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Your data matches 54 different statistics following compositions of up to 3 maps.
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Matching statistic: St000439
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000439: 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
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 2
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 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]
=> 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 5
[[],[],[],[],[]]
=> [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]
=> 2
[[],[],[],[[]]]
=> [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]
=> 3
[[],[],[[]],[]]
=> [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]
=> 3
[[],[],[[],[]]]
=> [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]
=> 3
[[],[],[[[]]]]
=> [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]
=> 4
[[],[[]],[],[]]
=> [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]
=> 3
[[],[[]],[[]]]
=> [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]
=> 3
[[],[[],[]],[]]
=> [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]
=> 3
[[],[[[]]],[]]
=> [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]
=> 4
[[],[[],[],[]]]
=> [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]
=> 3
[[],[[],[[]]]]
=> [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]
=> 4
[[],[[[]],[]]]
=> [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]
=> 3
[[],[[[],[]]]]
=> [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]
=> 4
[[],[[[[]]]]]
=> [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]
=> 5
[[[]],[],[],[]]
=> [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]
=> 3
[[[]],[],[[]]]
=> [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]
=> 3
[[[]],[[]],[]]
=> [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]
=> 3
[[[]],[[],[]]]
=> [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]
=> 3
[[[]],[[[]]]]
=> [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]
=> 4
[[[],[]],[],[]]
=> [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]
=> 3
[[[[]]],[],[]]
=> [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]
=> 4
[[[],[]],[[]]]
=> [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]
=> 3
[[[[]]],[[]]]
=> [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]
=> 4
[[[],[],[]],[]]
=> [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]
=> 3
[[[],[[]]],[]]
=> [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]
=> 4
[[[[]],[]],[]]
=> [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]
=> 3
[[[[],[]]],[]]
=> [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]
=> 4
[[[[[]]]],[]]
=> [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]
=> 5
Description
The position of the first down step of a Dyck path.
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 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1] => 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2] => 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2 = 3 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2 = 3 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 4 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2 = 3 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2 = 3 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2 = 3 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 4 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 4 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4 = 5 - 1
[[],[],[],[],[]]
=> [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 = 2 - 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 = 3 - 1
[[],[],[[]],[]]
=> [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 = 3 - 1
[[],[],[[],[]]]
=> [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 = 3 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 4 - 1
[[],[[]],[],[]]
=> [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 = 3 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2 = 3 - 1
[[],[[],[]],[]]
=> [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 = 3 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3 = 4 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2 = 3 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 4 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 4 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4 = 5 - 1
[[[]],[],[],[]]
=> [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 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2 = 3 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2 = 3 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3 = 4 - 1
[[[],[]],[],[]]
=> [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 = 3 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3 = 4 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2 = 3 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3 = 4 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3 = 4 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4 = 5 - 1
Description
The largest part of an integer composition.
