- FindStat

Your data matches 13 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000383

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The last part of an integer composition.

Matching statistic: St001107
(load all 7 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.

Matching statistic: St000974
(load all 2 compositions to match this statistic)

Mapping

Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00246: Ordered trees —rotate⟶ Ordered trees
St000974: Ordered trees ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the trunk of an ordered tree. This is the length of the path from the root to the first vertex which has not exactly one child.

Matching statistic: St000297

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 71% ●values known / values provided: 72%●distinct values known / distinct values provided: 71%

Values

[]
 => []
 => []
 =>  => ? = 1
[[]]
 => [1,0]
 => []
 =>  => ? = 2
[[],[]]
 => [1,0,1,0]
 => [1]
 => 10 => 1
[[[]]]
 => [1,1,0,0]
 => []
 =>  => ? = 3
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1]
 => 1010 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1]
 => 110 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2]
 => 100 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1]
 => 10 => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => []
 =>  => ? = 4
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 101010 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 11010 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 100110 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 10110 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1110 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 10100 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1100 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 10010 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1010 => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 110 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => 10 => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => []
 =>  => ? = 5
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 10101010 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 1101010 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 10011010 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 1011010 => 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 111010 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 10100110 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 1100110 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => 10010110 => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 10001110 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 1010110 => 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 110110 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 1001110 => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 101110 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 11110 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 1010100 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 110100 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1001100 => 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 101100 => 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 11100 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 1010010 => 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 101000 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 110010 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 11000 => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 1001010 => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 1000110 => 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 100100 => 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 100010 => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 10000 => 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => 101010 => 1
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => 11010 => 2
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => 100110 => 1
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 10110 => 1
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 =>  => ? = 6
[[[[[[[]]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 =>  => ? = 7
[[],[],[],[],[],[[]]]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,2,1]
 => 11010101010 => ? = 2
[[],[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,2,1]
 => 100110101010 => ? = 1
[[],[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2,1]
 => 10110101010 => ? = 1
[[],[],[],[[]],[],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,2,1]
 => 101001101010 => ? = 1
[[],[],[],[[]],[[]]]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,2,1]
 => 11001101010 => ? = 2
[[],[],[],[[],[]],[]]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,2,1]
 => 100101101010 => ? = 1
[[],[],[],[[[]]],[]]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,3,3,2,1]
 => 100011101010 => ? = 1
[[],[],[],[[],[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3,2,1]
 => 10101101010 => ? = 1
[[],[],[],[[],[[]]]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => 1101101010 => ? = 2
[[],[],[],[[[]],[]]]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,3,3,2,1]
 => 10011101010 => ? = 1
[[],[],[],[[[],[]]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,3,3,2,1]
 => 1011101010 => ? = 1
[[],[],[[]],[],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2,1]
 => 101010011010 => ? = 1
[[],[],[[]],[],[[]]]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,2,1]
 => 11010011010 => ? = 2
[[],[],[[]],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2,1]
 => 100110011010 => ? = 1
[[],[],[[]],[[],[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2,1]
 => 10110011010 => ? = 1
[[],[],[[]],[[[]]]]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,4,4,2,2,1]
 => 1110011010 => ? = 3
[[],[],[[],[]],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2,1]
 => 101001011010 => ? = 1
[[],[],[[[]]],[],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,2,2,1]
 => 101000111010 => ? = 1
[[],[],[[],[]],[[]]]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2,1]
 => 11001011010 => ? = 2
[[],[],[[[]]],[[]]]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,5,2,2,2,1]
 => 11000111010 => ? = 2
[[],[],[[],[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,2,1]
 => 100101011010 => ? = 1
[[],[],[[],[[]]],[]]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,3,3,2,2,1]
 => 100011011010 => ? = 1
[[],[],[[[]],[]],[]]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [6,4,2,2,2,1]
 => 100100111010 => ? = 1
[[],[],[[[],[]]],[]]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [6,3,2,2,2,1]
 => 100010111010 => ? = 1
[[],[],[[[[]]]],[]]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2,1]
 => 100001111010 => ? = 1
[[],[],[[],[],[],[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,2,1]
 => 10101011010 => ? = 1
[[],[],[[],[],[[]]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [4,4,3,2,2,1]
 => 1101011010 => ? = 2
[[],[],[[],[[]],[]]]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [5,3,3,2,2,1]
 => 10011011010 => ? = 1
[[],[],[[],[[],[]]]]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [4,3,3,2,2,1]
 => 1011011010 => ? = 1
[[],[],[[[]],[],[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [5,4,2,2,2,1]
 => 10100111010 => ? = 1
[[],[],[[[]],[[]]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [4,4,2,2,2,1]
 => 1100111010 => ? = 2
[[],[],[[[],[]],[]]]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [5,3,2,2,2,1]
 => 10010111010 => ? = 1
[[],[],[[[[]]],[]]]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,2,2,2,1]
 => 10001111010 => ? = 1
[[],[],[[[],[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,2,2,1]
 => 1010111010 => ? = 1
[[],[],[[[[]],[]]]]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,2,2,2,1]
 => 1001111010 => ? = 1
[[],[[]],[],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,1]
 => 101010100110 => ? = 1
[[],[[]],[],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,1,1]
 => 11010100110 => ? = 2
[[],[[]],[],[[]],[]]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,1,1]
 => 100110100110 => ? = 1
[[],[[]],[],[[],[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,1,1]
 => 10110100110 => ? = 1
[[],[[]],[],[[[]]]]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,1,1]
 => 1110100110 => ? = 3
[[],[[]],[[]],[],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,1,1]
 => 101001100110 => ? = 1
[[],[[]],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,1,1]
 => 11001100110 => ? = 2
[[],[[]],[[],[]],[]]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,1,1]
 => 100101100110 => ? = 1

Description

The number of leading ones in a binary word.

Matching statistic: St000993
(load all 2 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 71%

