- FindStat

Your data matches 50 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000013
(load all 4 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[]
 => []
 => []
 => 0
[[]]
 => [1,0]
 => [1,0]
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3

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: St000381
(load all 2 compositions to match this statistic)

Mapping

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: 89% ●values known / values provided: 99%●distinct values known / distinct values provided: 89%

Values

[]
 => []
 => []
 => [] => ? = 0
[[]]
 => [1,0]
 => [1,0]
 => [1] => 1
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1] => 1
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => [2] => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1] => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,2] => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1] => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,2] => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [3] => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 3
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4] => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 3
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 3
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => 4
[[],[[],[[],[[[],[]],[]]]]]
 => [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ? = 4
[[],[[],[[[],[[],[]]],[]]]]
 => [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? => ? = 4
[[],[[[],[]],[[],[[],[]]]]]
 => [1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? => ? = 4
[[],[[[],[[],[]]],[[],[]]]]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? => ? = 4
[[],[[[[],[]],[[],[]]],[]]]
 => [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? => ? = 4
[[],[[[[[],[]],[]],[]],[]]]
 => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? => ? = 4
[[[],[[],[[[],[]],[]]]],[]]
 => [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? => ? = 4
[[[],[[[],[[],[]]],[]]],[]]
 => [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [2,4,3,1] => ? = 4
[[[[],[]],[[],[[],[]]]],[]]
 => [1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? => ? = 4
[[[[],[[],[]]],[[],[]]],[]]
 => [1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => [2,4,3,1] => ? = 4
[[[[[],[]],[[],[]]],[]],[]]
 => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? => ? = 4
[[[[[[],[]],[]],[]],[]],[]]
 => [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? => ? = 4

Description

The largest part of an integer composition.

Matching statistic: St001039

Mapping

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: 89% ●values known / values provided: 95%●distinct values known / distinct values provided: 89%

Values

[]
 => []
 => []
 => []
 => ? = 0
[[]]
 => [1,0]
 => [1,0]
 => [1,0]
 => ? = 1
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 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
[[],[[]],[]]
 => [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
[[],[[],[]]]
 => [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
[[],[[[]]]]
 => [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
[[[]],[],[]]
 => [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
[[[]],[[]]]
 => [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
[[[],[]],[]]
 => [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
[[[[]]],[]]
 => [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
[[[],[],[]]]
 => [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
[[[],[[]]]]
 => [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
[[[[]],[]]]
 => [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
[[[[],[]]]]
 => [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
[[[[[]]]]]
 => [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
[[],[],[],[],[]]
 => [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
[[],[],[],[[]]]
 => [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
[[],[],[[]],[]]
 => [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
[[],[],[[],[]]]
 => [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
[[],[],[[[]]]]
 => [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
[[],[[]],[],[]]
 => [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
[[],[[]],[[]]]
 => [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
[[],[[],[]],[]]
 => [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
[[],[[[]]],[]]
 => [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
[[],[[],[],[]]]
 => [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
[[],[[],[[]]]]
 => [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
[[],[[[]],[]]]
 => [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
[[],[[[],[]]]]
 => [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
[[],[[[[]]]]]
 => [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
[[[]],[],[],[]]
 => [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
[[[]],[],[[]]]
 => [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
[[[]],[[]],[]]
 => [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
[[[]],[[],[]]]
 => [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
[[[]],[[[]]]]
 => [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
[[[],[]],[],[]]
 => [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
[[[[]]],[],[]]
 => [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
[[[],[]],[[]]]
 => [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
[[[[]]],[[]]]
 => [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
[[[],[],[]],[]]
 => [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
[[[],[[]]],[]]
 => [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
[[[[]],[]],[]]
 => [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
[[[[],[]]],[]]
 => [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
[[[[[]]]],[]]
 => [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
[[[],[],[],[]]]
 => [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
[[],[],[],[],[],[],[],[]]
 => [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]
 => ? = 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]
 => ? = 2
[[],[],[],[],[],[[]],[]]
 => [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]
 => ? = 2
[[],[],[],[],[],[[],[]]]
 => [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]
 => ? = 2
[[],[],[],[],[],[[[]]]]
 => [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]
 => ? = 3
[[],[],[],[],[[]],[],[]]
 => [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]
 => ? = 2
[[],[],[],[],[[]],[[]]]
 => [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]
 => ? = 2
[[],[],[],[],[[],[]],[]]
 => [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]
 => ? = 2
[[],[],[],[],[[[]]],[]]
 => [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]
 => ? = 3
[[],[],[],[],[[],[],[]]]
 => [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]
 => ? = 2
[[],[],[],[],[[],[[]]]]
 => [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]
 => ? = 3
[[],[],[],[],[[[]],[]]]
 => [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]
 => ? = 2
[[],[],[],[],[[[],[]]]]
 => [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]
 => ? = 3
[[],[],[],[],[[[[]]]]]
 => [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]
 => ? = 4
[[],[],[],[[]],[],[],[]]
 => [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]
 => ? = 2
[[],[],[],[[]],[],[[]]]
 => [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]
 => ? = 2
[[],[],[],[[]],[[],[]]]
 => [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]
 => ? = 2
[[],[],[],[[],[]],[],[]]
 => [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]
 => ? = 2
[[],[],[],[[],[]],[[]]]
 => [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]
 => ? = 2
[[],[],[],[[[]]],[[]]]
 => [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]
 => ? = 3
[[],[],[],[[],[[]]],[]]
 => [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]
 => ? = 3
[[],[],[],[[[],[]]],[]]
 => [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]
 => ? = 3
[[],[],[],[[],[],[],[]]]
 => [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]
 => ? = 2
[[],[],[],[[],[[]],[]]]
 => [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]
 => ? = 2
[[],[],[],[[],[[[]]]]]
 => [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]
 => ? = 4
[[],[],[],[[[]],[],[]]]
 => [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]
 => ? = 3
[[],[],[],[[[],[]],[]]]
 => [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]
 => ? = 3
[[],[],[],[[[[]]],[]]]
 => [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]
 => ? = 3
[[],[],[],[[[],[[]]]]]
 => [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]
 => ? = 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]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 4
[[],[],[],[[[[[]]]]]]
 => [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]
 => ? = 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]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 2
[[],[],[[]],[],[],[[]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [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]
 => ? = 2
[[],[],[[]],[],[[],[]]]
 => [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]
 => ? = 2
[[],[],[[]],[[]],[[]]]
 => [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]
 => ? = 2
[[],[],[[]],[[],[],[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[[]],[[[]],[]]]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[[],[]],[],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2
[[],[],[[],[]],[],[[]]]
 => [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]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 2
[[],[],[[],[]],[[],[]]]
 => [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]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 2
[[],[],[[],[],[]],[[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[[],[[]]],[[]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 3
[[],[],[[[]],[]],[[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[[[],[]]],[[]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 3
[[],[],[[],[],[],[],[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[[],[],[[]],[]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[[],[[]],[],[]]]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 3
[[],[],[[],[[],[]],[]]]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 3