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: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[[],[],[],[[]],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[],[],[[],[]],[[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[],[],[]],[],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[],[],[[[]],[]],[],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[],[],[[],[],[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[],[[]],[]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[[]],[],[[]],[[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[[]],[],[[],[],[]]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[[]],[],[[[]],[]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[[]],[[]],[],[],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[],[[]],[[],[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[[]],[[],[],[],[]]]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[[]],[[],[[]],[]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[[[]]],[],[[]],[]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[[],[]],[[]],[],[]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[],[[[]]],[[],[]],[]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[[],[],[]],[],[],[]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[],[[[]],[]],[],[],[]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[],[[],[],[],[]],[],[]]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[],[[],[[]],[]],[],[]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[[]],[],[],[[]],[[]]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[]],[],[],[[],[],[]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[]],[],[],[[[]],[]]]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[]],[],[[]],[],[],[]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[[]],[],[[]],[[]],[]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[[]],[],[[],[]],[[]]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[]],[],[[],[],[]],[]]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[[]],[],[[[]],[]],[]]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[[]],[],[[],[],[],[]]]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[]],[],[[],[[]],[]]]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[]],[[]],[[]],[],[]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[[]],[[],[]],[],[],[]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[[]],[[],[]],[[]],[]]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[[]],[[],[],[]],[],[]]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[[]],[[[]],[]],[],[]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[[]],[[],[],[],[]],[]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[[]],[[],[[]],[]],[]]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[[],[]],[],[[]],[[]]]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[],[]],[],[[],[],[]]]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[],[]],[],[[[]],[]]]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[],[]],[[]],[],[],[]]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[[],[]],[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[],[]],[[],[],[],[]]]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[],[]],[[],[[]],[]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[[],[[]]],[],[[]],[]]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[[[],[]]],[],[[]],[]]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[[[[]]]],[],[[[]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
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: St001039
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[],[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
[[],[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 4 - 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[],[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 4 - 1
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 4 - 1
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 3 - 1
[[],[],[],[[]],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[],[],[],[[],[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[],[[[],[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[],[[],[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[[],[[]],[]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[],[],[[]],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[]],[[],[],[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[]],[[[]],[]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[],[]],[],[],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 3 - 1
[[],[],[[],[]],[[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[[],[],[]],[],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[],[],[[[]],[]],[],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[],[],[[],[],[]],[[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[],[[]]],[[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[[]],[]],[[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[[],[]]],[[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[],[],[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[[],[[]],[]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[[],[],[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[],[],[[]],[]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[],[[]],[],[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[],[[],[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[],[[[]]],[]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[],[[[[]]]]]]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 6 - 1
[[],[],[[[]],[],[],[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000521
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000521: Ordered trees ⟶ ℤResult quality: 88% ●values known / values provided: 91%●distinct values known / distinct values provided: 88%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000521: Ordered trees ⟶ ℤResult quality: 88% ●values known / values provided: 91%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1,0]
=> [[]]
=> 2
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [[],[]]
=> 2
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [[[]]]
=> 3
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 3
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[[]],[]]
=> 3
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 3
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 4
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 3
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 4
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 3
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 4
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 5
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 2
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 3
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 3
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 4
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 3
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 3
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 4
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 3
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 4
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 3
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 4
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 5
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 3
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 3
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 4
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 3
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 4
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 3
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 4
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 4
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 4
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 5
[[[[]]],[[[[[]]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[[]]]],[],[],[],[]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[]]]],[],[],[],[]]
=> ? = 5
[[[[[]]]],[],[[[]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [[[[[]]]],[],[[[]]]]
=> ? = 5
[[[[[]]]],[[],[[]]]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [[[[[]]]],[],[[[]]]]
=> ? = 5
[[[[[]]]],[[[],[]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [[[[[]]]],[],[[[]]]]
=> ? = 5
[[[[[]]]],[[[[]]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[[[]]]]]],[],[]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[[[[[]]]]]],[],[]]
=> ? = 7
[[[[[[[]]]]]],[[]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[[[[[[]]]]]],[[]]]
=> ? = 7
[[[[[[[[]]]]]]],[]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[[[[[[[]]]]]]],[]]
=> ? = 8
[[[[]],[],[[[[]]]]]]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[]],[[],[[[]]]]]]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[]],[[[],[[]]]]]]
=> [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[]],[[[[],[]]]]]]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[[]]],[],[[[]]]]]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[],[]],[[[[]]]]]]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[[]]],[[],[[]]]]]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[]]],[[[],[]]]]]
=> [1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[]]],[[[[]]]]]]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[[]],[]],[[[]]]]]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[],[]]],[[[]]]]]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[[]]]],[[[]]]]]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[[[]]]]],[],[]]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[[[[[[]]]]]],[[]]]
=> ? = 7
[[[[[[[]]]],[]],[]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[[[[[[]]]]]],[[]]]
=> ? = 7
[[[[[[[]]],[]]],[]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[[[[[[]]]]]],[[]]]
=> ? = 7
[[[[[[[]],[]]]],[]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[[[[[[]]]]]],[[]]]
=> ? = 7
[[[[[[[],[]]]]],[]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[[[[[[]]]]]],[[]]]
=> ? = 7
[[[[[[[[]]]]]],[]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[[[[[[]]]]]],[[]]]
=> ? = 7
[[[[],[],[],[[[]]]]]]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[],[],[[],[[]]]]]]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[],[],[[[],[]]]]]]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[],[[]],[[[]]]]]]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[],[[],[],[[]]]]]]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[],[[],[[],[]]]]]]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[],[[[]],[[]]]]]]
=> [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[],[[[],[],[]]]]]]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[],[[[[]],[]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[[]],[],[],[[]]]]]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[]],[],[[],[]]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[]],[],[[[]]]]]]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[[]],[[]],[[]]]]]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[]],[[],[],[]]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[]],[[],[[]]]]]]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[[]],[[[]],[]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[]],[[[],[]]]]]]
=> [1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[[],[]],[],[[]]]]]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[[]]],[],[[]]]]]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[],[]],[[],[]]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[],[]],[[[]]]]]]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
[[[[[[]]],[[],[]]]]]
=> [1,1,1,1,1,0,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [[[[[]]]],[[[[]]]]]
=> ? = 5
[[[[[[]]],[[[]]]]]]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 6
Description
The number of distinct subtrees of an ordered tree.
