Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000700
(load all 3 compositions to match this statistic)
Mapping
St000700: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => 1 = 0 + 1
[[],[]]
 => 1 = 0 + 1
[[[]]]
 => 2 = 1 + 1
[[],[],[]]
 => 1 = 0 + 1
[[],[[]]]
 => 1 = 0 + 1
[[[]],[]]
 => 1 = 0 + 1
[[[],[]]]
 => 2 = 1 + 1
[[[[]]]]
 => 3 = 2 + 1
[[],[],[],[]]
 => 1 = 0 + 1
[[],[],[[]]]
 => 1 = 0 + 1
[[],[[]],[]]
 => 1 = 0 + 1
[[],[[],[]]]
 => 1 = 0 + 1
[[],[[[]]]]
 => 1 = 0 + 1
[[[]],[],[]]
 => 1 = 0 + 1
[[[]],[[]]]
 => 2 = 1 + 1
[[[],[]],[]]
 => 1 = 0 + 1
[[[[]]],[]]
 => 1 = 0 + 1
[[[],[],[]]]
 => 2 = 1 + 1
[[[],[[]]]]
 => 2 = 1 + 1
[[[[]],[]]]
 => 2 = 1 + 1
[[[[],[]]]]
 => 3 = 2 + 1
[[[[[]]]]]
 => 4 = 3 + 1
[[],[],[],[],[]]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => 1 = 0 + 1
[[],[],[[]],[]]
 => 1 = 0 + 1
[[],[],[[],[]]]
 => 1 = 0 + 1
[[],[],[[[]]]]
 => 1 = 0 + 1
[[],[[]],[],[]]
 => 1 = 0 + 1
[[],[[]],[[]]]
 => 1 = 0 + 1
[[],[[],[]],[]]
 => 1 = 0 + 1
[[],[[[]]],[]]
 => 1 = 0 + 1
[[],[[],[],[]]]
 => 1 = 0 + 1
[[],[[],[[]]]]
 => 1 = 0 + 1
[[],[[[]],[]]]
 => 1 = 0 + 1
[[],[[[],[]]]]
 => 1 = 0 + 1
[[],[[[[]]]]]
 => 1 = 0 + 1
[[[]],[],[],[]]
 => 1 = 0 + 1
[[[]],[],[[]]]
 => 1 = 0 + 1
[[[]],[[]],[]]
 => 1 = 0 + 1
[[[]],[[],[]]]
 => 2 = 1 + 1
[[[]],[[[]]]]
 => 2 = 1 + 1
[[[],[]],[],[]]
 => 1 = 0 + 1
[[[[]]],[],[]]
 => 1 = 0 + 1
[[[],[]],[[]]]
 => 2 = 1 + 1
[[[[]]],[[]]]
 => 2 = 1 + 1
[[[],[],[]],[]]
 => 1 = 0 + 1
[[[],[[]]],[]]
 => 1 = 0 + 1
[[[[]],[]],[]]
 => 1 = 0 + 1
[[[[],[]]],[]]
 => 1 = 0 + 1
[[[[[]]]],[]]
 => 1 = 0 + 1
Description
The protection number of an ordered tree. This is the minimal distance from the root to a leaf.
Matching statistic: St001107
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[[],[]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
[[[]]]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 0
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 0
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 0
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 0
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 0
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 0
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 0
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 0
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 0
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 0
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => 0
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 0
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 0
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 0
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 0
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 0
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 0
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 0
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 0
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 0
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 0
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 0
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: St001038
(load all 3 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1,0]
 => ? = 0 + 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[[[]]]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1 = 0 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1 = 0 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 0 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001316
(load all 2 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001316: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => ([],1)
 => 1 = 0 + 1
[[],[]]
 => [[.,.],.]
 => [1,2] => ([],2)
 => 1 = 0 + 1
[[[]]]
 => [.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => ([],3)
 => 1 = 0 + 1
[[],[[]]]
 => [[.,.],[.,.]]
 => [1,3,2] => ([(1,2)],3)
 => 1 = 0 + 1
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => 1 = 0 + 1
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => 1 = 0 + 1
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => 1 = 0 + 1
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => ([(2,3)],4)
 => 1 = 0 + 1
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 1 = 0 + 1
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => 1 = 0 + 1
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 1 = 0 + 1
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => ([],5)
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => ([(3,4)],5)
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => ([(3,4)],5)
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [1,3,2,4,5] => ([(3,4)],5)
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => ([(3,4)],5)
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[[[[]]]]]]]
 => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
 => [1,7,6,5,4,3,2] => ([(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)
 => ? = 0 + 1
Description
The domatic number of a graph. This is the maximal size of a partition of the vertices into dominating sets.
Matching statistic: St001829
(load all 2 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001829: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => ([],1)
 => 1 = 0 + 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[[[]]]
 => [[.,.],.]
 => [1,2] => ([],2)
 => 2 = 1 + 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => ([],3)
 => 3 = 2 + 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => 3 = 2 + 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => 4 = 3 + 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[],[[[],[],[]]]]
 => [.,[.,[[[.,[.,[.,.]]],.],.]]]
 => [5,4,3,6,7,2,1] => ([(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)
 => ? = 0 + 1
[[],[[[],[],[]]],[]]
 => [.,[[[.,[.,[.,.]]],.],[.,.]]]
 => [7,4,3,2,5,6,1] => ([(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)
 => ? = 0 + 1
Description
The common independence number of a graph. The common independence number of a graph $G$ is the greatest integer $r$ such that every vertex of $G$ belongs to some independent set $X$ of vertices of cardinality at least $r$.
