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Your data matches 69 different statistics following compositions of up to 3 maps.
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Matching statistic: St001058
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St001058: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> 1
[[],[]]
=> 2
[[[]]]
=> 1
[[],[],[]]
=> 3
[[],[[]]]
=> 2
[[[]],[]]
=> 2
[[[],[]]]
=> 2
[[[[]]]]
=> 1
[[],[],[],[]]
=> 4
[[],[],[[]]]
=> 3
[[],[[]],[]]
=> 3
[[],[[],[]]]
=> 2
[[],[[[]]]]
=> 2
[[[]],[],[]]
=> 3
[[[]],[[]]]
=> 2
[[[],[]],[]]
=> 2
[[[[]]],[]]
=> 2
[[[],[],[]]]
=> 3
[[[],[[]]]]
=> 2
[[[[]],[]]]
=> 2
[[[[],[]]]]
=> 2
[[[[[]]]]]
=> 1
[[],[],[],[],[]]
=> 5
[[],[],[],[[]]]
=> 4
[[],[],[[]],[]]
=> 4
[[],[],[[],[]]]
=> 3
[[],[],[[[]]]]
=> 3
[[],[[]],[],[]]
=> 4
[[],[[]],[[]]]
=> 3
[[],[[],[]],[]]
=> 3
[[],[[[]]],[]]
=> 3
[[],[[],[],[]]]
=> 3
[[],[[],[[]]]]
=> 2
[[],[[[]],[]]]
=> 2
[[],[[[],[]]]]
=> 2
[[],[[[[]]]]]
=> 2
[[[]],[],[],[]]
=> 4
[[[]],[],[[]]]
=> 3
[[[]],[[]],[]]
=> 3
[[[]],[[],[]]]
=> 3
[[[]],[[[]]]]
=> 2
[[[],[]],[],[]]
=> 3
[[[[]]],[],[]]
=> 3
[[[],[]],[[]]]
=> 3
[[[[]]],[[]]]
=> 2
[[[],[],[]],[]]
=> 3
[[[],[[]]],[]]
=> 2
[[[[]],[]],[]]
=> 2
[[[[],[]]],[]]
=> 2
[[[[[]]]],[]]
=> 2
Description
The breadth of the ordered tree.
This is the maximal number of nodes having the same depth.
Matching statistic: St000013
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
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: St000025
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000062
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1] => 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,2] => 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 4
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000166
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000166: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000166: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [[]]
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [[],[]]
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [[[]]]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[[]],[]]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 4
Description
The depth minus 1 of an ordered tree.
The ordered trees of size $n$ are bijection with the Dyck paths of size $n-1$, and this statistic then corresponds to [[St000013]].
Matching statistic: St000308
(load all 2 compositions to match this statistic)
(load all 2 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
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [.,.]
=> [1] => [1] => 1
[[],[]]
=> [[.,.],.]
=> [1,2] => [1,2] => 2
[[[]]]
=> [.,[.,.]]
=> [2,1] => [2,1] => 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 3
[[],[[]]]
=> [[.,.],[.,.]]
=> [1,3,2] => [3,1,2] => 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 2
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => 2
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 4
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [4,1,2,3] => 3
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [3,1,2,4] => 3
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,4,1,2] => 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [4,3,1,2] => 2
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 3
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,4,1,3] => 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => 2
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 2
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,4,1] => 3
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,3,1] => 2
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => 2
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => 2
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [5,1,2,3,4] => 4
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [4,1,2,3,5] => 4
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,5,1,2,3] => 3
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [5,4,1,2,3] => 3
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => [3,1,2,4,5] => 4
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [3,5,1,2,4] => 3
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [3,4,1,2,5] => 3
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => [4,3,1,2,5] => 3
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [3,4,5,1,2] => 3
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [5,3,4,1,2] => 2
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [4,3,5,1,2] => 2
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [4,5,3,1,2] => 2
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [5,4,3,1,2] => 2
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [2,1,3,4,5] => 4
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,5,1,3,4] => 3
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => [2,4,1,3,5] => 3
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,4,5,1,3] => 3
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [5,2,4,1,3] => 2
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [2,3,1,4,5] => 3
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [3,2,1,4,5] => 3
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,5,1,4] => 3
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,5,1,4] => 2
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [2,3,4,1,5] => 3
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [4,2,3,1,5] => 2
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [3,2,4,1,5] => 2
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1,5] => 2
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => 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: St000381
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1] => 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1] => 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2] => 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [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]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [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]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
Description
The largest part of an integer composition.
Matching statistic: St000392
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> 10 => 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1010 => 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1100 => 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 101010 => 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 111000 => 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 4
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000442
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1,1,0,0]
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 4
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000451
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1] => 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 4
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 3
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 3
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 4
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 3
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 3
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 3
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 3
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4
Description
The length of the longest pattern of the form k 1 2...(k-1).
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000628The balance of a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000982The length of the longest constant subword. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001530The depth of a Dyck path. St001652The length of a longest interval of consecutive numbers. St001809The index of the step at the first peak of maximal height in a Dyck path. St000094The depth of an ordered tree. St000141The maximum drop size of a permutation. St000209Maximum difference of elements in cycles. St000306The bounce count of a Dyck path. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000662The staircase size of the code of a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001090The number of pop-stack-sorts needed to sort a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000444The length of the maximal rise of a Dyck path. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000956The maximal displacement of a permutation. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000028The number of stack-sorts needed to sort a permutation. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001589The nesting number of a perfect matching. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000143The largest repeated part of a partition. 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. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St001118The acyclic chromatic index of a graph. St001545The second Elser number of a connected graph. St001875The number of simple modules with projective dimension at most 1. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000983The length of the longest alternating subword. St001730The number of times the path corresponding to a binary word crosses the base line. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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