A subtree is specified by a node of the tree. Thus, the tree consisting of a single path has as many subtrees as nodes, whereas the tree of height two, having all leaves attached to the root, has only two distinct subtrees. Because we consider ordered trees, the tree $[[[[]], []], [[], [[]]]]$ on nine nodes has five distinct subtrees.
Matching statistic: St000444
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 91%●distinct values known / distinct values provided: 88%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 91%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1,0]
=> ? = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[[]]],[],[],[],[],[]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 - 1
[[[[]]],[[[[[]]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[[]]]],[],[],[],[]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[[[[[]]]],[],[[[]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[[[[[]]]],[[],[[]]]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[[[[[]]]],[[[],[]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[[[[[]]]],[[[[]]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[[[[]]]]]],[],[]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 7 - 1
[[[[[[[]]]]]],[[]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[[[[[[[]]]]]]],[]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[[[[]],[],[[[[]]]]]]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[]],[[],[[[]]]]]]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[]],[[[],[[]]]]]]
=> [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[]],[[[[],[]]]]]]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[[]]],[],[[[]]]]]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[],[]],[[[[]]]]]]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[[]]],[[],[[]]]]]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[[]]],[[[],[]]]]]
=> [1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[[]]],[[[[]]]]]]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[[]],[]],[[[]]]]]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[[],[]]],[[[]]]]]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[[[]]]],[[[]]]]]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[[[[]]]]],[],[]]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[[[[[[]]]],[]],[]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[[[[[[]]],[]]],[]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[[[[[[]],[]]]],[]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[[[[[[],[]]]]],[]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[[[[[[[]]]]]],[]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 1
[[[[],[],[],[[[]]]]]]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[],[],[[],[[]]]]]]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[],[],[[[],[]]]]]]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[],[[]],[[[]]]]]]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[],[[],[],[[]]]]]]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[],[[],[[],[]]]]]]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[],[[[]],[[]]]]]]
=> [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[],[[[],[],[]]]]]]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[],[[[[]],[]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[[]],[],[],[[]]]]]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[[]],[],[[],[]]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[[]],[],[[[]]]]]]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[[]],[[]],[[]]]]]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[[]],[[],[],[]]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[[]],[[],[[]]]]]]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[[]],[[[]],[]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[[]],[[[],[]]]]]]
=> [1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
[[[[[],[]],[],[[]]]]]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[[[]]],[],[[]]]]]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[[],[]],[[],[]]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 1
[[[[[],[]],[[[]]]]]]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 1
Description
The length of the maximal rise of a Dyck path.