Description

The maximal height of a column in the parallelogram polyomino associated with a Dyck path.

Matching statistic: St000521

Mapping

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: 78% ●values known / values provided: 88%●distinct values known / distinct values provided: 78%

Values

[]
 => []
 => []
 => []
 => ? = 0 + 1
[[]]
 => [1,0]
 => [1,0]
 => [[]]
 => 2 = 1 + 1
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [[],[]]
 => 2 = 1 + 1
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => [[[]]]
 => 3 = 2 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [[],[],[]]
 => 2 = 1 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [[],[[]]]
 => 3 = 2 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [[[]],[]]
 => 3 = 2 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [[],[[]]]
 => 3 = 2 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 4 = 3 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 2 = 1 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => 3 = 2 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 3 = 2 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => 3 = 2 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 4 = 3 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 3 = 2 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 3 = 2 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 3 = 2 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 4 = 3 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 3 = 2 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 4 = 3 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 3 = 2 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 4 = 3 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 5 = 4 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => 2 = 1 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => 3 = 2 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => 3 = 2 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => 3 = 2 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 4 = 3 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => 3 = 2 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => 3 = 2 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => 3 = 2 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => 4 = 3 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => 3 = 2 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 4 = 3 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => 3 = 2 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 4 = 3 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => 5 = 4 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => 3 = 2 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => 3 = 2 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => 3 = 2 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => 3 = 2 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => 4 = 3 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => 3 = 2 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[],[]]
 => 4 = 3 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => 3 = 2 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[[[]]],[[]]]
 => 4 = 3 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => 3 = 2 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => 4 = 3 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => 3 = 2 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => 4 = 3 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[[[]]]],[]]
 => 5 = 4 + 1
[[[[]]],[],[[[]]],[]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [[[[]]],[],[[[]]],[]]
 => ? = 3 + 1
[[[[]]],[[]],[],[[]]]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[]]],[[]],[],[[]]]
 => ? = 3 + 1
[[[[]]],[[]],[[]],[]]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [[[[]]],[[]],[[]],[]]
 => ? = 3 + 1
[[[[]]],[[]],[[],[]]]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[]]],[[]],[],[[]]]
 => ? = 3 + 1
[[[[]]],[[[]]],[],[]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[[[]]],[],[]]
 => ? = 3 + 1
[[[[]]],[[],[],[]],[]]
 => [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [[[[]]],[[]],[[]],[]]
 => ? = 3 + 1
[[[[]]],[[],[[]]],[]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [[[[]]],[],[[[]]],[]]
 => ? = 3 + 1
[[[[]]],[[[]],[]],[]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [[[[]]],[[]],[[]],[]]
 => ? = 3 + 1
[[[[]]],[[[],[]]],[]]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [[[[]]],[],[[[]]],[]]
 => ? = 3 + 1
[[[[]]],[[[[]]]],[]]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [[[[]]],[[[[]]]],[]]
 => ? = 4 + 1
