 1 - 1
[[[[[],[]]]],[[],[]]]
=> [[[[.,[.,.]],.],.],[[.,[.,.]],.]]
=> [7,6,8,2,1,3,4,5] => ? => ? = 1 - 1
[[[[[]]],[[]]],[],[]]
=> [[[[.,.],.],[[.,.],.]],[.,[.,.]]]
=> [8,7,4,5,1,2,3,6] => ? => ? = 1 - 1
[[[[[[]],[]]]],[],[]]
=> [[[[[.,.],[.,.]],.],.],[.,[.,.]]]
=> [8,7,3,1,2,4,5,6] => ? => ? = 1 - 1
[[[[[]],[]],[]],[[]]]
=> [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [7,8,5,3,1,2,4,6] => ? => ? = 1 - 1
[[[[[]]],[],[[]]],[]]
=> [[[[.,.],.],[.,[[.,.],.]]],[.,.]]
=> [8,5,6,4,1,2,3,7] => ? => ? = 1 - 1
[[[[[]]],[[]],[]],[]]
=> [[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> [8,6,4,5,1,2,3,7] => ? => ? = 1 - 1
[[[[[]]],[[],[]]],[]]
=> [[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,1,2,3,7] => ? => ? = 1 - 1
[[[[[[]]],[]],[]],[]]
=> [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [8,6,4,1,2,3,5,7] => ? => ? = 1 - 1
[[[[[[]],[]]],[]],[]]
=> [[[[[.,.],[.,.]],.],[.,.]],[.,.]]
=> [8,6,3,1,2,4,5,7] => ? => ? = 1 - 1
[[[[[[],[]]]],[]],[]]
=> [[[[[.,[.,.]],.],.],[.,.]],[.,.]]
=> [8,6,2,1,3,4,5,7] => ? => ? = 1 - 1
[[[[[[]]],[],[]]],[]]
=> [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
=> [8,5,4,1,2,3,6,7] => ? => ? = 1 - 1
[[[[[[]]],[[]]]],[]]
=> [[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> [8,4,5,1,2,3,6,7] => ? => ? = 1 - 1
[[[[[[]],[]],[]]],[]]
=> [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [8,5,3,1,2,4,6,7] => ? => ? = 1 - 1
[[[[[[]],[],[]]]],[]]
=> [[[[[.,.],[.,[.,.]]],.],.],[.,.]]
=> [8,4,3,1,2,5,6,7] => ? => ? = 1 - 1
[[[[[[]],[[]]]]],[]]
=> [[[[[.,.],[[.,.],.]],.],.],[.,.]]
=> [8,3,4,1,2,5,6,7] => ? => ? = 1 - 1
[[[[]],[[[],[[]]]]]]
=> [[[.,.],[[[.,[[.,.],.]],.],.]],.]
=> [4,5,3,6,7,1,2,8] => ? => ? = 2 - 1
[[[[]],[[[[]],[]]]]]
=> [[[.,.],[[[[.,.],[.,.]],.],.]],.]
=> [5,3,4,6,7,1,2,8] => ? => ? = 2 - 1
[[[[[]]],[],[[],[]]]]
=> [[[[.,.],.],[.,[[.,[.,.]],.]]],.]
=> [6,5,7,4,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[]],[[]]]]
=> [[[[.,.],.],[[.,.],[[.,.],.]]],.]
=> [6,7,4,5,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[],[]],[]]]
=> [[[[.,.],.],[[.,[.,.]],[.,.]]],.]
=> [7,5,4,6,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[],[],[]]]]
=> [[[[.,.],.],[[.,[.,[.,.]]],.]],.]
=> [6,5,4,7,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[[]],[]]]]
=> [[[[.,.],.],[[[.,.],[.,.]],.]],.]