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
St000442: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 91%●distinct values known / distinct values provided: 88%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 91%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1,0]
=> ? = 2 - 2
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0 = 2 - 2
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1 = 3 - 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1 = 3 - 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 3 - 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2 = 4 - 2
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 5 - 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 3 - 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 3 - 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 4 - 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 3 - 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 4 - 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 4 - 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 3 - 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 4 - 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3 = 5 - 2
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 3 - 2
[[[[]]],[],[],[],[],[]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 - 2
[[[[]]],[[[[[]]]]]]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[[]]]],[],[],[],[]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5 - 2
[[[[[]]]],[],[[[]]]]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 2
[[[[[]]]],[[],[[]]]]
=> [1,1,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 2
[[[[[]]]],[[[],[]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 2
[[[[[]]]],[[[[]]]]]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[[[[]]]]]],[],[]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 7 - 2
[[[[[[[]]]]]],[[]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 2
[[[[[[[[]]]]]]],[]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 2
[[[[]],[],[[[[]]]]]]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[]],[[],[[[]]]]]]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[]],[[[],[[]]]]]]
=> [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[]],[[[[],[]]]]]]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[[]]],[],[[[]]]]]
=> [1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[],[]],[[[[]]]]]]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[[]]],[[],[[]]]]]
=> [1,1,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[[]]],[[[],[]]]]]
=> [1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[[]]],[[[[]]]]]]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[[]],[]],[[[]]]]]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[[],[]]],[[[]]]]]
=> [1,1,1,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[[[]]]],[[[]]]]]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[[[[]]]]],[],[]]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 2
[[[[[[[]]]],[]],[]]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 2
[[[[[[[]]],[]]],[]]]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 2
[[[[[[[]],[]]]],[]]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 2
[[[[[[[],[]]]]],[]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 2
[[[[[[[[]]]]]],[]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 2
[[[[],[],[],[[[]]]]]]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[],[],[[],[[]]]]]]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[],[],[[[],[]]]]]]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[],[[]],[[[]]]]]]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[],[[],[],[[]]]]]]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[],[[],[[],[]]]]]]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[],[[[]],[[]]]]]]
=> [1,1,1,0,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[],[[[],[],[]]]]]]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[],[[[[]],[]]]]]]
=> [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[[]],[],[],[[]]]]]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[[]],[],[[],[]]]]]
=> [1,1,1,1,0,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[[]],[],[[[]]]]]]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[[]],[[]],[[]]]]]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[[]],[[],[],[]]]]]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[[]],[[],[[]]]]]]
=> [1,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[[]],[[[]],[]]]]]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[[]],[[[],[]]]]]]
=> [1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
[[[[[],[]],[],[[]]]]]
=> [1,1,1,1,0,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[[[]]],[],[[]]]]]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[[],[]],[[],[]]]]]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 - 2
[[[[[],[]],[[[]]]]]]
=> [1,1,1,1,0,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6 - 2
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St001058
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St001058: Ordered trees ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St001058: Ordered trees ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [[]]
=> 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[[]]]
=> 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [[],[]]
=> 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [[[]],[]]