[[[[]]],[[[[[]]]]]]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[[[]]],[[[[[]]]]]]
 => ? = 5 + 1
[[[[[]]]],[],[],[],[]]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [[[[[]]]],[],[],[],[]]
 => ? = 4 + 1
[[[[[]]]],[],[],[[]]]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [[[[[]]]],[],[],[[]]]
 => ? = 4 + 1
[[[[[]]]],[],[[]],[]]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [[[[[]]]],[],[[]],[]]
 => ? = 4 + 1
[[[[[]]]],[],[[],[]]]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [[[[[]]]],[],[],[[]]]
 => ? = 4 + 1
[[[[[]]]],[[]],[],[]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [[[[[]]]],[[]],[],[]]
 => ? = 4 + 1
[[[[[]]]],[[]],[[]]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [[[[[]]]],[[]],[[]]]
 => ? = 4 + 1
[[[[[]]]],[[],[]],[]]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [[[[[]]]],[],[[]],[]]
 => ? = 4 + 1
[[[[[]]]],[[],[],[]]]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [[[[[]]]],[[]],[[]]]
 => ? = 4 + 1
[[[[[]]]],[[[]],[]]]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [[[[[]]]],[[]],[[]]]
 => ? = 4 + 1
[[[[[]]]],[[[[]]]]]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [[[[[]]]],[[[[]]]]]
 => ? = 4 + 1
[[[[[[]]]]],[],[],[]]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [[[[[[]]]]],[],[],[]]
 => ? = 5 + 1
[[[[]],[],[]],[],[[]]]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[]]],[[]],[],[[]]]
 => ? = 3 + 1
[[[[],[]],[]],[],[[]]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[]]],[[]],[],[[]]]
 => ? = 3 + 1
[[[[[]]],[]],[],[[]]]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[]]],[[]],[],[[]]]
 => ? = 3 + 1
[[[[]],[],[]],[[]],[]]
 => [1,1,1,0,0,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [[[[]]],[[]],[[]],[]]
 => ? = 3 + 1
[[[[],[]],[]],[[]],[]]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [[[[]]],[[]],[[]],[]]
 => ? = 3 + 1
[[[[[]]],[]],[[]],[]]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [[[[]]],[[]],[[]],[]]
 => ? = 3 + 1
[[[[]],[],[]],[[],[]]]
 => [1,1,1,0,0,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[]]],[[]],[],[[]]]
 => ? = 3 + 1
[[[[],[]],[]],[[],[]]]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[]]],[[]],[],[[]]]
 => ? = 3 + 1
[[[[[]]],[]],[[],[]]]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [[[[]]],[[]],[],[[]]]
 => ? = 3 + 1
[[[[[[]]]]],[[[]]]]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [[[[[[]]]]],[[[]]]]
 => ? = 5 + 1
[[[[]],[],[[]]],[],[]]
 => [1,1,1,0,0,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[[[]]],[],[]]
 => ? = 3 + 1
[[[[]],[[],[]]],[],[]]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[[[]]],[],[]]
 => ? = 3 + 1
[[[[[]]],[],[]],[],[]]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [[[[[]]]],[[]],[],[]]
 => ? = 4 + 1
[[[[],[]],[[]]],[],[]]
 => [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[[[]]],[],[]]
 => ? = 3 + 1
[[[[[]]],[[]]],[],[]]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[[[]]],[],[]]
 => ? = 3 + 1
[[[[[]],[]],[]],[],[]]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [[[[[]]]],[[]],[],[]]
 => ? = 4 + 1
[[[[[],[]]],[]],[],[]]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [[[[[]]]],[[]],[],[]]
 => ? = 4 + 1
[[[[[[]]]],[]],[],[]]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [[[[[]]]],[[]],[],[]]
 => ? = 4 + 1
[[[[],[],[],[]]],[],[]]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[[[]]],[],[]]
 => ? = 3 + 1
[[[[],[[]],[]]],[],[]]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[[[]]],[],[]]
 => ? = 3 + 1
[[[[[]],[],[]]],[],[]]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[[[]]],[],[]]
 => ? = 3 + 1
[[[[[],[]],[]]],[],[]]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[[[]]],[],[]]
 => ? = 3 + 1
[[[[[[]]],[]]],[],[]]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[[[]]],[],[]]
 => ? = 3 + 1
[[[[[[[]]]]]],[],[]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [[[[[[[]]]]]],[],[]]
 => ? = 6 + 1
[[[[[]]],[],[]],[[]]]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [[[[[]]]],[[]],[[]]]
 => ? = 4 + 1
[[[[[]],[]],[]],[[]]]
 => [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [[[[[]]]],[[]],[[]]]
 => ? = 4 + 1
[[[[[],[]]],[]],[[]]]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [[[[[]]]],[[]],[[]]]
 => ? = 4 + 1

Description

The number of distinct subtrees of an ordered tree. A subtree is specified by a node of the tree. Thus, the tree consisting of a single path has as many subtrees as nodes, whereas the tree of height two, having all leaves attached to the root, has only two distinct subtrees. Because we consider ordered trees, the tree

$[[[[]], []], [[], [[]]]]$ on nine nodes has five distinct subtrees.

