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Your data matches 54 different statistics following compositions of up to 3 maps.
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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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000521: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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: 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: 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: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
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
[[[[]]],[],[],[],[[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4 - 1
[[[[]]],[],[],[[]],[]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[[[]]],[],[],[[],[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4 - 1
[[[[]]],[],[],[[[]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[[[]]],[],[[]],[],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 1
[[[[]]],[],[[]],[[]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[[[]]],[],[[],[]],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[[[]]],[],[[],[],[]]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[[[]]],[],[[],[[]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[[[]]],[],[[[]],[]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[[[]]],[],[[[],[]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[[[]]],[[],[]],[],[]]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 1
[[[[]]],[[],[]],[[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[[[]]],[[],[],[],[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 1
[[[[]]],[[],[[]],[]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 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: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
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
[[[[]]],[],[],[],[[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4 - 2
[[[[]]],[],[],[[]],[]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 2
[[[[]]],[],[],[[],[]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4 - 2
[[[[]]],[],[],[[[]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 2
[[[[]]],[],[[]],[],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 2
[[[[]]],[],[[]],[[]]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 2
[[[[]]],[],[[],[]],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 - 2
[[[[]]],[],[[],[],[]]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 2
[[[[]]],[],[[],[[]]]]
=> [1,1,1,0,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 2
[[[[]]],[],[[[]],[]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 2
[[[[]]],[],[[[],[]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 2
[[[[]]],[[],[]],[],[]]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 2
[[[[]]],[[],[]],[[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 2
[[[[]]],[[],[],[],[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 2
[[[[]]],[[],[[]],[]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 4 - 2
Description
The maximal area to the right of an up step of a Dyck path.
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
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5 - 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
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 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
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 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
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 4 - 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
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 5 - 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
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 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
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 1
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5 - 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
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 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
[[],[],[],[[[],[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 1
[[],[],[],[[[],[[]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5 - 1
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 4 - 1
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5 - 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
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 3 - 1
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 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]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 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
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
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: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ 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]
=> [1,0,1,0]
=> 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
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 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]
=> ? = 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
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[],[],[[[],[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 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]
=> ? = 3 - 1
[[],[],[[]],[[],[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[],[[]],[[[]]],[]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 - 1
[[],[],[[]],[[],[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[],[[]],[[[],[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[],[[[]]],[],[[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 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]
=> ? = 3 - 1
[[],[],[[[]]],[[]],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[],[]],[[[]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[],[[[]]],[[],[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 4 - 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
[[],[],[[],[[]]],[],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 4 - 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
[[],[],[[[],[]]],[],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 4 - 1
[[],[],[[[[]]]],[],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 5 - 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
[[],[],[[],[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 - 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
[[],[],[[],[[],[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 - 1
[[],[],[[],[[[]]]],[]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[],[],[[[]],[],[]],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[[]],[[]]],[]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 - 1
[[],[],[[[],[]],[]],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[[[]]],[]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 4 - 1
[[],[],[[[],[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 - 1
[[],[],[[[],[[]]]],[]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[],[],[[[[]],[]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 - 1
[[],[],[[[[],[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 - 1
[[],[],[[],[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[],[[],[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[],[[],[[]],[[]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[],[[],[[],[],[]]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[],[[],[[[]],[]]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[[]],[],[],[[]],[]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 - 1