Values

[]
 => []
 => []
 => ? = 1
[[]]
 => [1,0]
 => []
 => ? = 2
[[],[]]
 => [1,0,1,0]
 => [1]
 => ? = 1
[[[]]]
 => [1,1,0,0]
 => []
 => ? = 3
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1]
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2]
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1]
 => ? = 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => []
 => ? = 4
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => []
 => ? = 5
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => 1
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => 2
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => 1
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 1
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 3
[[[[]],[],[]]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 1
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => 2
[[[[[],[]]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 1
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 6
[[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => ? = 1
[[],[],[],[],[[]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => ? = 2
[[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => ? = 1
[[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => ? = 1
[[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => ? = 1
[[],[],[[]],[[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => ? = 2
[[],[],[[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => ? = 1
[[],[[]],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => ? = 1
[[],[[]],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => ? = 2
[[],[[]],[[]],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => ? = 1
[[],[[],[]],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => ? = 1
[[[]],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => ? = 1
[[[]],[],[],[[]]]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => ? = 2
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2]
 => ? = 1
[[[]],[[]],[],[]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2]
 => ? = 1
[[[],[]],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => ? = 1
[[[[[[],[]]]]]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 1
[[[[[[[]]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 7
[[],[],[],[],[],[[]]]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,2,1]
 => ? = 2
[[],[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,2,1]
 => ? = 1
[[],[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2,1]
 => ? = 1
[[],[],[],[[]],[],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,2,1]
 => ? = 1
[[],[],[],[[]],[[]]]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,2,1]
 => ? = 2
[[],[],[],[[],[]],[]]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,2,1]
 => ? = 1
[[],[],[],[[[]]],[]]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,3,3,2,1]
 => ? = 1
[[],[],[],[[],[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3,2,1]
 => ? = 1
[[],[],[],[[],[[]]]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => ? = 2
[[],[],[],[[[]],[]]]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,3,3,2,1]
 => ? = 1
[[],[],[],[[[],[]]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,3,3,2,1]
 => ? = 1
[[],[],[[]],[],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2,1]
 => ? = 1
[[],[],[[]],[],[[]]]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,2,1]
 => ? = 2
[[],[],[[]],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2,1]
 => ? = 1
[[],[],[[]],[[],[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2,1]
 => ? = 1
[[],[],[[]],[[[]]]]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,4,4,2,2,1]
 => ? = 3
[[],[],[[],[]],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2,1]
 => ? = 1
[[],[],[[[]]],[],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,2,2,1]
 => ? = 1
[[],[],[[],[]],[[]]]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2,1]
 => ? = 2
[[],[],[[[]]],[[]]]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,5,2,2,2,1]
 => ? = 2
[[],[],[[],[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,2,1]
 => ? = 1
[[],[],[[],[[]]],[]]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,3,3,2,2,1]
 => ? = 1

Description

The multiplicity of the largest part of an integer partition.

Matching statistic: St000733

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 71%

Values

[]
 => []
 => []
 => []
 => ? = 1
[[]]
 => [1,0]
 => []
 => []
 => ? = 2
[[],[]]
 => [1,0,1,0]
 => [1]
 => [[1]]
 => 1
[[[]]]
 => [1,1,0,0]
 => []
 => []
 => ? = 3
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1]
 => [[1,3],[2]]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1]
 => [[1],[2]]
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2]
 => [[1,2]]
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1]
 => [[1]]
 => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => []
 => []
 => ? = 4
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [[1,3,4],[2]]
 => 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => [[1,2,3]]
 => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [[1,3],[2]]
 => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1],[2]]
 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => [[1,2]]
 => 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => [[1]]
 => 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? = 5
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [[1,3,6,10],[2,5,9],[4,8],[7]]
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8],[7]]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [[1,3,8,9],[2,5],[4,7],[6]]
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [[1,4,5,9],[2,7,8],[3],[6]]
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [[1,2,5,9],[3,4,8],[6,7]]
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[1,2,3,4]]
 => 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => 1
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 1
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 1
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? = 6
[[[[[[[]]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? = 7
[[],[],[],[],[],[[]]]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,2,1]
 => [[1,3,6,10,15],[2,5,9,14,20],[4,8,13,19],[7,12,18],[11,17],[16]]
 => ? = 2
[[],[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,2,1]
 => [[1,3,6,10,19,20],[2,5,9,14],[4,8,13,18],[7,12,17],[11,16],[15]]
 => ? = 1
[[],[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2,1]
 => [[1,3,6,10,19],[2,5,9,14],[4,8,13,18],[7,12,17],[11,16],[15]]
 => ? = 1
[[],[],[],[[]],[],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,2,1]
 => [[1,3,6,13,14,20],[2,5,9,18,19],[4,8,12],[7,11,17],[10,16],[15]]
 => ? = 1
[[],[],[],[[]],[[]]]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,2,1]
 => [[1,3,6,13,14],[2,5,9,18,19],[4,8,12],[7,11,17],[10,16],[15]]
 => ? = 2
[[],[],[],[[],[]],[]]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,2,1]
 => [[1,3,6,13,18,19],[2,5,9,17],[4,8,12],[7,11,16],[10,15],[14]]
 => ? = 1
[[],[],[],[[[]]],[]]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,3,3,2,1]
 => [[1,3,6,16,17,18],[2,5,9],[4,8,12],[7,11,15],[10,14],[13]]
 => ? = 1
[[],[],[],[[],[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3,2,1]
 => [[1,3,6,13,18],[2,5,9,17],[4,8,12],[7,11,16],[10,15],[14]]
 => ? = 1
[[],[],[],[[],[[]]]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => [[1,3,6,13],[2,5,9,17],[4,8,12],[7,11,16],[10,15],[14]]
 => ? = 2
[[],[],[],[[[]],[]]]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,3,3,2,1]
 => [[1,3,6,16,17],[2,5,9],[4,8,12],[7,11,15],[10,14],[13]]
 => ? = 1
[[],[],[],[[[],[]]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,3,3,2,1]
 => [[1,3,6,16],[2,5,9],[4,8,12],[7,11,15],[10,14],[13]]
 => ? = 1
[[],[],[[]],[],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2,1]
 => [[1,3,8,9,14,20],[2,5,12,13,19],[4,7,17,18],[6,11],[10,16],[15]]
 => ? = 1
[[],[],[[]],[],[[]]]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,2,1]
 => [[1,3,8,9,14],[2,5,12,13,19],[4,7,17,18],[6,11],[10,16],[15]]
 => ? = 2
[[],[],[[]],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2,1]
 => [[1,3,8,9,18,19],[2,5,12,13],[4,7,16,17],[6,11],[10,15],[14]]
 => ? = 1
[[],[],[[]],[[],[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2,1]
 => [[1,3,8,9,18],[2,5,12,13],[4,7,16,17],[6,11],[10,15],[14]]
 => ? = 1
[[],[],[[]],[[[]]]]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,4,4,2,2,1]
 => [[1,3,8,9],[2,5,12,13],[4,7,16,17],[6,11],[10,15],[14]]
 => ? = 3
[[],[],[[],[]],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2,1]
 => [[1,3,8,12,13,19],[2,5,11,17,18],[4,7,16],[6,10],[9,15],[14]]
 => ? = 1
[[],[],[[[]]],[],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,2,2,1]
 => [[1,3,10,11,12,18],[2,5,15,16,17],[4,7],[6,9],[8,14],[13]]
 => ? = 1
[[],[],[[],[]],[[]]]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2,1]
 => [[1,3,8,12,13],[2,5,11,17,18],[4,7,16],[6,10],[9,15],[14]]
 => ? = 2
[[],[],[[[]]],[[]]]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,5,2,2,2,1]
 => [[1,3,10,11,12],[2,5,15,16,17],[4,7],[6,9],[8,14],[13]]
 => ? = 2
[[],[],[[],[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,2,1]
 => [[1,3,8,12,17,18],[2,5,11,16],[4,7,15],[6,10],[9,14],[13]]
 => ? = 1
[[],[],[[],[[]]],[]]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,3,3,2,2,1]
 => [[1,3,8,15,16,17],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
 => ? = 1
[[],[],[[[]],[]],[]]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [6,4,2,2,2,1]
 => [[1,3,10,11,16,17],[2,5,14,15],[4,7],[6,9],[8,13],[12]]
 => ? = 1
[[],[],[[[],[]]],[]]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [6,3,2,2,2,1]
 => [[1,3,10,14,15,16],[2,5,13],[4,7],[6,9],[8,12],[11]]
 => ? = 1
[[],[],[[[[]]]],[]]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [6,2,2,2,2,1]
 => [[1,3,12,13,14,15],[2,5],[4,7],[6,9],[8,11],[10]]
 => ? = 1
[[],[],[[],[],[],[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,2,1]
 => [[1,3,8,12,17],[2,5,11,16],[4,7,15],[6,10],[9,14],[13]]
 => ? = 1
[[],[],[[],[],[[]]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [4,4,3,2,2,1]
 => [[1,3,8,12],[2,5,11,16],[4,7,15],[6,10],[9,14],[13]]
 => ? = 2
[[],[],[[],[[]],[]]]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [5,3,3,2,2,1]
 => [[1,3,8,15,16],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
 => ? = 1
[[],[],[[],[[],[]]]]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [4,3,3,2,2,1]
 => [[1,3,8,15],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
 => ? = 1
[[],[],[[],[[[]]]]]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [3,3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
 => ? = 3
[[],[],[[[]],[],[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [5,4,2,2,2,1]
 => [[1,3,10,11,16],[2,5,14,15],[4,7],[6,9],[8,13],[12]]
 => ? = 1
[[],[],[[[]],[[]]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [4,4,2,2,2,1]
 => [[1,3,10,11],[2,5,14,15],[4,7],[6,9],[8,13],[12]]
 => ? = 2
[[],[],[[[],[]],[]]]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [5,3,2,2,2,1]
 => [[1,3,10,14,15],[2,5,13],[4,7],[6,9],[8,12],[11]]
 => ? = 1
[[],[],[[[[]]],[]]]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,2,2,2,1]
 => [[1,3,12,13,14],[2,5],[4,7],[6,9],[8,11],[10]]
 => ? = 1
[[],[],[[[],[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [4,3,2,2,2,1]
 => [[1,3,10,14],[2,5,13],[4,7],[6,9],[8,12],[11]]
 => ? = 1
[[],[],[[[],[[]]]]]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [3,3,2,2,2,1]
 => [[1,3,10],[2,5,13],[4,7],[6,9],[8,12],[11]]
 => ? = 2
[[],[],[[[[]],[]]]]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,2,2,2,2,1]
 => [[1,3,12,13],[2,5],[4,7],[6,9],[8,11],[10]]
 => ? = 1
[[],[],[[[[],[]]]]]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [3,2,2,2,2,1]
 => [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
 => ? = 1
[[],[[]],[],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,1]
 => [[1,4,5,9,14,20],[2,7,8,13,19],[3,11,12,18],[6,16,17],[10],[15]]
 => ? = 1
[[],[[]],[],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,1,1]
 => [[1,4,5,9,14],[2,7,8,13,19],[3,11,12,18],[6,16,17],[10],[15]]
 => ? = 2
[[],[[]],[],[[]],[]]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,1,1]
 => [[1,4,5,9,18,19],[2,7,8,13],[3,11,12,17],[6,15,16],[10],[14]]
 => ? = 1
[[],[[]],[],[[],[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,1,1]
 => [[1,4,5,9,18],[2,7,8,13],[3,11,12,17],[6,15,16],[10],[14]]
 => ? = 1
[[],[[]],[],[[[]]]]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,1,1]
 => [[1,4,5,9],[2,7,8,13],[3,11,12,17],[6,15,16],[10],[14]]
 => ? = 3