Matching statistic: St000444
(load all 3 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 87%●distinct values known / distinct values provided: 78%

Values

[]
 => []
 => []
 => ? = 0
[[]]
 => [1,0]
 => [1,0]
 => ? = 1
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 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]
 => ? = 3
[[[[]]],[],[],[],[[]]]
 => [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]
 => ? = 3
[[[[]]],[],[],[[],[]]]
 => [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]
 => ? = 3
[[[[]]],[],[],[[[]]]]
 => [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]
 => ? = 3
[[[[]]],[],[[]],[[]]]
 => [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]
 => ? = 3
[[[[]]],[],[[[]]],[]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3
[[[[]]],[],[[],[],[]]]
 => [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]
 => ? = 3
[[[[]]],[],[[],[[]]]]
 => [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]
 => ? = 3
[[[[]]],[],[[[]],[]]]
 => [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]
 => ? = 3
[[[[]]],[],[[[],[]]]]
 => [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]
 => ? = 3
[[[[]]],[[]],[],[[]]]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[[[[]]],[[]],[[]],[]]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3
[[[[]]],[[]],[[],[]]]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[[[[]]],[[[]]],[],[]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
[[[[]]],[[],[]],[[]]]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3
[[[[]]],[[],[],[]],[]]
 => [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3
[[[[]]],[[],[[]]],[]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3
[[[[]]],[[[]],[]],[]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3
[[[[]]],[[[],[]]],[]]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3
[[[[]]],[[[[]]]],[]]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 4
[[[[]]],[[],[],[],[]]]
 => [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]
 => ? = 3
[[[[]]],[[],[[]],[]]]
 => [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]
 => ? = 3
[[[[]]],[[[[[]]]]]]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5
[[[[[]]]],[],[],[],[]]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
[[[[[]]]],[],[],[[]]]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4
[[[[[]]]],[],[[]],[]]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4
[[[[[]]]],[],[[],[]]]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4
[[[[[]]]],[[]],[],[]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4
[[[[[]]]],[[]],[[]]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
[[[[[]]]],[[],[]],[]]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4
[[[[[]]]],[[],[],[]]]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
[[[[[]]]],[[[]],[]]]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
[[[[[]]]],[[[[]]]]]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[[[[[[]]]]],[],[],[]]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 5
[[[[]],[],[]],[],[[]]]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[[[[],[]],[]],[],[[]]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[[[[[]]],[]],[],[[]]]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[[[[]],[],[]],[[]],[]]
 => [1,1,1,0,0,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3
[[[[],[]],[]],[[]],[]]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3
[[[[[]]],[]],[[]],[]]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3
[[[[]],[],[]],[[],[]]]
 => [1,1,1,0,0,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[[[[],[]],[]],[[],[]]]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[[[[[]]],[]],[[],[]]]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3
[[[[[[]]]]],[[[]]]]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5
[[[[]],[],[[]]],[],[]]
 => [1,1,1,0,0,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
[[[[]],[[],[]]],[],[]]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
[[[[[]]],[],[]],[],[]]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4
[[[[],[]],[[]]],[],[]]
 => [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3

Description

The length of the maximal rise of a Dyck path.

Matching statistic: St000442
(load all 4 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 87%●distinct values known / distinct values provided: 78%

Values

[]
 => []
 => []
 => ? = 0 - 1
[[]]
 => [1,0]
 => [1,0]
 => ? = 1 - 1
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 4 - 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 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]
 => ? = 3 - 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]
 => ? = 3 - 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]
 => ? = 3 - 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]
 => ? = 3 - 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]
 => ? = 3 - 1
[[[[]]],[],[[[]]],[]]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3 - 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]
 => ? = 3 - 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]
 => ? = 3 - 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]
 => ? = 3 - 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]
 => ? = 3 - 1
[[[[]]],[[]],[],[[]]]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 - 1
[[[[]]],[[]],[[]],[]]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3 - 1
[[[[]]],[[]],[[],[]]]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 - 1
[[[[]]],[[[]]],[],[]]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 - 1
[[[[]]],[[],[]],[[]]]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 3 - 1
[[[[]]],[[],[],[]],[]]
 => [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3 - 1
[[[[]]],[[],[[]]],[]]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3 - 1
[[[[]]],[[[]],[]],[]]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3 - 1
[[[[]]],[[[],[]]],[]]
 => [1,1,1,0,0,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 3 - 1
[[[[]]],[[[[]]]],[]]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 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]
 => ? = 3 - 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]
 => ? = 3 - 1
[[[[]]],[[[[[]]]]]]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 5 - 1
[[[[[]]]],[],[],[],[]]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 4 - 1
[[[[[]]]],[],[],[[]]]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4 - 1
[[[[[]]]],[],[[]],[]]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4 - 1
[[[[[]]]],[],[[],[]]]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 4 - 1
[[[[[]]]],[[]],[],[]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4 - 1
[[[[[]]]],[[]],[[]]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4 - 1
[[[[[]]]],[[],[]],[]]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4 - 1
[[[[[]]]],[[],[],[]]]
 => [1,1,1,1,0,0,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4 - 1
[[[[[]]]],[[[]],[]]]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4 - 1
[[[[[]]]],[[[[]]]]]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
[[[[[[]]]]],[],[],[]]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 5 - 1
[[[[]],[],[]],[],[[]]]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 - 1
[[[[],[]],[]],[],[[]]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 - 1
[[[[[]]],[]],[],[[]]]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 - 1
[[[[]],[],[]],[[]],[]]
 => [1,1,1,0,0,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3 - 1
[[[[],[]],[]],[[]],[]]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3 - 1
[[[[[]]],[]],[[]],[]]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3 - 1
[[[[]],[],[]],[[],[]]]
 => [1,1,1,0,0,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 - 1
[[[[],[]],[]],[[],[]]]
 => [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 - 1
[[[[[]]],[]],[[],[]]]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 3 - 1
[[[[[[]]]]],[[[]]]]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 5 - 1
[[[[]],[],[[]]],[],[]]
 => [1,1,1,0,0,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 - 1
[[[[]],[[],[]]],[],[]]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 - 1
[[[[[]]],[],[]],[],[]]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4 - 1
[[[[],[]],[[]]],[],[]]
 => [1,1,1,0,1,0,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3 - 1