=> [6,4,5,7,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[[[]]]]]]
=> [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [4,5,6,7,1,2,3,8] => ? => ? = 2 - 1
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: St001419
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001419: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001419: Binary words ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 00 => 0 = 1 - 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 00 => 0 = 1 - 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 01 => 1 = 2 - 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 000 => 0 = 1 - 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 000 => 0 = 1 - 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 000 => 0 = 1 - 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 000 => 0 = 1 - 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 000 => 0 = 1 - 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 000 => 0 = 1 - 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 000 => 0 = 1 - 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 000 => 0 = 1 - 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 001 => 1 = 2 - 1
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 001 => 1 = 2 - 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 001 => 1 = 2 - 1
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 011 => 2 = 3 - 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0000 => 0 = 1 - 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0000 => 0 = 1 - 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 0000 => 0 = 1 - 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0000 => 0 = 1 - 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0000 => 0 = 1 - 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 0000 => 0 = 1 - 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 0000 => 0 = 1 - 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0000 => 0 = 1 - 1
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 0000 => 0 = 1 - 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0000 => 0 = 1 - 1
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0000 => 0 = 1 - 1
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 0000 => 0 = 1 - 1
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 0000 => 0 = 1 - 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0000 => 0 = 1 - 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 0000 => 0 = 1 - 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 0000 => 0 = 1 - 1
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 0000 => 0 = 1 - 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 0000 => 0 = 1 - 1
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 0000 => 0 = 1 - 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 0000 => 0 = 1 - 1
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 0000 => 0 = 1 - 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 0000 => 0 = 1 - 1
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 0000 => 0 = 1 - 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 0000 => 0 = 1 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 0000 => 0 = 1 - 1
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 0000 => 0 = 1 - 1
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 0000 => 0 = 1 - 1
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 0001 => 1 = 2 - 1
[[[],[],[[]]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 0001 => 1 = 2 - 1
[[[],[[]],[]]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 0001 => 1 = 2 - 1
[[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 0001 => 1 = 2 - 1
[[[],[[[]]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 0001 => 1 = 2 - 1
[[[[]],[],[]]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 0001 => 1 = 2 - 1
[[[[]],[[]]]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 0001 => 1 = 2 - 1
[[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 0001 => 1 = 2 - 1
[[],[[[[[],[]]],[]]]]
=> [.,[[[[[.,[.,.]],.],[.,.]],.],.]]
=> [6,3,2,4,5,7,8,1] => ? => ? = 1 - 1
[[],[[[[],[],[[]]]]]]
=> [.,[[[[.,[.,[[.,.],.]]],.],.],.]]
=> [4,5,3,2,6,7,8,1] => ? => ? = 1 - 1
[[],[[[[],[[],[]]]]]]
=> [.,[[[[.,[[.,[.,.]],.]],.],.],.]]
=> [4,3,5,2,6,7,8,1] => ? => ? = 1 - 1
[[[]],[[[[],[]]]],[]]
=> [[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [8,4,3,5,6,7,1,2] => ? => ? = 1 - 1
[[[]],[[],[[[]],[]]]]
=> [[.,.],[[.,[[[.,.],[.,.]],.]],.]]
=> [6,4,5,7,3,8,1,2] => ? => ? = 1 - 1
[[[]],[[],[[[],[]]]]]
=> [[.,.],[[.,[[[.,[.,.]],.],.]],.]]
=> [5,4,6,7,3,8,1,2] => ? => ? = 1 - 1
[[[]],[[[]],[[[]]]]]
=> [[.,.],[[[.,.],[[[.,.],.],.]],.]]
=> [5,6,7,3,4,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[]]],[[]]]]
=> [[.,.],[[[[.,.],.],[[.,.],.]],.]]
=> [6,7,3,4,5,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[]],[]],[]]]
=> [[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> [7,5,3,4,6,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[],[]]],[]]]
=> [[.,.],[[[[.,[.,.]],.],[.,.]],.]]
=> [7,4,3,5,6,8,1,2] => ? => ? = 1 - 1
[[[]],[[[],[[]],[]]]]
=> [[.,.],[[[.,[[.,.],[.,.]]],.],.]]
=> [6,4,5,3,7,8,1,2] => ? => ? = 1 - 1
[[[]],[[[[]],[],[]]]]
=> [[.,.],[[[[.,.],[.,[.,.]]],.],.]]
=> [6,5,3,4,7,8,1,2] => ? => ? = 1 - 1
[[[[]]],[],[[[]],[]]]
=> [[[.,.],.],[.,[[[.,.],[.,.]],.]]]
=> [7,5,6,8,4,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[]],[[],[]]]
=> [[[.,.],.],[[.,.],[[.,[.,.]],.]]]
=> [7,6,8,4,5,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[]],[[]]]
=> [[[.,.],.],[[.,[.,.]],[[.,.],.]]]
=> [7,8,5,4,6,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[[]]],[]]
=> [[[.,.],.],[[.,[[.,.],.]],[.,.]]]
=> [8,5,6,4,7,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[[]],[]]]
=> [[[.,.],.],[[.,[[.,.],[.,.]]],.]]