Description

The row containing the largest entry of a standard tableau.

Matching statistic: St000907

Mapping

Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00246: Ordered trees —rotate⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%

Values

[]
 => []
 => []
 => ([],1)
 => 1
[[]]
 => [[]]
 => [[]]
 => ([(0,1)],2)
 => 2
[[],[]]
 => [[],[]]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => 1
[[[]]]
 => [[[]]]
 => [[[]]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[],[]]
 => [[],[],[]]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[[],[[]]]
 => [[[]],[]]
 => [[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2
[[[]],[]]
 => [[],[[]]]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[[[],[]]]
 => [[[],[]]]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[[[[]]]]
 => [[[[]]]]
 => [[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[],[],[],[]]
 => [[],[],[],[]]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[],[],[[]]]
 => [[[]],[],[]]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => 2
[[],[[]],[]]
 => [[],[[]],[]]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[],[[],[]]]
 => [[[],[]],[]]
 => [[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 1
[[],[[[]]]]
 => [[[[]]],[]]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 3
[[[]],[],[]]
 => [[],[],[[]]]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[[]],[[]]]
 => [[[]],[[]]]
 => [[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[[[],[]],[]]
 => [[],[[],[]]]
 => [[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 1
[[[[]]],[]]
 => [[],[[[]]]]
 => [[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[[[],[],[]]]
 => [[[],[],[]]]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[[[],[[]]]]
 => [[[[]],[]]]
 => [[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[[[[]],[]]]
 => [[[],[[]]]]
 => [[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 1
[[[[],[]]]]
 => [[[[],[]]]]
 => [[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[[[[[]]]]]
 => [[[[[]]]]]
 => [[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[[],[],[],[],[]]
 => [[],[],[],[],[]]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[[],[],[],[[]]]
 => [[[]],[],[],[]]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => 2
[[],[],[[]],[]]
 => [[],[[]],[],[]]
 => [[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[[],[],[[],[]]]
 => [[[],[]],[],[]]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => 1
[[],[],[[[]]]]
 => [[[[]]],[],[]]
 => [[[[],[],[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => 3
[[],[[]],[],[]]
 => [[],[],[[]],[]]
 => [[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[[],[[]],[[]]]
 => [[[]],[[]],[]]
 => [[[[]],[],[]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => 2
[[],[[],[]],[]]
 => [[],[[],[]],[]]
 => [[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[[],[[[]]],[]]
 => [[],[[[]]],[]]
 => [[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[[],[[],[],[]]]
 => [[[],[],[]],[]]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[[],[[],[[]]]]
 => [[[[]],[]],[]]
 => [[[],[[],[]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => 2
[[],[[[]],[]]]
 => [[[],[[]]],[]]
 => [[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[[],[[[],[]]]]
 => [[[[],[]]],[]]
 => [[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => 1
[[],[[[[]]]]]
 => [[[[[]]]],[]]
 => [[[[[],[]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => 4
[[[]],[],[],[]]
 => [[],[],[],[[]]]
 => [[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[[[]],[],[[]]]
 => [[[]],[],[[]]]
 => [[[],[[]],[]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => 2
[[[]],[[]],[]]
 => [[],[[]],[[]]]
 => [[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[[[]],[[],[]]]
 => [[[],[]],[[]]]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 1
[[[]],[[[]]]]
 => [[[[]]],[[]]]
 => [[[[[]],[]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => 3
[[[],[]],[],[]]
 => [[],[],[[],[]]]
 => [[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[[[[]]],[],[]]
 => [[],[],[[[]]]]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[[[],[]],[[]]]
 => [[[]],[[],[]]]
 => [[[[],[]],[]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => 2
[[[[]]],[[]]]
 => [[[]],[[[]]]]
 => [[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => 2
[[[],[],[]],[]]
 => [[],[[],[],[]]]
 => [[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => 1
[[[],[[]]],[]]
 => [[],[[[]],[]]]
 => [[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 1
[[[[]],[]],[]]
 => [[],[[],[[]]]]
 => [[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 1
[[[[],[]]],[]]
 => [[],[[[],[]]]]
 => [[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => 1
[[],[],[],[],[],[[]]]
 => [[[]],[],[],[],[],[]]
 => [[[],[],[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(7,6)],8)
 => ? = 2
[[],[],[],[],[[]],[]]
 => [[],[[]],[],[],[],[]]
 => [[[]],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 1
[[],[],[],[],[[],[]]]
 => [[[],[]],[],[],[],[]]
 => [[],[[],[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 1
[[],[],[],[[]],[],[]]
 => [[],[],[[]],[],[],[]]
 => [[],[[]],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 1
[[],[],[],[[]],[[]]]
 => [[[]],[[]],[],[],[]]
 => [[[[]],[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,7),(7,6)],8)
 => ? = 2
[[],[],[],[[],[]],[]]
 => [[],[[],[]],[],[],[]]
 => [[[],[]],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 1
[[],[],[],[[[]]],[]]
 => [[],[[[]]],[],[],[]]
 => [[[[]]],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? = 1
[[],[],[],[[],[],[]]]
 => [[[],[],[]],[],[],[]]
 => [[],[],[[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,6)],8)