Matching statistic: St001322
(load all 2 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => ([],1)
 => 1 = 0 + 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[[[]]]
 => [[.,.],.]
 => [1,2] => ([],2)
 => 2 = 1 + 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 0 + 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => ([],3)
 => 3 = 2 + 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => 3 = 2 + 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => 4 = 3 + 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[],[],[],[],[],[]]
 => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(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)
 => ? = 0 + 1
[[],[],[],[],[],[[]]]
 => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => [6,7,5,4,3,2,1] => ([(0,2),(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)
 => ? = 0 + 1
[[],[],[],[],[[]],[]]
 => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => [7,5,6,4,3,2,1] => ([(0,2),(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)
 => ? = 0 + 1
[[],[],[],[],[[],[]]]
 => [.,[.,[.,[.,[[.,[.,.]],.]]]]]
 => [6,5,7,4,3,2,1] => ([(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)
 => ? = 0 + 1
[[],[],[],[[]],[],[]]
 => [.,[.,[.,[[.,.],[.,[.,.]]]]]]
 => [7,6,4,5,3,2,1] => ([(0,2),(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)
 => ? = 0 + 1
[[],[],[],[[]],[[]]]
 => [.,[.,[.,[[.,.],[[.,.],.]]]]]
 => [6,7,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[],[],[[],[]],[]]
 => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => [7,5,4,6,3,2,1] => ([(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)
 => ? = 0 + 1
[[],[],[],[[],[],[]]]
 => [.,[.,[.,[[.,[.,[.,.]]],.]]]]
 => [6,5,4,7,3,2,1] => ([(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)
 => ? = 0 + 1
[[],[],[[]],[],[],[]]
 => [.,[.,[[.,.],[.,[.,[.,.]]]]]]
 => [7,6,5,3,4,2,1] => ([(0,2),(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)
 => ? = 0 + 1
[[],[],[[]],[],[[]]]
 => [.,[.,[[.,.],[.,[[.,.],.]]]]]
 => [6,7,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[],[[]],[[]],[]]
 => [.,[.,[[.,.],[[.,.],[.,.]]]]]
 => [7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[],[[]],[[],[]]]
 => [.,[.,[[.,.],[[.,[.,.]],.]]]]
 => [6,5,7,3,4,2,1] => ([(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),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[],[[],[]],[],[]]
 => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => [7,6,4,3,5,2,1] => ([(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)
 => ? = 0 + 1
[[],[],[[],[]],[[]]]
 => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => [6,7,4,3,5,2,1] => ([(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),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[],[[],[],[]],[]]
 => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => [7,5,4,3,6,2,1] => ([(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)
 => ? = 0 + 1
[[],[],[[],[],[],[]]]
 => [.,[.,[[.,[.,[.,[.,.]]]],.]]]
 => [6,5,4,3,7,2,1] => ([(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)
 => ? = 0 + 1
[[],[[]],[],[],[],[]]
 => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
 => [7,6,5,4,2,3,1] => ([(0,2),(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)
 => ? = 0 + 1
[[],[[]],[],[],[[]]]
 => [.,[[.,.],[.,[.,[[.,.],.]]]]]
 => [6,7,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[]],[],[[]],[]]
 => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => [7,5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[]],[],[[],[]]]
 => [.,[[.,.],[.,[[.,[.,.]],.]]]]
 => [6,5,7,4,2,3,1] => ([(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),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[]],[[]],[],[]]
 => [.,[[.,.],[[.,.],[.,[.,.]]]]]
 => [7,6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[]],[[]],[[]]]
 => [.,[[.,.],[[.,.],[[.,.],.]]]]
 => [6,7,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[]],[[],[]],[]]
 => [.,[[.,.],[[.,[.,.]],[.,.]]]]
 => [7,5,4,6,2,3,1] => ([(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),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[]],[[],[],[]]]
 => [.,[[.,.],[[.,[.,[.,.]]],.]]]
 => [6,5,4,7,2,3,1] => ([(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),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[],[]],[],[],[]]
 => [.,[[.,[.,.]],[.,[.,[.,.]]]]]
 => [7,6,5,3,2,4,1] => ([(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)
 => ? = 0 + 1
[[],[[],[]],[],[[]]]
 => [.,[[.,[.,.]],[.,[[.,.],.]]]]
 => [6,7,5,3,2,4,1] => ([(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),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[],[]],[[]],[]]
 => [.,[[.,[.,.]],[[.,.],[.,.]]]]
 => [7,5,6,3,2,4,1] => ([(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),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[],[]],[[],[]]]
 => [.,[[.,[.,.]],[[.,[.,.]],.]]]
 => [6,5,7,3,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[],[],[]],[],[]]
 => [.,[[.,[.,[.,.]]],[.,[.,.]]]]
 => [7,6,4,3,2,5,1] => ([(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)
 => ? = 0 + 1
[[],[[],[],[]],[[]]]
 => [.,[[.,[.,[.,.]]],[[.,.],.]]]
 => [6,7,4,3,2,5,1] => ([(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),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[],[],[],[]],[]]
 => [.,[[.,[.,[.,[.,.]]]],[.,.]]]
 => [7,5,4,3,2,6,1] => ([(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)
 => ? = 0 + 1
[[],[[],[],[],[],[]]]
 => [.,[[.,[.,[.,[.,[.,.]]]]],.]]