=> [7,5,6,4,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[],[[[]]]]]
=> [[[.,.],.],[[.,[[[.,.],.],.]],.]]
=> [5,6,7,4,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[]],[],[]]]
=> [[[.,.],.],[[[.,.],[.,[.,.]]],.]]
=> [7,6,4,5,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[],[]],[]]]
=> [[[.,.],.],[[[.,[.,.]],[.,.]],.]]
=> [7,5,4,6,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[[]]],[]]]
=> [[[.,.],.],[[[[.,.],.],[.,.]],.]]
=> [7,4,5,6,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[],[[]]]]]
=> [[[.,.],.],[[[.,[[.,.],.]],.],.]]
=> [5,6,4,7,8,1,2,3] => ? => ? = 1 - 1
[[[[]]],[[[[]],[]]]]
=> [[[.,.],.],[[[[.,.],[.,.]],.],.]]
=> [6,4,5,7,8,1,2,3] => ? => ? = 1 - 1
[[[[[]]],[]],[],[[]]]
=> [[[[.,.],.],[.,.]],[.,[[.,.],.]]]
=> [7,8,6,4,1,2,3,5] => ? => ? = 1 - 1
[[[[[]]],[]],[[]],[]]
=> [[[[.,.],.],[.,.]],[[.,.],[.,.]]]
=> [8,6,7,4,1,2,3,5] => ? => ? = 1 - 1
[[[[[],[]]]],[[]],[]]
=> [[[[.,[.,.]],.],.],[[.,.],[.,.]]]
=> [8,6,7,2,1,3,4,5] => ? => ? = 1 - 1
[[[[[]],[]]],[[],[]]]
=> [[[[.,.],[.,.]],.],[[.,[.,.]],.]]
=> [7,6,8,3,1,2,4,5] => ? => ? = 1 - 1
[[[[[],[]]]],[[],[]]]
=> [[[[.,[.,.]],.],.],[[.,[.,.]],.]]
=> [7,6,8,2,1,3,4,5] => ? => ? = 1 - 1
[[[[[]]],[[]]],[],[]]
=> [[[[.,.],.],[[.,.],.]],[.,[.,.]]]
=> [8,7,4,5,1,2,3,6] => ? => ? = 1 - 1
[[[[[[]],[]]]],[],[]]
=> [[[[[.,.],[.,.]],.],.],[.,[.,.]]]
=> [8,7,3,1,2,4,5,6] => ? => ? = 1 - 1
[[[[[]],[]],[]],[[]]]
=> [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> [7,8,5,3,1,2,4,6] => ? => ? = 1 - 1
[[[[[]]],[],[[]]],[]]
=> [[[[.,.],.],[.,[[.,.],.]]],[.,.]]
=> [8,5,6,4,1,2,3,7] => ? => ? = 1 - 1
[[[[[]]],[[]],[]],[]]
=> [[[[.,.],.],[[.,.],[.,.]]],[.,.]]
=> [8,6,4,5,1,2,3,7] => ? => ? = 1 - 1
[[[[[]]],[[],[]]],[]]
=> [[[[.,.],.],[[.,[.,.]],.]],[.,.]]
=> [8,5,4,6,1,2,3,7] => ? => ? = 1 - 1
[[[[[[]]],[]],[]],[]]
=> [[[[[.,.],.],[.,.]],[.,.]],[.,.]]
=> [8,6,4,1,2,3,5,7] => ? => ? = 1 - 1
[[[[[[]],[]]],[]],[]]
=> [[[[[.,.],[.,.]],.],[.,.]],[.,.]]
=> [8,6,3,1,2,4,5,7] => ? => ? = 1 - 1
[[[[[[],[]]]],[]],[]]
=> [[[[[.,[.,.]],.],.],[.,.]],[.,.]]
=> [8,6,2,1,3,4,5,7] => ? => ? = 1 - 1
[[[[[[]]],[],[]]],[]]
=> [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
=> [8,5,4,1,2,3,6,7] => ? => ? = 1 - 1
[[[[[[]]],[[]]]],[]]
=> [[[[[.,.],.],[[.,.],.]],.],[.,.]]
=> [8,4,5,1,2,3,6,7] => ? => ? = 1 - 1
[[[[[[]],[]],[]]],[]]
=> [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
=> [8,5,3,1,2,4,6,7] => ? => ? = 1 - 1
[[[[[[]],[],[]]]],[]]
=> [[[[[.,.],[.,[.,.]]],.],.],[.,.]]