 => ? = 1
[[],[],[],[[],[[]]]]
 => [[[[]],[]],[],[],[]]
 => [[[],[[],[],[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(6,5),(7,6)],8)
 => ? = 2
[[],[],[],[[[]],[]]]
 => [[[],[[]]],[],[],[]]
 => [[[]],[[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(7,6)],8)
 => ? = 1
[[],[],[],[[[],[]]]]
 => [[[[],[]]],[],[],[]]
 => [[],[[[],[],[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(7,5)],8)
 => ? = 1
[[],[],[[]],[],[],[]]
 => [[],[],[],[[]],[],[]]
 => [[],[],[[]],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 1
[[],[],[[]],[],[[]]]
 => [[[]],[],[[]],[],[]]
 => [[[],[[]],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,7),(7,6)],8)
 => ? = 2
[[],[],[[]],[[]],[]]
 => [[],[[]],[[]],[],[]]
 => [[[]],[[]],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 1
[[],[],[[]],[[],[]]]
 => [[[],[]],[[]],[],[]]
 => [[],[[[]],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(7,6)],8)
 => ? = 1
[[],[],[[]],[[[]]]]
 => [[[[]]],[[]],[],[]]
 => [[[[[]],[],[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,4),(4,7),(5,6),(7,5)],8)
 => ? = 3
[[],[],[[],[]],[],[]]
 => [[],[],[[],[]],[],[]]
 => [[],[[],[]],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 1
[[],[],[[[]]],[],[]]
 => [[],[],[[[]]],[],[]]
 => [[],[[[]]],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? = 1
[[],[],[[],[]],[[]]]
 => [[[]],[[],[]],[],[]]
 => [[[[],[]],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(6,7),(7,5)],8)
 => ? = 2
[[],[],[[[]]],[[]]]
 => [[[]],[[[]]],[],[]]
 => [[[[[]]],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,4),(4,6),(6,7),(7,5)],8)
 => ? = 2
[[],[],[[],[],[]],[]]
 => [[],[[],[],[]],[],[]]
 => [[[],[],[]],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ? = 1
[[],[],[[],[[]]],[]]
 => [[],[[[]],[]],[],[]]
 => [[[[]],[]],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ? = 1
[[],[],[[[]],[]],[]]
 => [[],[[],[[]]],[],[]]
 => [[[],[[]]],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ? = 1
[[],[],[[[],[]]],[]]
 => [[],[[[],[]]],[],[]]
 => [[[[],[]]],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,7),(6,5)],8)
 => ? = 1
[[],[],[[[[]]]],[]]
 => [[],[[[[]]]],[],[]]
 => [[[[[]]]],[],[],[]]
 => ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
 => ? = 1
[[],[],[[],[],[],[]]]
 => [[[],[],[],[]],[],[]]
 => [[],[],[],[[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ? = 1
[[],[],[[],[],[[]]]]
 => [[[[]],[],[]],[],[]]
 => [[[],[],[[],[],[]]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,6),(6,7),(7,5)],8)
 => ? = 2
[[],[],[[],[[]],[]]]
 => [[[],[[]],[]],[],[]]
 => [[[]],[],[[],[],[]]]
 => ([(0,7),(1,6),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 1
[[],[],[[],[[],[]]]]
 => [[[[],[]],[]],[],[]]
 => [[],[[],[[],[],[]]]]
 => ([(0,6),(1,7),(2,7),(3,7),(4,5),(5,6),(7,5)],8)
 => ? = 1
[[],[],[[],[[[]]]]]
 => [[[[[]]],[]],[],[]]
 => [[[[],[[],[],[]]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(6,4),(7,6)],8)
 => ? = 3
[[],[],[[[]],[],[]]]
 => [[[],[],[[]]],[],[]]
 => [[],[[]],[[],[],[]]]
 => ([(0,7),(1,6),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 1
[[],[],[[[]],[[]]]]
 => [[[[]],[[]]],[],[]]
 => [[[[]],[[],[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,4),(4,6),(6,5),(7,6)],8)
 => ? = 2
[[],[],[[[],[]],[]]]
 => [[[],[[],[]]],[],[]]
 => [[[],[]],[[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,5),(4,5),(5,6),(7,6)],8)
 => ? = 1
[[],[],[[[[]]],[]]]
 => [[[],[[[]]]],[],[]]
 => [[[[]]],[[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,4),(4,5),(5,6),(7,6)],8)
 => ? = 1
[[],[],[[[],[],[]]]]
 => [[[[],[],[]]],[],[]]
 => [[],[],[[[],[],[]]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,6),(5,7),(6,5)],8)
 => ? = 1
[[],[],[[[],[[]]]]]
 => [[[[[]],[]]],[],[]]
 => [[[],[[[],[],[]]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(6,5),(7,4)],8)
 => ? = 2
[[],[],[[[[]],[]]]]
 => [[[[],[[]]]],[],[]]
 => [[[]],[[[],[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(7,5)],8)
 => ? = 1
[[],[],[[[[],[]]]]]
 => [[[[[],[]]]],[],[]]
 => [[],[[[[],[],[]]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(7,4)],8)
 => ? = 1
[[],[[]],[],[],[],[]]
 => [[],[],[],[],[[]],[]]
 => [[],[],[],[[]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 1
[[],[[]],[],[],[[]]]
 => [[[]],[],[],[[]],[]]
 => [[[],[],[[]],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,7),(7,6)],8)
 => ? = 2
[[],[[]],[],[[]],[]]
 => [[],[[]],[],[[]],[]]
 => [[[]],[],[[]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 1
[[],[[]],[],[[],[]]]
 => [[[],[]],[],[[]],[]]
 => [[],[[],[[]],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(7,6)],8)
 => ? = 1
[[],[[]],[],[[[]]]]
 => [[[[]]],[],[[]],[]]
 => [[[[],[[]],[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,4),(4,7),(5,6),(7,5)],8)
 => ? = 3
[[],[[]],[[]],[],[]]
 => [[],[],[[]],[[]],[]]
 => [[],[[]],[[]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 1
[[],[[]],[[]],[[]]]
 => [[[]],[[]],[[]],[]]
 => [[[[]],[[]],[],[]]]
 => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,7),(7,6)],8)
 => ? = 2
[[],[[]],[[],[]],[]]
 => [[],[[],[]],[[]],[]]
 => [[[],[]],[[]],[],[]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 1
[[],[[]],[[[]]],[]]
 => [[],[[[]]],[[]],[]]
 => [[[[]]],[[]],[],[]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? = 1
[[],[[]],[[],[],[]]]
 => [[[],[],[]],[[]],[]]
 => [[],[],[[[]],[],[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(7,6)],8)
 => ? = 1
[[],[[]],[[],[[]]]]
 => [[[[]],[]],[[]],[]]
 => [[[],[[[]],[],[]]]]
 => ([(0,7),(1,7),(2,6),(3,4),(4,7),(6,5),(7,6)],8)
 => ? = 2
[[],[[]],[[[]],[]]]
 => [[[],[[]]],[[]],[]]
 => [[[]],[[[]],[],[]]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,7),(5,6),(7,6)],8)
 => ? = 1