 => [6,5,4,3,2,7,1] => ([(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)
 => ? = 0 + 1
[[[]],[],[],[],[],[]]
 => [[.,.],[.,[.,[.,[.,[.,.]]]]]]
 => [7,6,5,4,3,1,2] => ([(0,2),(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)
 => ? = 0 + 1
[[[]],[],[],[],[[]]]
 => [[.,.],[.,[.,[.,[[.,.],.]]]]]
 => [6,7,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[[]],[],[],[[]],[]]
 => [[.,.],[.,[.,[[.,.],[.,.]]]]]
 => [7,5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[[]],[],[],[[],[]]]
 => [[.,.],[.,[.,[[.,[.,.]],.]]]]
 => [6,5,7,4,3,1,2] => ([(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),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 1
Description
The size of a minimal independent dominating set in a graph.
Matching statistic: St000906
(load all 3 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000906: Posets ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => ([],1)
 => ? = 0 + 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => ([],2)
 => 1 = 0 + 1
[[[]]]
 => [[.,.],.]
 => [1,2] => ([(0,1)],2)
 => 2 = 1 + 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => ([],3)
 => 1 = 0 + 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => ([(1,2)],3)
 => 1 = 0 + 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [3,1,2] => ([(1,2)],3)
 => 1 = 0 + 1
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([],4)
 => 1 = 0 + 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(2,3)],4)
 => 1 = 0 + 1
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => ([(2,3)],4)
 => 1 = 0 + 1
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 1 = 0 + 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 1 = 0 + 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => ([(2,3)],4)
 => 1 = 0 + 1
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 2 = 1 + 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 1 = 0 + 1
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 1 = 0 + 1
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 3 = 2 + 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([],5)
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(3,4)],5)
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => ([(3,4)],5)
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => ([(3,4)],5)
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 1 = 0 + 1
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => 1 = 0 + 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => 2 = 1 + 1
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => 2 = 1 + 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => 2 = 1 + 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => 1 = 0 + 1
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => 1 = 0 + 1
[[[],[],[],[]]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[[],[],[],[[],[[]]]]
 => [.,[.,[.,[[.,[[.,.],.]],.]]]]
 => [5,6,4,7,3,2,1] => ([(3,6),(4,5),(5,6)],7)
 => ? = 0 + 1
[[],[],[],[[[]],[]]]
 => [.,[.,[.,[[[.,.],[.,.]],.]]]]
 => [6,4,5,7,3,2,1] => ([(3,6),(4,5),(5,6)],7)
 => ? = 0 + 1
[[],[],[[]],[[[]]]]
 => [.,[.,[[.,.],[[[.,.],.],.]]]]
 => [5,6,7,3,4,2,1] => ([(2,4),(3,5),(5,6)],7)
 => ? = 0 + 1
[[],[],[[[]]],[[]]]
 => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => [6,7,3,4,5,2,1] => ([(2,4),(3,5),(5,6)],7)
 => ? = 0 + 1
[[],[],[[],[[]]],[]]
 => [.,[.,[[.,[[.,.],.]],[.,.]]]]
 => [7,4,5,3,6,2,1] => ([(3,6),(4,5),(5,6)],7)
 => ? = 0 + 1
[[],[],[[[]],[]],[]]
 => [.,[.,[[[.,.],[.,.]],[.,.]]]]
 => [7,5,3,4,6,2,1] => ([(3,6),(4,5),(5,6)],7)
 => ? = 0 + 1
[[],[],[[],[[[]]]]]
 => [.,[.,[[.,[[[.,.],.],.]],.]]]
 => [4,5,6,3,7,2,1] => ([(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 0 + 1
[[],[],[[[]],[[]]]]
 => [.,[.,[[[.,.],[[.,.],.]],.]]]
 => [5,6,3,4,7,2,1] => ([(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[],[[[[]]],[]]]
 => [.,[.,[[[[.,.],.],[.,.]],.]]]
 => [6,3,4,5,7,2,1] => ([(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 0 + 1
[[],[],[[[],[[]]]]]
 => [.,[.,[[[.,[[.,.],.]],.],.]]]
 => [4,5,3,6,7,2,1] => ([(2,6),(3,4),(4,6),(6,5)],7)
 => ? = 0 + 1
[[],[],[[[[]],[]]]]
 => [.,[.,[[[[.,.],[.,.]],.],.]]]
 => [5,3,4,6,7,2,1] => ([(2,6),(3,4),(4,6),(6,5)],7)
 => ? = 0 + 1
[[],[],[[[[[]]]]]]
 => [.,[.,[[[[[.,.],.],.],.],.]]]
 => [3,4,5,6,7,2,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ? = 0 + 1
[[],[[]],[],[[[]]]]
 => [.,[[.,.],[.,[[[.,.],.],.]]]]
 => [5,6,7,4,2,3,1] => ([(2,4),(3,5),(5,6)],7)
 => ? = 0 + 1
[[],[[]],[[]],[[]]]
 => [.,[[.,.],[[.,.],[[.,.],.]]]]
 => [6,7,4,5,2,3,1] => ([(1,6),(2,5),(3,4)],7)
 => ? = 0 + 1
[[],[[]],[[[]]],[]]
 => [.,[[.,.],[[[.,.],.],[.,.]]]]
 => [7,4,5,6,2,3,1] => ([(2,4),(3,5),(5,6)],7)
 => ? = 0 + 1
[[],[[]],[[],[[]]]]
 => [.,[[.,.],[[.,[[.,.],.]],.]]]