=> 2 = 3 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2 = 3 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3 = 4 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 2 = 3 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 2 = 3 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2 = 3 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3 = 4 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3 = 4 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 2 = 3 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 2 = 3 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 2 = 3 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 3 = 4 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> 2 = 3 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 2 = 3 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> 2 = 3 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 3 = 4 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 2 = 3 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 3 = 4 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 3 = 4 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 4 = 5 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 2 = 3 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> 2 = 3 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[[[],[]]],[]]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 3 = 4 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> 2 = 3 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 3 = 4 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[[]],[[]]]]
=> 2 = 3 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> 3 = 4 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> 3 = 4 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> 4 = 5 - 1
[[],[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 2 - 1
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[],[[[[[[]]]]]]]]
=> ? = 3 - 1
[[],[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[[[[[[[]]]]]]],[]]
=> ? = 3 - 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[],[[[[[]]]]]]]]
=> ? = 3 - 1
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [[[[[[[[]]]]]],[]]]
=> ? = 3 - 1
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[],[],[[[[[]]]]]]]
=> ? = 4 - 1
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[],[[[[[]]]]]],[]]
=> ? = 3 - 1
[[],[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[[[[[]]]]]],[],[]]
=> ? = 4 - 1
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [[[[[],[[[[]]]]]]]]
=> ? = 3 - 1
[[],[],[],[[]],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[[],[[],[[[[]]]]]]]
=> ? = 3 - 1
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [[[[[[[[]]]]],[]]]]
=> ? = 3 - 1
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[]],[[[[[]]]]]]]
=> ? = 3 - 1
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [[[],[[[[[]]]]],[]]]
=> ? = 4 - 1
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> [[[[],[[[[]]]]],[]]]
=> ? = 3 - 1
[[],[],[],[[[],[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [[[[[[[]]]]],[],[]]]
=> ? = 4 - 1
[[],[],[],[[],[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[[[[[[]]]]]],[[]]]
=> ? = 3 - 1
[[],[],[],[[],[[]],[]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [[[[[[[]]]]],[]],[]]
=> ? = 3 - 1
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[],[]],[[[[[]]]]]]
=> ? = 4 - 1
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[[],[],[[[[]]]]],[]]
=> ? = 4 - 1
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[[[],[[],[[[]]]]]]]
=> ? = 3 - 1
[[],[],[[]],[[[]],[]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[[],[[],[[[]]]]],[]]
=> ? = 3 - 1
[[],[],[[],[]],[],[],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [[[[[[[[]]]],[]]]]]
=> ? = 3 - 1
[[],[],[[[]]],[],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [[[[[],[],[[[]]]]]]]
=> ? = 4 - 1
[[],[],[[],[]],[[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[[],[[[[[]]]],[]]]]
=> ? = 3 - 1
[[],[],[[],[],[]],[],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [[[[[]],[[[[]]]]]]]
=> ? = 3 - 1
[[],[],[[[]],[]],[],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> [[[[[],[[[]]]],[]]]]
=> ? = 3 - 1
[[],[],[[],[],[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [[[[[[[]]]]],[[]]]]
=> ? = 3 - 1
[[],[],[[],[[]],[]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [[[[[[[]]]],[]],[]]]
=> ? = 3 - 1
[[],[],[[],[],[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[[[]]],[[[[[]]]]]]
=> ? = 3 - 1
[[],[],[[],[],[[]],[]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[]],[[[[]]]]],[]]
=> ? = 3 - 1
[[],[],[[],[[],[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[[],[[[[]]]]],[[]]]
=> ? = 4 - 1
[[],[],[[],[[[]]],[]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [[[],[[[[]]]],[]],[]]
=> ? = 4 - 1
[[],[],[[[]],[],[],[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[[[],[[[]]]]],[[]]]
=> ? = 3 - 1
[[],[],[[[]],[[]],[]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
=> [[[[],[[[]]]],[]],[]]
=> ? = 3 - 1
[[],[],[[[],[]],[],[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [[[[[[]]]]],[[],[]]]
=> ? = 4 - 1
[[],[],[[[],[],[]],[]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [[[[[[]]]],[]],[[]]]
=> ? = 4 - 1
[[],[],[[[],[[]]],[]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [[[[[[]]]],[],[]],[]]
=> ? = 4 - 1
[[],[],[[[[[],[]]]]]]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[]]]],[],[],[],[]]
=> ? = 6 - 1
[[],[[]],[],[],[],[],[]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [[[[[[[],[[]]]]]]]]
=> ? = 3 - 1
[[],[[]],[],[[[]],[]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[[],[[[],[[]]]]],[]]
=> ? = 3 - 1
[[],[[]],[[]],[],[],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [[[[[],[[],[[]]]]]]]
=> ? = 3 - 1
[[],[[]],[[]],[[]],[]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [[[],[[],[[],[[]]]]]]
=> ? = 3 - 1
[[],[[]],[[],[],[]],[]]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
=> [[[[]],[[[],[[]]]]]]
=> ? = 3 - 1
[[],[[]],[[[]],[]],[]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
=> [[[[],[[],[[]]]],[]]]
=> ? = 3 - 1
[[],[[]],[[],[],[],[]]]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [[[[[],[[]]]]],[[]]]
=> ? = 3 - 1
[[],[[]],[[],[[]],[]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> [[[[[],[[]]]],[]],[]]
=> ? = 3 - 1
[[],[[[]]],[],[],[],[]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [[[[[[],[],[[]]]]]]]
=> ? = 4 - 1
[[],[[[]]],[],[[]],[]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [[[],[[[],[],[[]]]]]]
=> ? = 4 - 1
[[],[[],[]],[[]],[],[]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[[[],[[[[]]],[]]]]]
=> ? = 3 - 1
[[],[[[]]],[[],[]],[]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [[[[[],[],[[]]]],[]]]
=> ? = 4 - 1
Description
The breadth of the ordered tree.