Description

The number of maximal antichains of minimal length in a poset.

Matching statistic: St000234

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%

Values

[]
 => []
 => [1] => [1] => 0 = 1 - 1
[[]]
 => [1,0]
 => [2,1] => [1,2] => 1 = 2 - 1
[[],[]]
 => [1,0,1,0]
 => [3,1,2] => [3,1,2] => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => [2,3,1] => [1,2,3] => 2 = 3 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [3,4,1,2] => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [3,1,2,4] => 1 = 2 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => [4,2,1,3] => 0 = 1 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => [4,1,3,2] => 0 = 1 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 3 = 4 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [3,4,5,1,2] => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [3,4,1,2,5] => 1 = 2 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [3,5,2,1,4] => 0 = 1 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [3,5,1,4,2] => 0 = 1 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,1,2,4,5] => 2 = 3 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [4,2,5,1,3] => 0 = 1 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [4,2,1,3,5] => 1 = 2 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [4,5,3,1,2] => 0 = 1 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,2,3,1,4] => 0 = 1 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,5,1,3,2] => 0 = 1 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [4,1,3,2,5] => 1 = 2 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [5,2,1,4,3] => 0 = 1 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,1,3,4,2] => 0 = 1 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 4 = 5 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [3,4,5,6,1,2] => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [3,4,5,1,2,6] => 1 = 2 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [3,4,6,2,1,5] => 0 = 1 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [3,4,6,1,5,2] => 0 = 1 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [3,4,1,2,5,6] => 2 = 3 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [3,5,2,6,1,4] => 0 = 1 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [3,5,2,1,4,6] => 1 = 2 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [3,5,6,4,1,2] => 0 = 1 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [3,6,2,4,1,5] => 0 = 1 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [3,5,6,1,4,2] => 0 = 1 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [3,5,1,4,2,6] => 1 = 2 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [3,6,2,1,5,4] => 0 = 1 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [3,6,1,4,5,2] => 0 = 1 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [3,1,2,4,5,6] => 3 = 4 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [4,2,5,6,1,3] => 0 = 1 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [4,2,5,1,3,6] => 1 = 2 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [4,2,6,3,1,5] => 0 = 1 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [4,2,6,1,5,3] => 0 = 1 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [4,2,1,3,5,6] => 2 = 3 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [4,5,3,6,1,2] => 0 = 1 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [5,2,3,6,1,4] => 0 = 1 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [4,5,3,1,2,6] => 1 = 2 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => [5,2,3,1,4,6] => 1 = 2 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,5,6,3,1,2] => 0 = 1 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [4,6,3,2,1,5] => 0 = 1 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => [5,2,6,4,1,3] => 0 = 1 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [5,6,3,4,1,2] => 0 = 1 - 1
[[],[],[],[],[],[[]]]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,1,2,3,4,5,8,6] => [3,4,5,6,7,1,2,8] => ? = 2 - 1
[[],[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,1,2,3,4,8,5,7] => [3,4,5,6,8,2,1,7] => ? = 1 - 1
[[],[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [8,1,2,3,4,7,5,6] => [3,4,5,6,8,1,7,2] => ? = 1 - 1
[[],[],[],[[]],[],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,1,2,3,8,4,6,7] => [3,4,5,7,2,8,1,6] => ? = 1 - 1
[[],[],[],[[]],[[]]]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,1,2,3,7,4,8,6] => [3,4,5,7,2,1,6,8] => ? = 2 - 1
[[],[],[],[[],[]],[]]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [8,1,2,3,6,4,5,7] => [3,4,5,7,8,6,1,2] => ? = 1 - 1
[[],[],[],[[[]]],[]]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,1,2,3,6,8,4,7] => [3,4,5,8,2,6,1,7] => ? = 1 - 1
[[],[],[],[[],[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [8,1,2,3,7,4,5,6] => [3,4,5,7,8,1,6,2] => ? = 1 - 1
[[],[],[],[[],[[]]]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [7,1,2,3,6,4,8,5] => [3,4,5,7,1,6,2,8] => ? = 2 - 1
[[],[],[],[[[]],[]]]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,1,2,3,8,7,4,6] => [3,4,5,8,2,1,7,6] => ? = 1 - 1
[[],[],[],[[[],[]]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [8,1,2,3,6,7,4,5] => [3,4,5,8,1,6,7,2] => ? = 1 - 1
[[],[],[[]],[],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [4,1,2,8,3,5,6,7] => [3,4,6,2,7,8,1,5] => ? = 1 - 1
[[],[],[[]],[],[[]]]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,1,2,7,3,5,8,6] => [3,4,6,2,7,1,5,8] => ? = 2 - 1
[[],[],[[]],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [4,1,2,6,3,8,5,7] => [3,4,6,2,8,5,1,7] => ? = 1 - 1
[[],[],[[]],[[],[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,1,2,8,3,7,5,6] => [3,4,6,2,8,1,7,5] => ? = 1 - 1
[[],[],[[]],[[[]]]]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,1,2,6,3,7,8,5] => [3,4,6,2,1,5,7,8] => ? = 3 - 1
[[],[],[[],[]],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [8,1,2,5,3,4,6,7] => [3,4,6,7,5,8,1,2] => ? = 1 - 1
[[],[],[[[]]],[],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [4,1,2,5,8,3,6,7] => [3,4,7,2,5,8,1,6] => ? = 1 - 1
[[],[],[[],[]],[[]]]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [7,1,2,5,3,4,8,6] => [3,4,6,7,5,1,2,8] => ? = 2 - 1
[[],[],[[[]]],[[]]]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,1,2,5,7,3,8,6] => [3,4,7,2,5,1,6,8] => ? = 2 - 1
[[],[],[[],[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [8,1,2,6,3,4,5,7] => [3,4,6,7,8,5,1,2] => ? = 1 - 1
[[],[],[[],[[]]],[]]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,1,2,5,3,8,4,7] => [3,4,6,8,5,2,1,7] => ? = 1 - 1
[[],[],[[[]],[]],[]]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,8,6,3,5,7] => [3,4,7,2,8,6,1,5] => ? = 1 - 1
[[],[],[[[],[]]],[]]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [8,1,2,5,6,3,4,7] => [3,4,7,8,5,6,1,2] => ? = 1 - 1
[[],[],[[[[]]]],[]]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,5,6,8,3,7] => [3,4,8,2,5,6,1,7] => ? = 1 - 1
[[],[],[[],[],[],[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [7,1,2,8,3,4,5,6] => [3,4,6,7,8,1,2,5] => ? = 1 - 1
[[],[],[[],[],[[]]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [7,1,2,6,3,4,8,5] => [3,4,6,7,1,5,2,8] => ? = 2 - 1
[[],[],[[],[[]],[]]]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [8,1,2,5,3,7,4,6] => [3,4,6,8,5,1,7,2] => ? = 1 - 1
[[],[],[[],[[],[]]]]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [8,1,2,6,3,7,4,5] => [3,4,6,8,1,5,7,2] => ? = 1 - 1
[[],[],[[],[[[]]]]]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [6,1,2,5,3,7,8,4] => [3,4,6,1,5,2,7,8] => ? = 3 - 1
[[],[],[[[]],[],[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,1,2,8,7,3,5,6] => [3,4,7,2,8,1,6,5] => ? = 1 - 1
[[],[],[[[]],[[]]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [4,1,2,7,6,3,8,5] => [3,4,7,2,1,6,5,8] => ? = 2 - 1
[[],[],[[[],[]],[]]]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [8,1,2,5,7,3,4,6] => [3,4,7,8,5,1,6,2] => ? = 1 - 1
[[],[],[[[[]]],[]]]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,8,7,3,6] => [3,4,8,2,5,1,7,6] => ? = 1 - 1
[[],[],[[[],[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [8,1,2,7,6,3,4,5] => [3,4,7,8,1,6,5,2] => ? = 1 - 1
[[],[],[[[],[[]]]]]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [7,1,2,5,6,3,8,4] => [3,4,7,1,5,6,2,8] => ? = 2 - 1
[[],[],[[[[]],[]]]]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [4,1,2,8,6,7,3,5] => [3,4,8,2,1,6,7,5] => ? = 1 - 1
[[],[],[[[[],[]]]]]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [8,1,2,5,6,7,3,4] => [3,4,8,1,5,6,7,2] => ? = 1 - 1
[[],[[]],[],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [3,1,8,2,4,5,6,7] => [3,5,2,6,7,8,1,4] => ? = 1 - 1
[[],[[]],[],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [3,1,7,2,4,5,8,6] => [3,5,2,6,7,1,4,8] => ? = 2 - 1
[[],[[]],[],[[]],[]]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [3,1,6,2,4,8,5,7] => [3,5,2,6,8,4,1,7] => ? = 1 - 1
[[],[[]],[],[[],[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [3,1,8,2,4,7,5,6] => [3,5,2,6,8,1,7,4] => ? = 1 - 1
[[],[[]],[],[[[]]]]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,1,6,2,4,7,8,5] => [3,5,2,6,1,4,7,8] => ? = 3 - 1
[[],[[]],[[]],[],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [3,1,5,2,8,4,6,7] => [3,5,2,7,4,8,1,6] => ? = 1 - 1
[[],[[]],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,7,4,8,6] => [3,5,2,7,4,1,6,8] => ? = 2 - 1
[[],[[]],[[],[]],[]]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [3,1,8,2,6,4,5,7] => [3,5,2,7,8,6,1,4] => ? = 1 - 1
[[],[[]],[[[]]],[]]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [3,1,5,2,6,8,4,7] => [3,5,2,8,4,6,1,7] => ? = 1 - 1
[[],[[]],[[],[],[]]]
 => [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [3,1,8,2,7,4,5,6] => [3,5,2,7,8,1,6,4] => ? = 1 - 1
[[],[[]],[[],[[]]]]
 => [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [3,1,7,2,6,4,8,5] => [3,5,2,7,1,6,4,8] => ? = 2 - 1
[[],[[]],[[[]],[]]]
 => [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [3,1,5,2,8,7,4,6] => [3,5,2,8,4,1,7,6] => ? = 1 - 1