=> [8,4,3,1,2,5,6,7] => ? => ? = 1 - 1
[[[[[[]],[[]]]]],[]]
=> [[[[[.,.],[[.,.],.]],.],.],[.,.]]
=> [8,3,4,1,2,5,6,7] => ? => ? = 1 - 1
[[[[]],[[[],[[]]]]]]
=> [[[.,.],[[[.,[[.,.],.]],.],.]],.]
=> [4,5,3,6,7,1,2,8] => ? => ? = 2 - 1
[[[[]],[[[[]],[]]]]]
=> [[[.,.],[[[[.,.],[.,.]],.],.]],.]
=> [5,3,4,6,7,1,2,8] => ? => ? = 2 - 1
[[[[[]]],[],[[],[]]]]
=> [[[[.,.],.],[.,[[.,[.,.]],.]]],.]
=> [6,5,7,4,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[]],[[]]]]
=> [[[[.,.],.],[[.,.],[[.,.],.]]],.]
=> [6,7,4,5,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[],[]],[]]]
=> [[[[.,.],.],[[.,[.,.]],[.,.]]],.]
=> [7,5,4,6,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[],[],[]]]]
=> [[[[.,.],.],[[.,[.,[.,.]]],.]],.]
=> [6,5,4,7,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[[]],[]]]]
=> [[[[.,.],.],[[[.,.],[.,.]],.]],.]
=> [6,4,5,7,1,2,3,8] => ? => ? = 2 - 1
[[[[[]]],[[[[]]]]]]
=> [[[[.,.],.],[[[[.,.],.],.],.]],.]
=> [4,5,6,7,1,2,3,8] => ? => ? = 2 - 1
Description
The length of the longest palindromic factor beginning with a one of a binary word.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00246: Ordered trees —rotate⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[[],[],[]]
=> [[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[[],[[]]]
=> [[[]],[]]
=> [1,1,0,0,1,0]
=> [2,1] => 1
[[[],[]]]
=> [[],[[]]]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[[],[],[],[]]
=> [[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[[],[],[[]]]
=> [[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[[],[[]],[]]
=> [[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1
[[],[[],[]]]
=> [[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 1
[[],[[[]]]]
=> [[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1
[[[]],[],[]]
=> [[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 1
[[[]],[[]]]
=> [[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 1
[[[],[]],[]]
=> [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 1
[[[],[],[]]]
=> [[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[[[],[[]]]]
=> [[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[[[]],[]]]
=> [[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[[[[],[]]]]
=> [[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[[],[],[],[],[]]
=> [[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[[],[],[],[[]]]
=> [[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[[],[],[[]],[]]
=> [[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[[],[],[[],[]]]
=> [[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 1
[[],[],[[[]]]]
=> [[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[[],[[]],[],[]]
=> [[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 1
[[],[[]],[[]]]
=> [[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 1
[[],[[],[]],[]]
=> [[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 1
[[],[[[]]],[]]
=> [[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 1
[[],[[],[],[]]]
=> [[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 1
[[],[[],[[]]]]
=> [[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 1
[[],[[[]],[]]]
=> [[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 1
[[],[[[],[]]]]
=> [[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 1
[[],[[[[]]]]]
=> [[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 1
[[[]],[],[],[]]
=> [[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 1
[[[]],[],[[]]]
=> [[[],[[]],[]]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 1
[[[]],[[]],[]]
=> [[[[]],[],[]]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => 1
[[[]],[[],[]]]
=> [[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => 1
[[[]],[[[]]]]
=> [[[[[]]],[]]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 1
[[[],[]],[],[]]
=> [[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 1
[[[[]]],[],[]]
=> [[[[],[],[]]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 1
[[[],[]],[[]]]
=> [[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 1
[[[[]]],[[]]]
=> [[[[[]],[]]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 1
[[[],[],[]],[]]
=> [[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 1
[[[],[[]]],[]]
=> [[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 1
[[[[]],[]],[]]
=> [[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 1
[[[[],[]]],[]]
=> [[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 1
[[[],[],[],[]]]
=> [[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2
[[[],[],[[]]]]
=> [[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[[[],[[]],[]]]