 => [5,6,4,7,2,3,1] => ([(1,6),(2,4),(3,5),(5,6)],7)
 => ? = 0 + 1
[[],[[]],[[[]],[]]]
 => [.,[[.,.],[[[.,.],[.,.]],.]]]
 => [6,4,5,7,2,3,1] => ([(1,6),(2,4),(3,5),(5,6)],7)
 => ? = 0 + 1
[[],[[]],[[[[]]]]]
 => [.,[[.,.],[[[[.,.],.],.],.]]]
 => [4,5,6,7,2,3,1] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 0 + 1
[[],[[[]]],[],[[]]]
 => [.,[[[.,.],.],[.,[[.,.],.]]]]
 => [6,7,5,2,3,4,1] => ([(2,4),(3,5),(5,6)],7)
 => ? = 0 + 1
[[],[[[]]],[[]],[]]
 => [.,[[[.,.],.],[[.,.],[.,.]]]]
 => [7,5,6,2,3,4,1] => ([(2,4),(3,5),(5,6)],7)
 => ? = 0 + 1
[[],[[[]]],[[[]]]]
 => [.,[[[.,.],.],[[[.,.],.],.]]]
 => [5,6,7,2,3,4,1] => ([(1,6),(2,5),(5,3),(6,4)],7)
 => ? = 0 + 1
[[],[[],[[]]],[],[]]
 => [.,[[.,[[.,.],.]],[.,[.,.]]]]
 => [7,6,3,4,2,5,1] => ([(3,6),(4,5),(5,6)],7)
 => ? = 0 + 1
[[],[[[]],[]],[],[]]
 => [.,[[[.,.],[.,.]],[.,[.,.]]]]
 => [7,6,4,2,3,5,1] => ([(3,6),(4,5),(5,6)],7)
 => ? = 0 + 1
[[],[[],[[]]],[[]]]
 => [.,[[.,[[.,.],.]],[[.,.],.]]]
 => [6,7,3,4,2,5,1] => ([(1,6),(2,4),(3,5),(5,6)],7)
 => ? = 0 + 1
[[],[[[]],[]],[[]]]
 => [.,[[[.,.],[.,.]],[[.,.],.]]]
 => [6,7,4,2,3,5,1] => ([(1,6),(2,4),(3,5),(5,6)],7)
 => ? = 0 + 1
[[],[[[[]]]],[[]]]
 => [.,[[[[.,.],.],.],[[.,.],.]]]
 => [6,7,2,3,4,5,1] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 0 + 1
[[],[[],[[[]]]],[]]
 => [.,[[.,[[[.,.],.],.]],[.,.]]]
 => [7,3,4,5,2,6,1] => ([(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 0 + 1
[[],[[[]],[[]]],[]]
 => [.,[[[.,.],[[.,.],.]],[.,.]]]
 => [7,4,5,2,3,6,1] => ([(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[[[]]],[]],[]]
 => [.,[[[[.,.],.],[.,.]],[.,.]]]
 => [7,5,2,3,4,6,1] => ([(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 0 + 1
[[],[[[],[[]]]],[]]
 => [.,[[[.,[[.,.],.]],.],[.,.]]]
 => [7,3,4,2,5,6,1] => ([(2,6),(3,4),(4,6),(6,5)],7)
 => ? = 0 + 1
[[],[[[[]],[]]],[]]
 => [.,[[[[.,.],[.,.]],.],[.,.]]]
 => [7,4,2,3,5,6,1] => ([(2,6),(3,4),(4,6),(6,5)],7)
 => ? = 0 + 1
[[],[[[[[]]]]],[]]
 => [.,[[[[[.,.],.],.],.],[.,.]]]
 => [7,2,3,4,5,6,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ? = 0 + 1
[[],[[],[[]],[[]]]]
 => [.,[[.,[[.,.],[[.,.],.]]],.]]
 => [5,6,3,4,2,7,1] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[],[[],[[]]]]]
 => [.,[[.,[[.,[[.,.],.]],.]],.]]
 => [4,5,3,6,2,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => ? = 0 + 1
[[],[[],[[[]],[]]]]
 => [.,[[.,[[[.,.],[.,.]],.]],.]]
 => [5,3,4,6,2,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => ? = 0 + 1
[[],[[[]],[],[[]]]]
 => [.,[[[.,.],[.,[[.,.],.]]],.]]
 => [5,6,4,2,3,7,1] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[[]],[[]],[]]]
 => [.,[[[.,.],[[.,.],[.,.]]],.]]
 => [6,4,5,2,3,7,1] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[[]],[[[]]]]]
 => [.,[[[.,.],[[[.,.],.],.]],.]]
 => [4,5,6,2,3,7,1] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[[[]]],[[]]]]
 => [.,[[[[.,.],.],[[.,.],.]],.]]
 => [5,6,2,3,4,7,1] => ([(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[[],[[]]],[]]]
 => [.,[[[.,[[.,.],.]],[.,.]],.]]
 => [6,3,4,2,5,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => ? = 0 + 1
[[],[[[[]],[]],[]]]
 => [.,[[[[.,.],[.,.]],[.,.]],.]]
 => [6,4,2,3,5,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
 => ? = 0 + 1
[[],[[[],[[[]]]]]]
 => [.,[[[.,[[[.,.],.],.]],.],.]]
 => [3,4,5,2,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
 => ? = 0 + 1
[[],[[[[]],[[]]]]]
 => [.,[[[[.,.],[[.,.],.]],.],.]]