Description

The number of global ascents of a permutation. The global ascents are the integers

$i$ such that

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$ Equivalently, by the pigeonhole principle,

$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$ For

$n > 1$ it can also be described as an occurrence of the mesh pattern

$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$ or equivalently

$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$ see [3]. According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].

Matching statistic: St000221
(load all 2 compositions to match this statistic)

Mapping

Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%

Values

[]
 => .
 => ? => ? => ? = 1 - 1
[[]]
 => [.,.]
 => [1] => [1] => 1 = 2 - 1
[[],[]]
 => [[.,.],.]
 => [1,2] => [2,1] => 0 = 1 - 1
[[[]]]
 => [.,[.,.]]
 => [2,1] => [1,2] => 2 = 3 - 1
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => [2,3,1] => 0 = 1 - 1
[[],[[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => [3,2,1] => 0 = 1 - 1
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => [3,1,2] => 0 = 1 - 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => 3 = 4 - 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,3,4,2] => 1 = 2 - 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => [4,2,3,1] => 0 = 1 - 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,1,3,2] => 0 = 1 - 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,2,4,3] => 2 = 3 - 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [3,2,4,1] => 0 = 1 - 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,4,3,2] => 1 = 2 - 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,4,2,1] => 0 = 1 - 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,3,2,1] => 0 = 1 - 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [3,4,1,2] => 0 = 1 - 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,4,2,3] => 1 = 2 - 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,1,2] => 0 = 1 - 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,2,3] => 0 = 1 - 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => 4 = 5 - 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [2,3,4,5,1] => 0 = 1 - 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,3,4,5,2] => 1 = 2 - 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => [5,2,3,4,1] => 0 = 1 - 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,1,3,4,2] => 0 = 1 - 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,2,4,5,3] => 2 = 3 - 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => [4,2,3,5,1] => 0 = 1 - 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,5,3,4,2] => 1 = 2 - 1
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => [4,5,2,3,1] => 0 = 1 - 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => [5,4,2,3,1] => 0 = 1 - 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [4,5,1,3,2] => 0 = 1 - 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,5,2,4,3] => 1 = 2 - 1
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [5,4,1,3,2] => 0 = 1 - 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [5,1,2,4,3] => 0 = 1 - 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,2,3,5,4] => 3 = 4 - 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [3,2,4,5,1] => 0 = 1 - 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,4,3,5,2] => 1 = 2 - 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => [5,3,2,4,1] => 0 = 1 - 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [5,1,4,3,2] => 0 = 1 - 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,2,5,4,3] => 2 = 3 - 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [3,4,2,5,1] => 0 = 1 - 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [4,3,2,5,1] => 0 = 1 - 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [1,4,5,3,2] => 1 = 2 - 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [1,5,4,3,2] => 1 = 2 - 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [3,4,5,2,1] => 0 = 1 - 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => [5,3,4,2,1] => 0 = 1 - 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [4,3,5,2,1] => 0 = 1 - 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [4,5,3,2,1] => 0 = 1 - 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [5,4,3,2,1] => 0 = 1 - 1
[[],[],[],[],[],[[]]]
 => [[[[[[.,.],.],.],.],.],[.,.]]
 => [7,1,2,3,4,5,6] => [1,3,4,5,6,7,2] => ? = 2 - 1
[[],[],[],[],[[]],[]]
 => [[[[[[.,.],.],.],.],[.,.]],.]
 => [6,1,2,3,4,5,7] => [7,2,3,4,5,6,1] => ? = 1 - 1
[[],[],[],[],[[],[]]]
 => [[[[[.,.],.],.],.],[[.,.],.]]
 => [6,7,1,2,3,4,5] => [7,1,3,4,5,6,2] => ? = 1 - 1
[[],[],[],[[]],[],[]]
 => [[[[[[.,.],.],.],[.,.]],.],.]
 => [5,1,2,3,4,6,7] => [6,2,3,4,5,7,1] => ? = 1 - 1
[[],[],[],[[]],[[]]]
 => [[[[[.,.],.],.],[.,.]],[.,.]]