Description

The maximal area to the right of an up step of a Dyck path.

Matching statistic: St001058

Mapping

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: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 89%

Values

[]
 => []
 => []
 => []
 => ? = 0
[[]]
 => [1,0]
 => [1,0]
 => [[]]
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => [[[]]]
 => 1
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [[],[]]
 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [[],[[]]]
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [[[],[]]]
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [[[]],[]]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [[],[],[]]
 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [[[],[[]]]]
 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 3
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[[],[[[]]]]]
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[[[]]]],[]]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[[[],[[]]]]]
 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[],[[],[[]]]]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[[[]]],[]]]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[[],[],[[]]]]
 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => 3
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[[],[[]]],[]]
 => 2
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[],[]]
 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[[],[]]]]]
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[[],[[],[]]]]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[[[],[]]],[]]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[[[[]],[]]]]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[[[],[],[]]]]
 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[[[]],[[]]]]
 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[[],[[]],[]]]
 => 3
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[[[],[]],[]]]
 => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[[[]],[],[]]]
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => 4
[[],[],[],[],[],[],[],[]]
 => [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]
 => [[[[[[[[[]]]]]]]]]
 => ? = 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]
 => [[[],[[[[[[]]]]]]]]
 => ? = 2
[[],[],[],[],[],[[],[]]]
 => [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]
 => [[[[[[[[]]]]]]],[]]
 => ? = 2
[[],[],[],[],[[]],[],[]]
 => [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]
 => [[[[],[[[[[]]]]]]]]
 => ? = 2
[[],[],[],[],[[],[]],[]]
 => [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]
 => [[[[[[[[]]]]]],[]]]
 => ? = 2
[[],[],[],[],[[[]]],[]]
 => [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]
 => [[[],[],[[[[[]]]]]]]
 => ? = 3
[[],[],[],[],[[[]],[]]]
 => [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]
 => [[[],[[[[[]]]]]],[]]
 => ? = 2
[[],[],[],[],[[[],[]]]]
 => [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]
 => [[[[[[[]]]]]],[],[]]
 => ? = 3
[[],[],[],[[]],[],[],[]]
 => [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]
 => [[[[[],[[[[]]]]]]]]
 => ? = 2
[[],[],[],[[]],[[],[]]]
 => [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]
 => [[[[],[[[[]]]]]],[]]
 => ? = 2
[[],[],[],[[],[]],[],[]]
 => [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]
 => [[[[[[[[]]]]],[]]]]
 => ? = 2
[[],[],[],[[],[[]]],[]]
 => [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]
 => [[[],[[[[[]]]]],[]]]
 => ? = 3
[[],[],[],[[[],[]]],[]]
 => [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]
 => [[[[[[[]]]]],[],[]]]
 => ? = 3
[[],[],[],[[],[],[],[]]]
 => [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]
 => [[[[[[[]]]]]],[[]]]
 => ? = 2
[[],[],[],[[],[[]],[]]]
 => [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]
 => [[[[[[[]]]]],[]],[]]
 => ? = 2
[[],[],[],[[[],[]],[]]]
 => [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]
 => [[[],[]],[[[[[]]]]]]
 => ? = 3
[[],[],[],[[[[]]],[]]]
 => [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]
 => [[[],[],[[[[]]]]],[]]
 => ? = 3
[[],[],[],[[[[],[]]]]]
 => [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]
 => [[[[[[]]]]],[],[],[]]
 => ? = 4
[[],[],[[]],[],[],[],[]]
 => [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]
 => [[[[[[],[[[]]]]]]]]
 => ? = 2
[[],[],[[]],[],[[],[]]]
 => [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]
 => [[[[[],[[[]]]]]],[]]
 => ? = 2
[[],[],[[]],[[[]],[]]]
 => [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]
 => [[[],[[],[[[]]]]],[]]
 => ? = 2
[[],[],[[],[]],[],[],[]]
 => [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]
 => [[[[[[[[]]]],[]]]]]
 => ? = 2
[[],[],[[[]]],[],[],[]]
 => [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]
 => [[[[[],[],[[[]]]]]]]
 => ? = 3
[[],[],[[],[]],[[],[]]]
 => [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]
 => [[[[[[[]]]],[]]],[]]
 => ? = 2
[[],[],[[[[[]]]]],[]]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [[[],[],[],[],[[[]]]]]
 => ? = 5
[[],[],[[],[],[],[],[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [[[[]]],[[[[[]]]]]]
 => ? = 2
[[],[],[[],[],[[]],[]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
 => [[[[]],[[[[]]]]],[]]
 => ? = 2
[[],[],[[],[[],[]],[]]]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [[[],[[[[]]]]],[[]]]
 => ? = 3
[[],[],[[],[[[]]],[]]]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0]
 => [[[],[[[[]]]],[]],[]]
 => ? = 3
[[],[],[[[]],[],[],[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
 => [[[[],[[[]]]]],[[]]]
 => ? = 2
[[],[],[[[]],[[]],[]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
 => [[[[],[[[]]]],[]],[]]
 => ? = 2
[[],[],[[[],[]],[],[]]]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
 => [[[[[[]]]]],[[],[]]]
 => ? = 3
[[],[],[[[],[],[]],[]]]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
 => [[[[[[]]]],[]],[[]]]
 => ? = 3
[[],[],[[[],[[]]],[]]]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
 => [[[[[[]]]],[],[]],[]]
 => ? = 3
[[],[],[[[[]],[]],[]]]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [[[],[]],[[],[[[]]]]]
 => ? = 4
[[],[],[[[[],[]]],[]]]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [[[],[],[]],[[[[]]]]]
 => ? = 4
[[],[],[[[[[]]]],[]]]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [[[],[],[],[[[]]]],[]]
 => ? = 4
[[],[],[[[],[[]],[]]]]
 => [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [[[],[]],[],[[[[]]]]]
 => ? = 3
[[],[],[[[[],[]],[]]]]
 => [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [[[],[]],[[[[]]]],[]]
 => ? = 3
[[],[],[[[[[]]],[]]]]
 => [1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [[[],[],[[[]]]],[],[]]
 => ? = 3
[[],[],[[[[[]],[]]]]]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [[[],[[[]]]],[],[],[]]
 => ? = 4
[[],[],[[[[[],[]]]]]]
 => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [[[[[]]]],[],[],[],[]]
 => ? = 5
[[],[[]],[],[],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [[[[[[[],[[]]]]]]]]
 => ? = 2
[[],[[]],[],[],[[],[]]]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [[[[[[],[[]]]]]],[]]
 => ? = 2
[[],[[]],[[]],[[]],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [[[],[[],[[],[[]]]]]]
 => ? = 2
[[],[[]],[[],[],[]],[]]
 => [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
 => [[[[]],[[[],[[]]]]]]
 => ? = 2
[[],[[]],[[[]],[]],[]]
 => [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,1,0,0]
 => [[[[],[[],[[]]]],[]]]
 => ? = 2
[[],[[]],[[[[]]]],[]]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [[[],[],[],[[],[[]]]]]
 => ? = 4
[[],[[]],[[[],[]],[]]]
 => [1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [[[],[]],[[[],[[]]]]]
 => ? = 3