This is the maximal number of nodes having the same depth.
Matching statistic: St001809
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001809: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 88%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001809: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 88%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3 = 4 - 1
[[],[],[],[]]
=> [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 = 2 - 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 = 3 - 1
[[],[[]],[]]
=> [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 = 3 - 1
[[],[[],[]]]
=> [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 = 3 - 1
[[],[[[]]]]
=> [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 = 4 - 1
[[[]],[],[]]
=> [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 = 3 - 1
[[[]],[[]]]
=> [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 = 3 - 1
[[[],[]],[]]
=> [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 = 3 - 1
[[[[]]],[]]
=> [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 = 4 - 1
[[[],[],[]]]
=> [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 = 3 - 1
[[[],[[]]]]
=> [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 = 4 - 1
[[[[]],[]]]
=> [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 = 3 - 1
[[[[],[]]]]
=> [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 = 4 - 1
[[[[[]]]]]
=> [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 = 5 - 1
[[],[],[],[],[]]
=> [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 = 2 - 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 = 3 - 1
[[],[],[[]],[]]
=> [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 = 3 - 1
[[],[],[[],[]]]
=> [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 = 3 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[[],[[]],[],[]]
=> [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 = 3 - 1
[[],[[]],[[]]]
=> [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 = 3 - 1
[[],[[],[]],[]]
=> [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 = 3 - 1
[[],[[[]]],[]]
=> [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 = 4 - 1
[[],[[],[],[]]]
=> [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 = 3 - 1
[[],[[],[[]]]]
=> [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 = 4 - 1
[[],[[[]],[]]]
=> [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 = 3 - 1
[[],[[[],[]]]]
=> [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 = 4 - 1
[[],[[[[]]]]]
=> [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 = 5 - 1
[[[]],[],[],[]]
=> [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 = 3 - 1
[[[]],[],[[]]]
=> [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 = 3 - 1
[[[]],[[]],[]]
=> [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 = 3 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [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 = 4 - 1
[[[],[]],[],[]]
=> [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 = 3 - 1
[[[[]]],[],[]]
=> [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 = 4 - 1
[[[],[]],[[]]]
=> [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 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[[[],[],[]],[]]
=> [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 = 3 - 1
[[[],[[]]],[]]
=> [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 = 4 - 1
[[[[]],[]],[]]
=> [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 = 3 - 1
[[[[],[]]],[]]
=> [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 = 4 - 1
[[[[[]]]],[]]
=> [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 = 5 - 1
[[],[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 4 - 1
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
[[],[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 - 1
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 3 - 1
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 4 - 1
[[],[],[],[[[],[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 4 - 1
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[],[[[[[]]]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 6 - 1
[[],[],[[],[]],[],[],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 - 1
[[],[],[[[]]],[],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4 - 1
[[],[],[[],[[]]],[[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[[[],[]]],[[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[[],[[]],[],[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[[],[[],[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[[],[[[]]],[]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[[],[[[[]]]]]]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 6 - 1
[[],[],[[[],[]],[],[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[[[],[],[]],[]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[[[],[[]]],[]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[[[],[[[]]]]]]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 6 - 1
[[],[],[[[[],[[]]]]]]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 6 - 1
[[],[],[[[[[],[]]]]]]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 6 - 1
[[],[],[[[[[[]]]]]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
[[],[[[]]],[],[],[],[]]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4 - 1
[[],[[[]]],[],[[]],[]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 4 - 1
[[],[[[]]],[],[[[]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[],[[[]]],[[]],[[]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[[[]]],[[],[]],[]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> ? = 4 - 1
[[],[[[]]],[[],[],[]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[[[]]],[[],[[]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[],[[[]]],[[[]],[]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[[[]]],[[[],[]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[],[[],[[]]],[],[],[]]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4 - 1
[[],[[[],[]]],[],[],[]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4 - 1
[[],[[[]],[],[]],[[]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[[[],[]],[]],[[]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[[[[]]],[]],[[]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[],[[[[[[]]]]]],[]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 7 - 1
[[],[[],[[[[[]]]]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 7 - 1
[[],[[[]],[],[],[],[]]]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
Description
The index of the step at the first peak of maximal height in a Dyck path.