 => [7,5,1,2,3,4,6] => [1,7,3,4,5,6,2] => ? = 2 - 1
[[],[],[],[[],[]],[]]
 => [[[[[.,.],.],.],[[.,.],.]],.]
 => [5,6,1,2,3,4,7] => [6,7,2,3,4,5,1] => ? = 1 - 1
[[],[],[],[[[]]],[]]
 => [[[[[.,.],.],.],[.,[.,.]]],.]
 => [6,5,1,2,3,4,7] => [7,6,2,3,4,5,1] => ? = 1 - 1
[[],[],[],[[],[],[]]]
 => [[[[.,.],.],.],[[[.,.],.],.]]
 => [5,6,7,1,2,3,4] => [6,7,1,3,4,5,2] => ? = 1 - 1
[[],[],[],[[],[[]]]]
 => [[[[.,.],.],.],[[.,.],[.,.]]]
 => [7,5,6,1,2,3,4] => [1,7,2,4,5,6,3] => ? = 2 - 1
[[],[],[],[[[]],[]]]
 => [[[[.,.],.],.],[[.,[.,.]],.]]
 => [6,5,7,1,2,3,4] => [7,6,1,3,4,5,2] => ? = 1 - 1
[[],[],[],[[[],[]]]]
 => [[[[.,.],.],.],[.,[[.,.],.]]]
 => [6,7,5,1,2,3,4] => [7,1,2,4,5,6,3] => ? = 1 - 1
[[],[],[[]],[],[],[]]
 => [[[[[[.,.],.],[.,.]],.],.],.]
 => [4,1,2,3,5,6,7] => [5,2,3,4,6,7,1] => ? = 1 - 1
[[],[],[[]],[],[[]]]
 => [[[[[.,.],.],[.,.]],.],[.,.]]
 => [7,4,1,2,3,5,6] => [1,6,3,4,5,7,2] => ? = 2 - 1
[[],[],[[]],[[]],[]]
 => [[[[[.,.],.],[.,.]],[.,.]],.]
 => [6,4,1,2,3,5,7] => [7,5,2,3,4,6,1] => ? = 1 - 1
[[],[],[[]],[[],[]]]
 => [[[[.,.],.],[.,.]],[[.,.],.]]
 => [6,7,4,1,2,3,5] => [7,1,6,3,4,5,2] => ? = 1 - 1
[[],[],[[]],[[[]]]]
 => [[[[.,.],.],[.,.]],[.,[.,.]]]
 => [7,6,4,1,2,3,5] => [1,2,7,4,5,6,3] => ? = 3 - 1
[[],[],[[],[]],[],[]]
 => [[[[[.,.],.],[[.,.],.]],.],.]
 => [4,5,1,2,3,6,7] => [5,6,2,3,4,7,1] => ? = 1 - 1
[[],[],[[[]]],[],[]]
 => [[[[[.,.],.],[.,[.,.]]],.],.]
 => [5,4,1,2,3,6,7] => [6,5,2,3,4,7,1] => ? = 1 - 1
[[],[],[[],[]],[[]]]
 => [[[[.,.],.],[[.,.],.]],[.,.]]
 => [7,4,5,1,2,3,6] => [1,6,7,3,4,5,2] => ? = 2 - 1
[[],[],[[[]]],[[]]]
 => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => [7,5,4,1,2,3,6] => [1,7,6,3,4,5,2] => ? = 2 - 1
[[],[],[[],[],[]],[]]
 => [[[[.,.],.],[[[.,.],.],.]],.]
 => [4,5,6,1,2,3,7] => [5,6,7,2,3,4,1] => ? = 1 - 1
[[],[],[[],[[]]],[]]
 => [[[[.,.],.],[[.,.],[.,.]]],.]
 => [6,4,5,1,2,3,7] => [7,5,6,2,3,4,1] => ? = 1 - 1
[[],[],[[[]],[]],[]]
 => [[[[.,.],.],[[.,[.,.]],.]],.]
 => [5,4,6,1,2,3,7] => [6,5,7,2,3,4,1] => ? = 1 - 1
[[],[],[[[],[]]],[]]
 => [[[[.,.],.],[.,[[.,.],.]]],.]
 => [5,6,4,1,2,3,7] => [6,7,5,2,3,4,1] => ? = 1 - 1
[[],[],[[[[]]]],[]]
 => [[[[.,.],.],[.,[.,[.,.]]]],.]
 => [6,5,4,1,2,3,7] => [7,6,5,2,3,4,1] => ? = 1 - 1
[[],[],[[],[],[],[]]]
 => [[[.,.],.],[[[[.,.],.],.],.]]
 => [4,5,6,7,1,2,3] => [5,6,7,1,3,4,2] => ? = 1 - 1
[[],[],[[],[],[[]]]]
 => [[[.,.],.],[[[.,.],.],[.,.]]]
 => [7,4,5,6,1,2,3] => [1,6,7,2,4,5,3] => ? = 2 - 1
[[],[],[[],[[]],[]]]
 => [[[.,.],.],[[[.,.],[.,.]],.]]
 => [6,4,5,7,1,2,3] => [7,5,6,1,3,4,2] => ? = 1 - 1
[[],[],[[],[[],[]]]]
 => [[[.,.],.],[[.,.],[[.,.],.]]]
 => [6,7,4,5,1,2,3] => [7,1,6,2,4,5,3] => ? = 1 - 1
[[],[],[[],[[[]]]]]
 => [[[.,.],.],[[.,.],[.,[.,.]]]]
 => [7,6,4,5,1,2,3] => [1,2,7,3,5,6,4] => ? = 3 - 1
[[],[],[[[]],[],[]]]
 => [[[.,.],.],[[[.,[.,.]],.],.]]
 => [5,4,6,7,1,2,3] => [6,5,7,1,3,4,2] => ? = 1 - 1
[[],[],[[[]],[[]]]]
 => [[[.,.],.],[[.,[.,.]],[.,.]]]
 => [7,5,4,6,1,2,3] => [1,7,6,2,4,5,3] => ? = 2 - 1
[[],[],[[[],[]],[]]]
 => [[[.,.],.],[[.,[[.,.],.]],.]]
 => [5,6,4,7,1,2,3] => [6,7,5,1,3,4,2] => ? = 1 - 1
[[],[],[[[[]]],[]]]
 => [[[.,.],.],[[.,[.,[.,.]]],.]]
 => [6,5,4,7,1,2,3] => [7,6,5,1,3,4,2] => ? = 1 - 1
[[],[],[[[],[],[]]]]
 => [[[.,.],.],[.,[[[.,.],.],.]]]
 => [5,6,7,4,1,2,3] => [6,7,1,2,4,5,3] => ? = 1 - 1
[[],[],[[[],[[]]]]]
 => [[[.,.],.],[.,[[.,.],[.,.]]]]
 => [7,5,6,4,1,2,3] => [1,7,2,3,5,6,4] => ? = 2 - 1
[[],[],[[[[]],[]]]]
 => [[[.,.],.],[.,[[.,[.,.]],.]]]
 => [6,5,7,4,1,2,3] => [7,6,1,2,4,5,3] => ? = 1 - 1
[[],[],[[[[],[]]]]]
 => [[[.,.],.],[.,[.,[[.,.],.]]]]
 => [6,7,5,4,1,2,3] => [7,1,2,3,5,6,4] => ? = 1 - 1
[[],[[]],[],[],[],[]]
 => [[[[[[.,.],[.,.]],.],.],.],.]
 => [3,1,2,4,5,6,7] => [4,2,3,5,6,7,1] => ? = 1 - 1
[[],[[]],[],[],[[]]]
 => [[[[[.,.],[.,.]],.],.],[.,.]]
 => [7,3,1,2,4,5,6] => [1,5,3,4,6,7,2] => ? = 2 - 1
[[],[[]],[],[[]],[]]
 => [[[[[.,.],[.,.]],.],[.,.]],.]
 => [6,3,1,2,4,5,7] => [7,4,2,3,5,6,1] => ? = 1 - 1
[[],[[]],[],[[],[]]]
 => [[[[.,.],[.,.]],.],[[.,.],.]]
 => [6,7,3,1,2,4,5] => [7,1,5,3,4,6,2] => ? = 1 - 1
[[],[[]],[],[[[]]]]
 => [[[[.,.],[.,.]],.],[.,[.,.]]]
 => [7,6,3,1,2,4,5] => [1,2,6,4,5,7,3] => ? = 3 - 1
[[],[[]],[[]],[],[]]
 => [[[[[.,.],[.,.]],[.,.]],.],.]
 => [5,3,1,2,4,6,7] => [6,4,2,3,5,7,1] => ? = 1 - 1
[[],[[]],[[]],[[]]]
 => [[[[.,.],[.,.]],[.,.]],[.,.]]
 => [7,5,3,1,2,4,6] => [1,7,5,3,4,6,2] => ? = 2 - 1
[[],[[]],[[],[]],[]]
 => [[[[.,.],[.,.]],[[.,.],.]],.]
 => [5,6,3,1,2,4,7] => [6,7,4,2,3,5,1] => ? = 1 - 1
[[],[[]],[[[]]],[]]
 => [[[[.,.],[.,.]],[.,[.,.]]],.]
 => [6,5,3,1,2,4,7] => [7,6,4,2,3,5,1] => ? = 1 - 1
[[],[[]],[[],[],[]]]
 => [[[.,.],[.,.]],[[[.,.],.],.]]
 => [5,6,7,3,1,2,4] => [6,7,1,5,3,4,2] => ? = 1 - 1
[[],[[]],[[],[[]]]]
 => [[[.,.],[.,.]],[[.,.],[.,.]]]
 => [7,5,6,3,1,2,4] => [1,7,2,6,4,5,3] => ? = 2 - 1