=> [[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[[],[[],[]]]]
=> [[[],[]],[[]]]
=> [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] => 2
[[[[]],[],[]]]
=> [[[],[],[[]]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 2
[[[[]],[[]]]]
=> [[[[]],[[]]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 2
[[[[],[]],[]]]
=> [[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 2
[[],[[],[[[[[]]]]]]]
=> [[[],[[[[[]]]]]],[]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,5,1] => ? = 1
[[],[[[[],[[]],[]]]]]
=> [[[[[],[[]],[]]]],[]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [4,2,1,1] => ? = 1
[[],[[[[],[[],[]]]]]]
=> [[[[[],[[],[]]]]],[]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [4,2,1,1] => ? = 1
[[[]],[],[[[[],[]]]]]
=> [[[],[[[[],[]]]],[]]]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [2,4,1,1] => ? = 1
[[[]],[],[[[[[]]]]]]
=> [[[],[[[[[]]]]],[]]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [2,5,1] => ? = 1
[[[]],[[[],[[]]],[]]]
=> [[[[[],[[]]],[]],[]]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> [4,2,1,1] => ? = 1
[[[]],[[[],[],[],[]]]]
=> [[[[[],[],[],[]]],[]]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [4,1,1,1,1] => ? = 1
[[[]],[[[],[[]],[]]]]
=> [[[[[],[[]],[]]],[]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,1,0,0]
=> [4,2,1,1] => ? = 1
[[[]],[[[],[[],[]]]]]
=> [[[[[],[[],[]]]],[]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [4,2,1,1] => ? = 1
[[[[]]],[],[[],[],[]]]
=> [[[[],[[],[],[]],[]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> [3,2,1,1,1] => ? = 1
[[[[]]],[[]],[[],[]]]
=> [[[[[]],[[],[]],[]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> [4,2,1,1] => ? = 1
[[[[]]],[[],[],[]],[]]
=> [[[[[],[],[]],[],[]]]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [4,1,1,1,1] => ? = 1
[[[[]]],[[],[[]]],[]]
=> [[[[[],[[]]],[],[]]]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,1,0,0,0]
=> [4,2,1,1] => ? = 1
[[[[]]],[[],[],[],[]]]
=> [[[[[],[],[],[]],[]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [4,1,1,1,1] => ? = 1
[[[[]]],[[],[[]],[]]]
=> [[[[[],[[]],[]],[]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
=> [4,2,1,1] => ? = 1
[[[[]]],[[],[[],[]]]]
=> [[[[[],[[],[]]],[]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [4,2,1,1] => ? = 1
[[[[[]]]],[],[],[],[]]
=> [[[[[],[],[],[],[]]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,1,1,1,1] => ? = 1
[[[[[]]]],[],[[]],[]]
=> [[[[[],[[]],[],[]]]]]
=> [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
=> [4,2,1,1] => ? = 1
[[[[[]]]],[],[[],[]]]
=> [[[[[],[[],[]],[]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,1] => ? = 1
[[[[]],[]],[[[]],[]]]
=> [[[],[[[[]],[]],[]]]]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [2,4,1,1] => ? = 1
[[[[]],[]],[[[],[]]]]
=> [[[],[[[[],[]]],[]]]]
=> [1,1,0,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [2,4,1,1] => ? = 1
[[[[]],[]],[[[[]]]]]
=> [[[],[[[[[]]]],[]]]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,5,1] => ? = 1
[[[[[]]],[]],[],[],[]]
=> [[[[],[[],[],[],[]]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1] => ? = 1
[[[[[]],[]]],[[]],[]]
=> [[[],[[[[]],[],[]]]]]
=> [1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [2,4,1,1] => ? = 1
[[[[[]],[]]],[[],[]]]
=> [[[],[[[[],[]],[]]]]]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [2,4,1,1] => ? = 1
[[[[[]],[]]],[[[]]]]
=> [[[],[[[[[]]],[]]]]]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,5,1] => ? = 1
[[[[[]]],[],[]],[],[]]
=> [[[[],[],[[],[],[]]]]]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,1,2,1,1] => ? = 1
[[[[[]]],[[]]],[],[]]
=> [[[[[]],[[],[],[]]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [4,2,1,1] => ? = 1
[[[[[[]]]],[]],[],[]]
=> [[[[[],[[],[],[]]]]]]
=> [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [4,2,1,1] => ? = 1
[[[[[[]],[]]]],[],[]]
=> [[[],[[[[],[],[]]]]]]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,4,1,1] => ? = 1
[[[[[[]],[]]]],[[]]]
=> [[[],[[[[[]],[]]]]]]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,5,1] => ? = 1
[[[[[[[]],[]]]]],[]]
=> [[[],[[[[[],[]]]]]]]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,5,1] => ? = 1
[[[],[[[[[[]]]]]]]]
=> [[[[[[[]]]]]],[[]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => ? = 2
[[[[]],[[[],[[]]]]]]
=> [[[[[],[[]]]],[[]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> [4,2,2] => ? = 2
[[[[]],[[[[]],[]]]]]
=> [[[[[[]],[]]],[[]]]]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> [5,1,2] => ? = 2
[[[[]],[[[[],[]]]]]]