 => [4,5,2,3,6,7,1] => ([(1,4),(2,3),(3,6),(4,6),(6,5)],7)
 => ? = 0 + 1
[[],[[[[[]]],[]]]]
 => [.,[[[[[.,.],.],[.,.]],.],.]]
 => [5,2,3,4,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
 => ? = 0 + 1
[[],[[[[],[[]]]]]]
 => [.,[[[[.,[[.,.],.]],.],.],.]]
 => [3,4,2,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 0 + 1
[[],[[[[[]],[]]]]]
 => [.,[[[[[.,.],[.,.]],.],.],.]]
 => [4,2,3,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
 => ? = 0 + 1
[[],[[[[[[]]]]]]]
 => [.,[[[[[[.,.],.],.],.],.],.]]
 => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
 => ? = 0 + 1
[[[]],[],[],[[[]]]]
 => [[.,.],[.,[.,[[[.,.],.],.]]]]
 => [5,6,7,4,3,1,2] => ([(2,4),(3,5),(5,6)],7)
 => ? = 0 + 1
Description
The length of the shortest maximal chain in a poset.
Matching statistic: St000908
(load all 3 compositions to match this statistic)
Mapping
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [[]]
 => [1,0]
 => ([],1)
 => 1 = 0 + 1
[[],[]]
 => [[],[]]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[[]]]
 => [[[]]]
 => [1,1,0,0]
 => ([],2)
 => 2 = 1 + 1
[[],[],[]]
 => [[],[],[]]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[],[[]]]
 => [[[]],[]]
 => [1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => 1 = 0 + 1
[[[]],[]]
 => [[],[[]]]
 => [1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[[[],[]]]
 => [[[],[]]]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => 2 = 1 + 1
[[[[]]]]
 => [[[[]]]]
 => [1,1,1,0,0,0]
 => ([],3)
 => 3 = 2 + 1
[[],[],[],[]]
 => [[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[],[],[[]]]
 => [[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => 1 = 0 + 1
[[],[[]],[]]
 => [[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[],[[],[]]]
 => [[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => 1 = 0 + 1
[[],[[[]]]]
 => [[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => 1 = 0 + 1
[[[]],[],[]]
 => [[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => 1 = 0 + 1
[[[]],[[]]]
 => [[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[[[],[]],[]]
 => [[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[[[[]]],[]]
 => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[[],[],[]]]
 => [[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
[[[],[[]]]]
 => [[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => 2 = 1 + 1
[[[[]],[]]]
 => [[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[[[[],[]]]]
 => [[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => 3 = 2 + 1
[[[[[]]]]]
 => [[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => ([],4)
 => 4 = 3 + 1
[[],[],[],[],[]]
 => [[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[],[[]]]
 => [[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 1 = 0 + 1
[[],[],[[]],[]]
 => [[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 1 = 0 + 1
[[],[],[[[]]]]
 => [[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 1 = 0 + 1
[[],[[[]]],[]]
 => [[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[],[],[]]]
 => [[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 1 = 0 + 1
[[],[[],[[]]]]
 => [[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 1 = 0 + 1
[[],[[[]],[]]]
 => [[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[[[],[]]]]
 => [[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => 1 = 0 + 1
[[],[[[[]]]]]
 => [[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => 2 = 1 + 1
[[[]],[[[]]]]
 => [[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => 2 = 1 + 1
[[[],[]],[],[]]
 => [[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 1 = 0 + 1
[[[[]]],[],[]]
 => [[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
[[[],[],[]],[]]
 => [[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[],[[]]],[]]
 => [[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[[[]],[]],[]]
 => [[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 1 = 0 + 1
[[[[],[]]],[]]
 => [[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[[[[[]]]],[]]
 => [[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[[],[],[],[],[],[],[]]
 => [[],[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? = 0 + 1
[[],[],[],[],[],[[]]]
 => [[[]],[],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
 => ? = 0 + 1
[[],[],[],[],[[]],[]]
 => [[],[[]],[],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => ? = 0 + 1
[[],[],[],[],[[],[]]]
 => [[[],[]],[],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => ? = 0 + 1
[[],[],[],[],[[[]]]]
 => [[[[]]],[],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
 => ? = 0 + 1
[[],[],[],[[]],[[]]]
 => [[[]],[[]],[],[],[]]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => ? = 0 + 1
[[],[],[],[[],[]],[]]
 => [[],[[],[]],[],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
 => ? = 0 + 1
[[],[],[],[[],[],[]]]
 => [[[],[],[]],[],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => ? = 0 + 1
[[],[],[],[[],[[]]]]
 => [[[[]],[]],[],[],[]]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(4,6),(5,2),(5,3),(6,1),(6,5)],7)
 => ? = 0 + 1
[[],[],[],[[[],[]]]]