Description

The number of strong fixed points of a permutation.

$i$ is called a strong fixed point of

$\pi$ if 1.

$j < i$ implies

$\pi_j < \pi_i$ , and 2.

$j > i$ implies

$\pi_j > \pi_i$ This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3]. The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].

Matching statistic: St000315

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000315: Graphs ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 100%

Values

[]
 => []
 => [] => ?
 => ? = 1 - 1
[[]]
 => [1,0]
 => [1] => ([],1)
 => 1 = 2 - 1
[[],[]]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => [2] => ([],2)
 => 2 = 3 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [3] => ([],3)
 => 3 = 4 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => 2 = 3 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => 4 = 5 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => 3 = 4 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 3 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[],[],[],[],[[]]]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[[],[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[],[[]],[],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[],[[]],[[]]]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[[],[],[],[[],[]],[]]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[],[[[]]],[]]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[],[[],[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[],[[],[[]]]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[[],[],[],[[[]],[]]]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[],[[[],[]]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[]],[],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[]],[],[[]]]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[[],[],[[]],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[]],[[],[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[]],[[[]]]]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[[],[],[[],[]],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[[]]],[],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[],[]],[[]]]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[[],[],[[[]]],[[]]]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[[],[],[[],[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[],[[]]],[]]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[[]],[]],[]]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[[],[]]],[]]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[[[]]]],[]]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[],[],[],[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[],[],[[]]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[[],[],[[],[[]],[]]]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[],[[],[]]]]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[],[[[]]]]]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[[],[],[[[]],[],[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[[]],[[]]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[[],[],[[[],[]],[]]]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[[[]]],[]]]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[[],[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[[],[[]]]]]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[[],[],[[[[]],[]]]]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[],[[[[],[]]]]]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[[]],[],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[[]],[],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[[],[[]],[],[[]],[]]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[[]],[],[[],[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[[]],[],[[[]]]]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[[],[[]],[[]],[],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[[]],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1
[[],[[]],[[],[]],[]]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[[]],[[[]]],[]]
 => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[[]],[[],[],[]]]
 => [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 - 1
[[],[[]],[[],[[]]]]
 => [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 - 1

Description

The number of isolated vertices of a graph.

The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000056The decomposition (or block) number of a permutation. St000654The first descent of a permutation.