Matching statistic: St000684
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 3 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[[],[],[],[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
[[],[],[],[],[],[],[[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[],[],[[]],[]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[],[],[],[[],[]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[],[],[[[]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 4 - 1
[[],[],[],[],[[]],[],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[[],[],[],[],[[]],[[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[],[[],[]],[]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[],[],[[[]]],[]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[],[],[[],[],[]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[],[[],[[]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 4 - 1
[[],[],[],[],[[[]],[]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[],[[[],[]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 4 - 1
[[],[],[],[[]],[],[],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[[],[],[],[[]],[[]],[]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[],[[],[]],[],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[[],[],[],[[],[]],[[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[[[]]],[[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[],[[],[],[]],[]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[],[[],[[]]],[]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[],[[[]],[]],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[],[[[],[]]],[]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 4 - 1
[[],[],[],[[],[],[],[]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[[],[[]],[]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[[[]],[],[]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[],[[[],[]],[]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[],[[[[]]],[]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[]],[[]],[],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[[],[],[[]],[[]],[[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[]],[[],[],[]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[]],[[[]],[]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[],[]],[],[],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 - 1
[[],[],[[[]]],[],[],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 1
[[],[],[[],[]],[[]],[]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[[],[],[]],[],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[[],[],[[[]],[]],[],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 - 1
[[],[],[[],[],[]],[[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[],[[]]],[[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[[]],[]],[[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[[],[]]],[[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[],[],[],[]],[]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[[],[[]],[]],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[[],[],[],[],[]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[],[],[[]],[]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[],[[]],[],[]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[],[[],[]],[]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[],[[[]]],[]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[[]],[],[],[]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[[]],[[]],[]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[[],[]],[],[]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
Description
The global dimension of the LNakayama algebra associated to a Dyck path.
An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with $n$ points for $n \geq 2$. Number those points from the left to the right by $0,1,\ldots,n-1$.
The algebra is then uniquely determined by the dimension $c_i$ of the projective indecomposable modules at point $i$. Such algebras are then uniquely determined by lists of the form $[c_0,c_1,...,c_{n-1}]$ with the conditions: $c_{n-1}=1$ and $c_i -1 \leq c_{i+1}$ for all $i$. The number of such algebras is then the $n-1$-st Catalan number $C_{n-1}$.
One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0].
Conjecture: that there is an explicit bijection between $n$-LNakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$.
Examples:
* For $m=2$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192.
* For $m=3$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.
The following 44 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000306The bounce count of a Dyck path. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St000025The number of initial rises of a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St000662The staircase size of the code of a permutation. St000141The maximum drop size of a permutation. St000209Maximum difference of elements in cycles. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St000308The height of the tree associated to a permutation. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001372The length of a longest cyclic run of ones of a binary word. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001652The length of a longest interval of consecutive numbers. St000094The depth of an ordered tree. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. 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. St001530The depth of a Dyck path. 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. St000028The number of stack-sorts needed to sort a permutation. St001330The hat guessing number of a graph. St001589The nesting number of a perfect matching. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001875The number of simple modules with projective dimension at most 1. 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. St001877Number of indecomposable injective modules with projective dimension 2. St000983The length of the longest alternating subword. 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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