 => [[[[],[]]],[],[],[]]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => ([(0,5),(4,3),(5,6),(6,1),(6,2),(6,4)],7)
 => ? = 0 + 1
[[],[],[[]],[],[],[]]
 => [[],[],[],[[]],[],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ? = 0 + 1
[[],[],[[]],[],[[]]]
 => [[[]],[],[[]],[],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => ? = 0 + 1
[[],[],[[]],[[]],[]]
 => [[],[[]],[[]],[],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[],[[]],[[],[]]]
 => [[[],[]],[[]],[],[]]
 => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
 => ? = 0 + 1
[[],[],[[]],[[[]]]]
 => [[[[]]],[[]],[],[]]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
 => ? = 0 + 1
[[],[],[[],[]],[],[]]
 => [[],[],[[],[]],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => ? = 0 + 1
[[],[],[[[]]],[],[]]
 => [[],[],[[[]]],[],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ? = 0 + 1
[[],[],[[],[]],[[]]]
 => [[[]],[[],[]],[],[]]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
 => ? = 0 + 1
[[],[],[[],[],[]],[]]
 => [[],[[],[],[]],[],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 0 + 1
[[],[],[[],[[]]],[]]
 => [[],[[[]],[]],[],[]]
 => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ? = 0 + 1
[[],[],[[[]],[]],[]]
 => [[],[[],[[]]],[],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => ? = 0 + 1
[[],[],[[[],[]]],[]]
 => [[],[[[],[]]],[],[]]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ? = 0 + 1
[[],[],[[],[],[],[]]]
 => [[[],[],[],[]],[],[]]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
 => ? = 0 + 1
[[],[],[[],[],[[]]]]
 => [[[[]],[],[]],[],[]]
 => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,5),(2,6),(3,1),(3,5),(3,6),(4,2),(4,3)],7)
 => ? = 0 + 1
[[],[],[[],[[]],[]]]
 => [[[],[[]],[]],[],[]]
 => [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(4,3),(5,2),(5,4)],7)
 => ? = 0 + 1
[[],[],[[],[[],[]]]]
 => [[[[],[]],[]],[],[]]
 => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => ([(0,5),(3,6),(4,1),(4,2),(4,6),(5,3),(5,4)],7)
 => ? = 0 + 1
[[],[],[[],[[[]]]]]
 => [[[[[]]],[]],[],[]]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => ([(0,5),(5,4),(5,6),(6,1),(6,2),(6,3)],7)
 => ? = 0 + 1
[[],[],[[[],[]],[]]]
 => [[[],[[],[]]],[],[]]
 => [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(4,6),(5,2),(5,3),(5,4)],7)
 => ? = 0 + 1
[[],[],[[[],[],[]]]]
 => [[[[],[],[]]],[],[]]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7)
 => ? = 0 + 1
[[],[],[[[],[[]]]]]
 => [[[[[]],[]]],[],[]]
 => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => ([(0,6),(5,3),(5,4),(6,1),(6,2),(6,5)],7)
 => ? = 0 + 1
[[],[],[[[[],[]]]]]
 => [[[[[],[]]]],[],[]]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => ([(0,6),(5,4),(6,1),(6,2),(6,3),(6,5)],7)
 => ? = 0 + 1
[[],[[]],[],[],[],[]]
 => [[],[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ? = 0 + 1
[[],[[]],[],[],[[]]]
 => [[[]],[],[],[[]],[]]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
 => ? = 0 + 1
[[],[[]],[],[[],[]]]
 => [[[],[]],[],[[]],[]]
 => [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => ([(0,3),(0,4),(3,6),(4,6),(5,1),(6,2),(6,5)],7)
 => ? = 0 + 1
[[],[[]],[[],[]],[]]
 => [[],[[],[]],[[]],[]]
 => [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
 => ? = 0 + 1
[[],[[],[]],[],[],[]]
 => [[],[],[],[[],[]],[]]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => ? = 0 + 1
[[],[[[]]],[],[],[]]
 => [[],[],[],[[[]]],[]]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => ? = 0 + 1
[[],[[],[]],[[]],[]]
 => [[],[[]],[[],[]],[]]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[],[]],[[],[]]]
 => [[[],[]],[[],[]],[]]
 => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7)
 => ? = 0 + 1
[[],[[],[]],[[[]]]]
 => [[[[]]],[[],[]],[]]
 => [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1)],7)
 => ? = 0 + 1
[[],[[],[],[]],[],[]]
 => [[],[],[[],[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 0 + 1
[[],[[],[[]]],[],[]]
 => [[],[],[[[]],[]],[]]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
 => ? = 0 + 1
[[],[[],[],[]],[[]]]
 => [[[]],[[],[],[]],[]]
 => [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
 => ? = 0 + 1
[[],[[[],[]]],[[]]]
 => [[[]],[[[],[]]],[]]
 => [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,1)],7)
 => ? = 0 + 1
[[],[[],[],[],[]],[]]
 => [[],[[],[],[],[]],[]]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[],[],[[]]],[]]
 => [[],[[[]],[],[]],[]]
 => [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 0 + 1
[[],[[],[[]],[]],[]]
 => [[],[[],[[]],[]],[]]
 => [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
 => ? = 0 + 1
[[],[[[],[]],[]],[]]
 => [[],[[],[[],[]]],[]]
 => [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
 => ? = 0 + 1
[[],[[[],[],[]]],[]]
 => [[],[[[],[],[]]],[]]
 => [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => ? = 0 + 1
[[],[[],[],[],[],[]]]
 => [[[],[],[],[],[]],[]]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(1,6),(2,4),(2,5),(3,1),(3,5),(5,6)],7)
 => ? = 0 + 1
Description
The length of the shortest maximal antichain in a poset.