Description

The breadth of the ordered tree. This is the maximal number of nodes having the same depth.

Matching statistic: St000439

Mapping

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: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 89%

Values

[]
 => []
 => []
 => []
 => ? = 0 + 1
[[]]
 => [1,0]
 => [1,0]
 => [1,0]
 => 2 = 1 + 1
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 3 = 2 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 3 = 2 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 3 = 2 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 4 = 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]
 => 2 = 1 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 4 = 3 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 4 = 3 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 4 = 3 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 4 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 4 = 3 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 4 = 3 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 4 = 3 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 4 = 3 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 4 = 3 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 5 = 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]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 4 + 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]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 2 + 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]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 2 + 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]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 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]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 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]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 4 + 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]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 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]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2 + 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]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => ? = 2 + 1
[[],[],[[]],[[[[]]]]]
 => [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]
 => ? = 4 + 1
[[],[],[[],[]],[],[[]]]
 => [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]
 => ? = 2 + 1
[[],[],[[],[]],[[],[]]]
 => [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]
 => ? = 2 + 1
[[],[],[[[]]],[[[]]]]
 => [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]
 => ? = 3 + 1
[[],[],[[[[]]]],[[]]]
 => [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]
 => ? = 4 + 1
[[],[],[[[[[]]]]],[]]
 => [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]
 => ? = 5 + 1
[[],[],[[],[],[[[]]]]]
 => [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]
 => ? = 4 + 1
[[],[],[[],[[],[[]]]]]
 => [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [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]
 => ? = 4 + 1
[[],[],[[],[[[],[]]]]]
 => [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [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]
 => ? = 4 + 1
[[],[],[[[]],[],[[]]]]
 => [1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [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]
 => ? = 3 + 1
[[],[],[[[]],[[],[]]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [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]
 => ? = 3 + 1
[[],[],[[[]],[[[]]]]]
 => [1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [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]
 => ? = 4 + 1
[[],[],[[[[]]],[],[]]]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [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]
 => ? = 4 + 1
[[],[],[[[],[]],[[]]]]
 => [1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [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]
 => ? = 3 + 1
[[],[],[[[[]]],[[]]]]
 => [1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [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]
 => ? = 3 + 1
[[],[],[[[[]],[]],[]]]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [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]
 => ? = 4 + 1
[[],[],[[[[],[]]],[]]]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [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]
 => ? = 4 + 1
[[],[],[[[[[]]]],[]]]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [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]
 => ? = 4 + 1
[[],[],[[[],[],[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [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]
 => ? = 3 + 1
[[],[],[[[],[],[[]]]]]
 => [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [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]
 => ? = 4 + 1
[[],[],[[[],[[]],[]]]]
 => [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [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]
 => ? = 3 + 1
[[],[],[[[],[[],[]]]]]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [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]
 => ? = 4 + 1
[[],[],[[[[]],[],[]]]]
 => [1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [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]
 => ? = 3 + 1
[[],[],[[[[]],[[]]]]]
 => [1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [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]
 => ? = 4 + 1
[[],[],[[[[],[]],[]]]]
 => [1,0,1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [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]
 => ? = 3 + 1
[[],[],[[[[[]]],[]]]]
 => [1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [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]
 => ? = 3 + 1
[[],[],[[[[],[],[]]]]]
 => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [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]
 => ? = 4 + 1
[[],[],[[[[[]],[]]]]]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [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]
 => ? = 4 + 1
[[],[[]],[],[],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 2 + 1
[[],[[]],[],[],[[],[]]]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => ? = 2 + 1
[[],[[]],[],[[[[]]]]]
 => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
 => ? = 4 + 1
[[],[[]],[[]],[[[]]]]
 => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 3 + 1
[[],[[]],[[[]]],[[]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
 => ? = 3 + 1
[[],[[]],[[[[]]]],[]]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
 => ? = 4 + 1
[[],[[]],[[],[],[[]]]]
 => [1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 3 + 1
[[],[[]],[[],[[],[]]]]
 => [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 3 + 1
[[],[[]],[[],[[[]]]]]
 => [1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
 => ? = 4 + 1
[[],[[]],[[[]],[],[]]]
 => [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
 => ? = 3 + 1
[[],[[]],[[[]],[[]]]]
 => [1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 3 + 1