=> [[[[[[],[]]]],[[]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> [5,1,2] => ? = 2
[[[[]],[[[[[]]]]]]]
=> [[[[[[[]]]]],[[]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [6,2] => ? = 2
[[[[[]]],[[]],[[]]]]
=> [[[[[]],[[]],[[]]]]]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [4,2,2] => ? = 2
[[[[[]]],[[],[]],[]]]
=> [[[[[],[]],[],[[]]]]]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [4,1,1,2] => ? = 2
[[[[[]]],[[[]]],[]]]
=> [[[[[[]]],[],[[]]]]]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [5,1,2] => ? = 2
[[[[[]]],[[],[],[]]]]
=> [[[[[],[],[]],[[]]]]]
=> [1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [4,1,1,2] => ? = 2
[[[[[]]],[[],[[]]]]]
=> [[[[[],[[]]],[[]]]]]
=> [1,1,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> [4,2,2] => ? = 2
[[[[[]]],[[[]],[]]]]
=> [[[[[[]],[]],[[]]]]]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [5,1,2] => ? = 2
[[[[[]]],[[[],[]]]]]
=> [[[[[[],[]]],[[]]]]]
=> [1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
=> [5,1,2] => ? = 2
[[[[[]]],[[[[]]]]]]
=> [[[[[[[]]]],[[]]]]]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [6,2] => ? = 2
[[[[[[]]]],[],[],[]]]
=> [[[[[],[],[],[[]]]]]]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [4,1,1,2] => ? = 2
[[[[[[]]]],[],[[]]]]
=> [[[[[],[[]],[[]]]]]]
=> [1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [4,2,2] => ? = 2
[[[[[[]]]],[[]],[]]]
=> [[[[[[]],[],[[]]]]]]
=> [1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0]
=> [5,1,2] => ? = 2
[[[[[[]]]],[[],[]]]]
=> [[[[[[],[]],[[]]]]]]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [5,1,2] => ? = 2
[[[[[[]]]],[[[]]]]]
=> [[[[[[[]]],[[]]]]]]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [6,2] => ? = 2
Description
The last part of an integer composition.
The following 57 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000974The length of the trunk of an ordered tree. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000674The number of hills of a Dyck path. St000729The minimal arc length of a set partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000234The number of global ascents of a permutation. St000916The packing number of a graph. St000918The 2-limited packing number of a graph. St000546The number of global descents of a permutation. St000989The number of final rises of a permutation. St000717The number of ordinal summands of a poset. St000654The first descent of a permutation. St000990The first ascent of a permutation. St000907The number of maximal antichains of minimal length in a poset. St000056The decomposition (or block) number of a permutation. St000221The number of strong fixed points of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St000160The multiplicity of the smallest part of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000312The number of leaves in a graph. St000314The number of left-to-right-maxima of a permutation. St000475The number of parts equal to 1 in a partition. St000991The number of right-to-left minima of a permutation. St001176The size of a partition minus its first part. St001201The grade of the simple module S0 in the special CNakayama algebra corresponding to the Dyck path. St001933The largest multiplicity of a part in an integer partition. St000010The length of the partition. St000258The burning number of a graph. St000262The vertex connectivity of a graph. St000273The domination number of a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000315The number of isolated vertices of a graph. St000461The rix statistic of a permutation. St000544The cop number of a graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001322The size of a minimal independent dominating set in a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001829The common independence number of a graph. St001342The number of vertices in the center of a graph. St001368The number of vertices of maximal degree in a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000264The girth of a graph, which is not a tree. St000260The radius of a connected graph. St001330The hat guessing number of a graph. St000181The number of connected components of the Hasse diagram for the poset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001826The maximal number of leaves on a vertex of a graph. St000553The number of blocks of a graph. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001672The restrained domination number of a graph. St001479The number of bridges of a graph.
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