Matching statistic: St001481
(load all 5 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 1 = 0 + 1
[[],[]]
 => [1,0,1,0]
 => 1 = 0 + 1
[[[]]]
 => [1,1,0,0]
 => 2 = 1 + 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[[],[],[],[],[]]
 => [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
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1 = 0 + 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 0 + 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1 = 0 + 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[[],[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0 + 1
[[],[],[],[],[],[[]]]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 1
[[],[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 0 + 1
[[],[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 1
[[],[],[],[],[[[]]]]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 0 + 1
[[],[],[],[[]],[],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 0 + 1
[[],[],[],[[]],[[]]]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
[[],[],[],[[],[]],[]]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => ? = 0 + 1
[[],[],[],[[[]]],[]]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 0 + 1
[[],[],[],[[],[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 0 + 1
[[],[],[],[[],[[]]]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => ? = 0 + 1
[[],[],[],[[[]],[]]]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 0 + 1
[[],[],[],[[[],[]]]]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => ? = 0 + 1
[[],[],[],[[[[]]]]]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 0 + 1
[[],[],[[]],[],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 0 + 1
[[],[],[[]],[],[[]]]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 0 + 1
[[],[],[[]],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 0 + 1
[[],[],[[]],[[],[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => ? = 0 + 1
[[],[],[[]],[[[]]]]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 0 + 1
[[],[],[[],[]],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 0 + 1
[[],[],[[[]]],[],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 0 + 1
[[],[],[[],[]],[[]]]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => ? = 0 + 1
[[],[],[[[]]],[[]]]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 0 + 1
[[],[],[[],[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => ? = 0 + 1
[[],[],[[],[[]]],[]]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => ? = 0 + 1
[[],[],[[[]],[]],[]]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => ? = 0 + 1
[[],[],[[[],[]]],[]]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => ? = 0 + 1
[[],[],[[[[]]]],[]]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0 + 1
[[],[],[[],[],[],[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[[],[],[[],[],[[]]]]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => ? = 0 + 1
[[],[],[[],[[]],[]]]
 => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => ? = 0 + 1
[[],[],[[],[[],[]]]]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => ? = 0 + 1
[[],[],[[],[[[]]]]]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => ? = 0 + 1
[[],[],[[[]],[],[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => ? = 0 + 1
[[],[],[[[]],[[]]]]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 0 + 1
[[],[],[[[],[]],[]]]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 0 + 1
[[],[],[[[[]]],[]]]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 0 + 1
[[],[],[[[],[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 0 + 1
[[],[],[[[],[[]]]]]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 0 + 1
[[],[],[[[[]],[]]]]
 => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 0 + 1
[[],[],[[[[],[]]]]]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 0 + 1
[[],[],[[[[[]]]]]]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[[],[[]],[],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 0 + 1
[[],[[]],[],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? = 0 + 1
[[],[[]],[],[[]],[]]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 0 + 1
[[],[[]],[],[[],[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => ? = 0 + 1
[[],[[]],[],[[[]]]]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 0 + 1
[[],[[]],[[]],[],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 0 + 1
[[],[[]],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 0 + 1
[[],[[]],[[],[]],[]]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => ? = 0 + 1
Description
The minimal height of a peak of a Dyck path.
Matching statistic: St000310
(load all 2 compositions to match this statistic)
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000310: Graphs ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => ([],1)
 => 0
[[],[]]
 => [[.,.],.]
 => [1,2] => ([],2)
 => 0
[[[]]]
 => [.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => 1
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => ([],3)
 => 0
[[],[[]]]
 => [[.,.],[.,.]]
 => [1,3,2] => ([(1,2)],3)
 => 0
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => 0
[[[],[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[[[[]]]]
 => [.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => 0
[[],[],[[]]]
 => [[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => 0
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => ([(2,3)],4)
 => 0
[[],[[],[]]]
 => [[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[[],[[[]]]]
 => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => 0
[[[]],[[]]]
 => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 0
[[[],[],[]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[[[],[[]]]]
 => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[[]],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[[],[]]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[[[[]]]]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => ([],5)
 => 0
[[],[],[],[[]]]
 => [[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => ([(3,4)],5)
 => 0
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => ([(3,4)],5)
 => 0
[[],[],[[],[]]]
 => [[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[[],[],[[[]]]]
 => [[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 0
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [1,3,2,4,5] => ([(3,4)],5)
 => 0
[[],[[]],[[]]]
 => [[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 0
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 0
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 0
[[],[[],[],[]]]
 => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[[],[[],[[]]]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[],[[[]],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[],[[[],[]]]]
 => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[],[[[[]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => ([(3,4)],5)
 => 0
[[[]],[],[[]]]
 => [[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[[[]],[[],[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 1
[[[]],[[[]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 0
[[[],[]],[[]]]
 => [[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 1
[[[[]]],[[]]]
 => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => 1
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[[],[],[],[],[],[],[]]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => [1,2,3,4,5,6,7] => ([],7)
 => ? = 0
[[],[],[],[],[],[[]]]
 => [[[[[[.,.],.],.],.],.],[.,.]]
 => [1,2,3,4,5,7,6] => ([(5,6)],7)
 => ? = 0
[[],[],[],[],[[]],[]]
 => [[[[[[.,.],.],.],.],[.,.]],.]
 => [1,2,3,4,6,5,7] => ([(5,6)],7)
 => ? = 0
[[],[],[],[],[[],[]]]
 => [[[[[.,.],.],.],.],[[.,.],.]]
 => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
 => ? = 0
[[],[],[],[],[[[]]]]
 => [[[[[.,.],.],.],.],[.,[.,.]]]
 => [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[],[[]],[],[]]
 => [[[[[[.,.],.],.],[.,.]],.],.]
 => [1,2,3,5,4,6,7] => ([(5,6)],7)
 => ? = 0
[[],[],[],[[]],[[]]]
 => [[[[[.,.],.],.],[.,.]],[.,.]]
 => [1,2,3,5,4,7,6] => ([(3,6),(4,5)],7)
 => ? = 0
[[],[],[],[[],[]],[]]
 => [[[[[.,.],.],.],[[.,.],.]],.]