[[],[[]],[[[],[]],[]]]
 => [1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
 => ? = 3 + 1

Description

The position of the first down step of a Dyck path.

Matching statistic: St000684

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 89%

Values

[]
 => []
 => []
 => []
 => ? = 0
[[]]
 => [1,0]
 => [1,0]
 => [1,0]
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[[],[[]],[]]
 => [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
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[[],[[[]]]]
 => [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
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[[[]],[[]]]
 => [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
[[[],[]],[]]
 => [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
[[[[]]],[]]
 => [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
[[[],[],[]]]
 => [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
[[[],[[]]]]
 => [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
[[[[]],[]]]
 => [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
[[[[],[]]]]
 => [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
[[[[[]]]]]
 => [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
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[],[[]],[[]]]
 => [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
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[],[[[]]],[]]
 => [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
[[],[[],[],[]]]
 => [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
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[],[[[]],[]]]
 => [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
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[],[[[[]]]]]
 => [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
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[[[]],[[]],[]]
 => [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
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[[[]],[[[]]]]
 => [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
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[[],[]],[[]]]
 => [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
[[[[]]],[[]]]
 => [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
[[[],[],[]],[]]
 => [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
[[[],[[]]],[]]
 => [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
[[[[]],[]],[]]
 => [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
[[[[],[]]],[]]
 => [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
[[[[[]]]],[]]
 => [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
[[],[],[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[[],[],[],[],[],[],[[]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
[[],[],[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
[[],[],[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
[[],[],[],[],[],[[[]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 3
[[],[],[],[],[[]],[],[]]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
[[],[],[],[],[[]],[[]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[],[],[[],[]],[]]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
[[],[],[],[],[[[]]],[]]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 3
[[],[],[],[],[[],[],[]]]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[],[],[[],[[]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 3
[[],[],[],[],[[[]],[]]]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[],[],[[[],[]]]]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 3
[[],[],[],[],[[[[]]]]]
 => [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]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
[[],[],[],[[]],[],[],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[[],[],[],[[]],[],[[]]]
 => [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]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
[[],[],[],[[]],[[],[]]]
 => [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]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
[[],[],[],[[],[]],[],[]]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
[[],[],[],[[],[]],[[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[],[[[]]],[[]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 3
[[],[],[],[[],[[]]],[]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 3
[[],[],[],[[[],[]]],[]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 3
[[],[],[],[[],[],[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[],[[],[[]],[]]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[],[[],[[[]]]]]
 => [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]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
[[],[],[],[[[]],[],[]]]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 3
[[],[],[],[[[],[]],[]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 3
[[],[],[],[[[[]]],[]]]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 3
[[],[],[],[[[],[[]]]]]
 => [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]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 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]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 4
[[],[],[[]],[],[],[],[]]
 => [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]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[[],[],[[]],[],[],[[]]]
 => [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]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2
[[],[],[[]],[],[[],[]]]
 => [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]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2
[[],[],[[]],[[]],[[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[[]],[[],[],[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[[]],[[[]],[]]]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[[]],[[[[]]]]]
 => [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]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
[[],[],[[],[]],[],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[[],[],[[[]]],[],[],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[[],[],[[],[]],[],[[]]]
 => [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]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
[[],[],[[],[]],[[],[]]]
 => [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]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
[[],[],[[[]]],[[[]]]]
 => [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]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[[],[],[[],[],[]],[[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[[],[[]]],[[]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 3
[[],[],[[[]],[]],[[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[[[],[]]],[[]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 3
[[],[],[[[[]]]],[[]]]
 => [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]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[[],[],[[],[],[],[],[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[[],[],[[],[],[[]],[]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2