[[],[[]],[],[],[[[]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4 - 1
[[],[[]],[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 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
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: 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: 61% ●values known / values provided: 61%●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: 61% ●values known / values provided: 61%●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
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[[[],[[[[]]]]]],[]]
=> ? = 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
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[[[],[],[[[[]]]]]]]
=> ? = 4 - 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
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[],[],[],[[[[]]]]]]
=> ? = 5 - 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
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[[],[[[[]]]]],[],[]]
=> ? = 4 - 1
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[[[]]]]],[],[],[]]
=> ? = 5 - 1
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [[[[[[],[[[]]]]]]]]
=> ? = 3 - 1
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[[],[[[],[[[]]]]]]]
=> ? = 3 - 1
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [[[[[],[[[]]]]]],[]]
=> ? = 3 - 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
[[],[],[[]],[[],[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> [[[[[],[[[]]]]],[]]]
=> ? = 3 - 1
[[],[],[[]],[[[]]],[]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [[[],[],[[],[[[]]]]]]
=> ? = 4 - 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
[[],[],[[]],[[[],[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[[[],[[[]]]]],[],[]]
=> ? = 4 - 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
[[],[],[[[]]],[[]],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [[[],[[],[],[[[]]]]]]
=> ? = 4 - 1
[[],[],[[],[]],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [[[[[[[]]]],[]]],[]]
=> ? = 3 - 1
[[],[],[[[]]],[[],[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[[[],[],[[[]]]]],[]]
=> ? = 4 - 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
[[],[],[[],[[]]],[],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[[[],[[[[]]]],[]]]]
=> ? = 4 - 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
[[],[],[[[],[]]],[],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
=> [[[[[[[]]]],[],[]]]]
=> ? = 4 - 1
[[],[],[[[[]]]],[],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [[[[],[],[],[[[]]]]]]
=> ? = 5 - 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
[[],[],[[],[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [[[],[[]],[[[[]]]]]]
=> ? = 4 - 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
[[],[],[[],[[],[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [[[[]],[],[[[[]]]]]]
=> ? = 4 - 1
[[],[],[[],[[[]]]],[]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [[[],[],[[[[]]]],[]]]
=> ? = 5 - 1
[[],[],[[[]],[],[]],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> [[[[]],[[],[[[]]]]]]
=> ? = 4 - 1
[[],[],[[[]],[[]]],[]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [[[],[[],[[[]]]],[]]]
=> ? = 4 - 1
Description
The breadth of the ordered tree.
This is the maximal number of nodes having the same depth.
Matching statistic: St000451
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1] => 1 = 2 - 1
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 1 = 2 - 1
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 2 = 3 - 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 2 - 1
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 3 - 1
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2 = 3 - 1
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2 = 3 - 1
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 3 = 4 - 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 2 - 1
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 3 - 1
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 3 - 1
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 3 - 1
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3 = 4 - 1
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2 = 3 - 1
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 3 - 1
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 3 - 1
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 3 = 4 - 1
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 3 - 1
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3 = 4 - 1
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 3 - 1
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 3 = 4 - 1
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 4 = 5 - 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = 2 - 1
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2 = 3 - 1
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2 = 3 - 1
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2 = 3 - 1
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 3 = 4 - 1
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2 = 3 - 1
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 3 - 1
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2 = 3 - 1
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 3 = 4 - 1
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 3 - 1
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 3 = 4 - 1
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 3 - 1
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 3 = 4 - 1
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 4 = 5 - 1
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2 = 3 - 1
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 3 - 1
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2 = 3 - 1
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 3 - 1
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 3 = 4 - 1
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2 = 3 - 1
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 3 = 4 - 1
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 3 - 1
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 3 = 4 - 1
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2 = 3 - 1
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 3 = 4 - 1
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2 = 3 - 1
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 3 = 4 - 1
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 4 = 5 - 1
[[],[[[[]]]],[],[]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 5 - 1
[[],[[[[[]]]]],[]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => ? = 6 - 1
[[[]],[],[],[],[[]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 3 - 1
[[[]],[],[],[[]],[]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 3 - 1
[[[]],[],[],[[],[]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 3 - 1
[[[]],[],[[]],[],[]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 3 - 1
[[[]],[],[[]],[[]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 1
[[[]],[],[[],[]],[]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 3 - 1
[[[]],[],[[[]]],[]]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,4,5,7] => ? = 4 - 1
[[[]],[],[[],[],[]]]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 1
[[[]],[],[[[]],[]]]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 1
[[[]],[[]],[],[],[]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 3 - 1
[[[]],[[]],[],[[]]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 3 - 1