Description

The global dimension of the LNakayama algebra associated to a Dyck path. An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with

$n$ points for

$n \geq 2$ . Number those points from the left to the right by

$0,1,\ldots,n-1$ . The algebra is then uniquely determined by the dimension

$c_i$ of the projective indecomposable modules at point

$i$ . Such algebras are then uniquely determined by lists of the form

$[c_0,c_1,...,c_{n-1}]$ with the conditions:

$c_{n-1}=1$ and

$c_i -1 \leq c_{i+1}$ for all

$i$ . The number of such algebras is then the

$n-1$ -st Catalan number

$C_{n-1}$ . One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0]. Conjecture: that there is an explicit bijection between

$n$ -LNakayama algebras with global dimension bounded by

$m$ and Dyck paths with height at most

$m$ . Examples: * For

$m=2$ , the number of Dyck paths with global dimension at most

$m$ starts for

$n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192. * For

$m=3$ , the number of Dyck paths with global dimension at most

$m$ starts for

$n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.

Matching statistic: St000306

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 78%

Values

[]
 => []
 => []
 => []
 => ? = 0 - 1
[[]]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
[[],[]]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 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]
 => 3 = 4 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1 = 2 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2 = 3 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1 = 2 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 4 - 1
[[],[],[],[[]],[[]]]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[],[],[[],[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[],[],[[[]],[]]]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[],[],[[[[]]]]]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
[[],[],[[]],[],[[]]]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[],[[]],[[],[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[],[[]],[[[]]]]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[],[[],[]],[[]]]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[],[[[[]]]],[]]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 4 - 1
[[],[],[[],[],[],[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[],[[],[],[[]]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[],[[],[[]],[]]]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[],[[],[[],[]]]]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[],[[],[[[]]]]]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
[[],[],[[[]],[[]]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[],[[[],[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[],[[[],[[]]]]]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
[[],[],[[[[]],[]]]]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[],[[[[],[]]]]]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
[[],[],[[[[[]]]]]]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5 - 1
[[],[[]],[],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[[]],[],[[]],[]]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[[]],[],[[],[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[[]],[[]],[],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[[]],[[],[]],[]]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[[]],[[[]]],[]]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[[],[]],[],[[]]]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[[],[]],[[],[]]]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[[],[]],[[[]]]]
 => [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[[],[],[]],[],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 2 - 1
[[],[[[]],[]],[],[]]
 => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => ? = 2 - 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]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 4 - 1
[[],[[],[],[[]]],[]]
 => [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[[],[[],[]]],[]]
 => [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[[],[[[]]]],[]]
 => [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 4 - 1
[[],[[[]],[[]]],[]]
 => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[[[],[],[]]],[]]
 => [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[[[],[[]]]],[]]
 => [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 4 - 1
[[],[[[[]],[]]],[]]
 => [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[[[[],[]]]],[]]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 4 - 1
[[],[[],[],[],[[]]]]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[[],[],[[],[]]]]
 => [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[[],[[]],[[]]]]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[[],[[],[],[]]]]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[[],[[[]],[]]]]
 => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3 - 1
[[],[[],[[[[]]]]]]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5 - 1
[[],[[[],[[[]]]]]]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5 - 1
[[],[[[[],[[]]]]]]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5 - 1
[[],[[[[[],[]]]]]]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 5 - 1

Description

The bounce count of a Dyck path. For a Dyck path

$D$ of length

$2n$ , this is the number of points

$(i,i)$ for

$1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of

$D$ .

The following 40 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001809The index of the step at the first peak of maximal height in a Dyck path. St000141The maximum drop size of a permutation. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. 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. St000094The depth of an ordered tree. 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. 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

$K[x]/(x^n)$ .