[[[]],[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3 - 1
[[[]],[[]],[[],[]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 3 - 1
[[[]],[[]],[[[]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 4 - 1
[[[]],[[],[]],[],[]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 3 - 1
[[[]],[[[]]],[],[]]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,3,4,6,7] => ? = 4 - 1
[[[]],[[],[]],[[]]]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 1
[[[]],[[[]]],[[]]]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 4 - 1
[[[]],[[],[],[]],[]]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3 - 1
[[[]],[[],[[]]],[]]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,4,5,7] => ? = 4 - 1
[[[]],[[[]],[]],[]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3 - 1
[[[]],[[[],[]]],[]]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,4,5,7] => ? = 4 - 1
[[[]],[[],[],[],[]]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 1
[[[]],[[],[],[[]]]]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 4 - 1
[[[]],[[],[[]],[]]]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 1
[[[]],[[],[[],[]]]]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 4 - 1
[[[]],[[[]],[],[]]]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 4 - 1
[[[]],[[[]],[[]]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 4 - 1
[[[]],[[[],[]],[]]]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 4 - 1
[[[]],[[[[]]],[]]]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 4 - 1
[[[]],[[[],[],[]]]]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 4 - 1
[[[]],[[[[]],[]]]]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 4 - 1
[[[[]]],[],[[]],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,1,2,4,6,5,7] => ? = 4 - 1
[[[[]]],[],[[[]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 4 - 1
[[[[]]],[[]],[],[]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => ? = 4 - 1
[[[[]]],[[]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 4 - 1
[[[[]]],[[],[]],[]]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,1,2,4,6,5,7] => ? = 4 - 1
[[[[]]],[[[]]],[]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 4 - 1
[[[[]]],[[],[],[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 4 - 1
[[[[]]],[[],[[]]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 4 - 1
[[[[]]],[[[]],[]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 4 - 1
[[[[]]],[[[],[]]]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 4 - 1
[[[[]]],[[[[]]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 5 - 1
[[[],[],[]],[],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 3 - 1
[[[[]],[]],[],[],[]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 3 - 1
[[[],[],[]],[],[[]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 3 - 1
[[[[]],[]],[],[[]]]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 3 - 1
[[[],[],[]],[[]],[]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3 - 1
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St001090
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001090: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001090: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
[[]]
=> [1,0]
=> [1,0]
=> [1] => 0 = 2 - 2
[[],[]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 0 = 2 - 2
[[[]]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 1 = 3 - 2
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 2 - 2
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 3 - 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1 = 3 - 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1 = 3 - 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 2 = 4 - 2
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 2 - 2
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 3 - 2
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 3 - 2
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1 = 3 - 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 4 - 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 3 - 2
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1 = 3 - 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1 = 3 - 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 2 = 4 - 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1 = 3 - 2
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 4 - 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1 = 3 - 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2 = 4 - 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 3 = 5 - 2
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 2 - 2
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 3 - 2
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1 = 3 - 2
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1 = 3 - 2
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 2 = 4 - 2
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1 = 3 - 2
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1 = 3 - 2
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1 = 3 - 2
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 2 = 4 - 2
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1 = 3 - 2
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 2 = 4 - 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1 = 3 - 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 2 = 4 - 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 3 = 5 - 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 3 - 2
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1 = 3 - 2
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1 = 3 - 2
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1 = 3 - 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 2 = 4 - 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1 = 3 - 2
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 2 = 4 - 2
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1 = 3 - 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 2 = 4 - 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1 = 3 - 2
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 2 = 4 - 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1 = 3 - 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 2 = 4 - 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 3 = 5 - 2
[[],[[[[]]]],[],[]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 5 - 2
[[],[[[[[]]]]],[]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => ? = 6 - 2
[[[]],[],[],[],[[]]]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 3 - 2
[[[]],[],[],[[]],[]]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 3 - 2
[[[]],[],[],[[],[]]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 3 - 2
[[[]],[],[[]],[],[]]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 3 - 2
[[[]],[],[[]],[[]]]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 2
[[[]],[],[[],[]],[]]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 3 - 2
[[[]],[],[[[]]],[]]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,4,5,7] => ? = 4 - 2
[[[]],[],[[],[],[]]]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 2
[[[]],[],[[[]],[]]]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 2
[[[]],[[]],[],[],[]]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 3 - 2
[[[]],[[]],[],[[]]]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 3 - 2
[[[]],[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3 - 2
[[[]],[[]],[[],[]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 3 - 2
[[[]],[[]],[[[]]]]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 4 - 2
[[[]],[[],[]],[],[]]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 3 - 2
[[[]],[[[]]],[],[]]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,3,4,6,7] => ? = 4 - 2
[[[]],[[],[]],[[]]]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 2
[[[]],[[[]]],[[]]]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 4 - 2
[[[]],[[],[],[]],[]]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3 - 2
[[[]],[[],[[]]],[]]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,4,5,7] => ? = 4 - 2
[[[]],[[[]],[]],[]]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3 - 2
[[[]],[[[],[]]],[]]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,4,5,7] => ? = 4 - 2
[[[]],[[],[],[],[]]]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 2
[[[]],[[],[],[[]]]]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 4 - 2
[[[]],[[],[[]],[]]]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3 - 2
[[[]],[[],[[],[]]]]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 4 - 2
[[[]],[[[]],[],[]]]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 4 - 2
[[[]],[[[]],[[]]]]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 4 - 2
[[[]],[[[],[]],[]]]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 4 - 2
[[[]],[[[[]]],[]]]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 4 - 2
[[[]],[[[],[],[]]]]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 4 - 2
[[[]],[[[[]],[]]]]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 4 - 2
[[[[]]],[],[[]],[]]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,1,2,4,6,5,7] => ? = 4 - 2
[[[[]]],[],[[[]]]]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 4 - 2
[[[[]]],[[]],[],[]]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => ? = 4 - 2
[[[[]]],[[]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 4 - 2
[[[[]]],[[],[]],[]]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,1,2,4,6,5,7] => ? = 4 - 2
[[[[]]],[[[]]],[]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 4 - 2
[[[[]]],[[],[],[]]]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 4 - 2
[[[[]]],[[],[[]]]]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 4 - 2
[[[[]]],[[[]],[]]]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 4 - 2
[[[[]]],[[[],[]]]]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 4 - 2
[[[[]]],[[[[]]]]]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 5 - 2
[[[],[],[]],[],[],[]]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 3 - 2
[[[[]],[]],[],[],[]]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 3 - 2
[[[],[],[]],[],[[]]]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 3 - 2
[[[[]],[]],[],[[]]]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 3 - 2
[[[],[],[]],[[]],[]]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3 - 2
Description
The number of pop-stack-sorts needed to sort a permutation.
The pop-stack sorting operator is defined as follows. Process the permutation $\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order.
A permutation is $t$-pop-stack sortable if it is sortable using $t$ pop-stacks in series.
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: 50% ●values known / values provided: 50%●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: 50% ●values known / values provided: 50%●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
[[],[],[],[],[[[[]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 5
[[],[],[],[[]],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[[],[],[],[[]],[[],[]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[[],[],[],[[]],[[[]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 4
[[],[],[],[[[]]],[],[]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
[[],[],[],[[[[]]]],[]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 5
[[],[],[],[[],[],[[]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 4
[[],[],[],[[],[[],[]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 4
[[],[],[],[[],[[[]]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 5
[[],[],[],[[[]],[[]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 4
[[],[],[],[[[],[],[]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 4
[[],[],[],[[[],[[]]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 5
[[],[],[],[[[[]],[]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 4
[[],[],[],[[[[],[]]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 5
[[],[],[[]],[],[],[],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[[],[],[[]],[],[],[[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 3
[[],[],[[]],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 3
[[],[],[[]],[],[[],[]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 3
[[],[],[[]],[],[[[]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 4
[[],[],[[]],[[],[]],[]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 3
[[],[],[[]],[[[]]],[]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 4
[[],[],[[]],[[],[[]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 4
[[],[],[[]],[[[],[]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 4
[[],[],[[]],[[[[]]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 5
[[],[],[[],[]],[],[[]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[[],[],[[[]]],[],[[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 4
[[],[],[[[]]],[[]],[]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 4
[[],[],[[],[]],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[[],[],[[],[]],[[[]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 4
[[],[],[[[]]],[[],[]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 4
[[],[],[[[]]],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 4
[[],[],[[],[[]]],[],[]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
[[],[],[[[],[]]],[],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 4
[[],[],[[[[]]]],[],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 5
[[],[],[[[[]]]],[[]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 5
[[],[],[[],[],[[]]],[]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 4
[[],[],[[],[[],[]]],[]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 4
[[],[],[[],[[[]]]],[]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 5
[[],[],[[[]],[],[]],[]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 4
[[],[],[[[]],[[]]],[]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 4
[[],[],[[[],[]],[]],[]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 4
[[],[],[[[[]]],[]],[]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 4
[[],[],[[[],[],[]]],[]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 4
[[],[],[[[],[[]]]],[]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 5
[[],[],[[[[]],[]]],[]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 4
[[],[],[[[[],[]]]],[]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 5
[[],[],[[[[[]]]]],[]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> ? = 6
[[],[],[[],[],[],[[]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 4
[[],[],[[],[],[[],[]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 4
[[],[],[[],[],[[[]]]]]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 5
Description
The position of the first down step of a Dyck path.
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. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000141The maximum drop size of a permutation. St000094The depth of an ordered tree. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St000662The staircase size of the code of 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:
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. 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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