 => [1,2,3,5,6,4,7] => ([(4,6),(5,6)],7)
 => ? = 0
[[],[],[],[[[]]],[]]
 => [[[[[.,.],.],.],[.,[.,.]]],.]
 => [1,2,3,6,5,4,7] => ([(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[],[[],[],[]]]
 => [[[[.,.],.],.],[[[.,.],.],.]]
 => [1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 0
[[],[],[],[[],[[]]]]
 => [[[[.,.],.],.],[[.,.],[.,.]]]
 => [1,2,3,5,7,6,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[],[[[]],[]]]
 => [[[[.,.],.],.],[[.,[.,.]],.]]
 => [1,2,3,6,5,7,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[],[[[],[]]]]
 => [[[[.,.],.],.],[.,[[.,.],.]]]
 => [1,2,3,6,7,5,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[],[[[[]]]]]
 => [[[[.,.],.],.],[.,[.,[.,.]]]]
 => [1,2,3,7,6,5,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[]],[],[],[]]
 => [[[[[[.,.],.],[.,.]],.],.],.]
 => [1,2,4,3,5,6,7] => ([(5,6)],7)
 => ? = 0
[[],[],[[]],[],[[]]]
 => [[[[[.,.],.],[.,.]],.],[.,.]]
 => [1,2,4,3,5,7,6] => ([(3,6),(4,5)],7)
 => ? = 0
[[],[],[[]],[[]],[]]
 => [[[[[.,.],.],[.,.]],[.,.]],.]
 => [1,2,4,3,6,5,7] => ([(3,6),(4,5)],7)
 => ? = 0
[[],[],[[]],[[],[]]]
 => [[[[.,.],.],[.,.]],[[.,.],.]]
 => [1,2,4,3,6,7,5] => ([(2,3),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[]],[[[]]]]
 => [[[[.,.],.],[.,.]],[.,[.,.]]]
 => [1,2,4,3,7,6,5] => ([(2,3),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[],[]],[],[]]
 => [[[[[.,.],.],[[.,.],.]],.],.]
 => [1,2,4,5,3,6,7] => ([(4,6),(5,6)],7)
 => ? = 0
[[],[],[[[]]],[],[]]
 => [[[[[.,.],.],[.,[.,.]]],.],.]
 => [1,2,5,4,3,6,7] => ([(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[],[]],[[]]]
 => [[[[.,.],.],[[.,.],.]],[.,.]]
 => [1,2,4,5,3,7,6] => ([(2,3),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[[]]],[[]]]
 => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => [1,2,5,4,3,7,6] => ([(2,3),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[],[],[]],[]]
 => [[[[.,.],.],[[[.,.],.],.]],.]
 => [1,2,4,5,6,3,7] => ([(3,6),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[],[[]]],[]]
 => [[[[.,.],.],[[.,.],[.,.]]],.]
 => [1,2,4,6,5,3,7] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[[]],[]],[]]
 => [[[[.,.],.],[[.,[.,.]],.]],.]
 => [1,2,5,4,6,3,7] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[[],[]]],[]]
 => [[[[.,.],.],[.,[[.,.],.]]],.]
 => [1,2,5,6,4,3,7] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[[[]]]],[]]
 => [[[[.,.],.],[.,[.,[.,.]]]],.]
 => [1,2,6,5,4,3,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[],[],[],[]]]
 => [[[.,.],.],[[[[.,.],.],.],.]]
 => [1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[],[],[[]]]]
 => [[[.,.],.],[[[.,.],.],[.,.]]]
 => [1,2,4,5,7,6,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[],[[]],[]]]
 => [[[.,.],.],[[[.,.],[.,.]],.]]
 => [1,2,4,6,5,7,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[],[[],[]]]]
 => [[[.,.],.],[[.,.],[[.,.],.]]]
 => [1,2,4,6,7,5,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[],[[[]]]]]
 => [[[.,.],.],[[.,.],[.,[.,.]]]]
 => [1,2,4,7,6,5,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[[]],[],[]]]
 => [[[.,.],.],[[[.,[.,.]],.],.]]
 => [1,2,5,4,6,7,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[[]],[[]]]]
 => [[[.,.],.],[[.,[.,.]],[.,.]]]
 => [1,2,5,4,7,6,3] => ([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[[],[]],[]]]
 => [[[.,.],.],[[.,[[.,.],.]],.]]
 => [1,2,5,6,4,7,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[[[]]],[]]]
 => [[[.,.],.],[[.,[.,[.,.]]],.]]
 => [1,2,6,5,4,7,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[[],[],[]]]]
 => [[[.,.],.],[.,[[[.,.],.],.]]]
 => [1,2,5,6,7,4,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[[],[[]]]]]
 => [[[.,.],.],[.,[[.,.],[.,.]]]]
 => [1,2,5,7,6,4,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[[[]],[]]]]
 => [[[.,.],.],[.,[[.,[.,.]],.]]]
 => [1,2,6,5,7,4,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[[[],[]]]]]
 => [[[.,.],.],[.,[.,[[.,.],.]]]]
 => [1,2,6,7,5,4,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[],[[[[[]]]]]]
 => [[[.,.],.],[.,[.,[.,[.,.]]]]]
 => [1,2,7,6,5,4,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[[]],[],[],[],[]]
 => [[[[[[.,.],[.,.]],.],.],.],.]
 => [1,3,2,4,5,6,7] => ([(5,6)],7)
 => ? = 0
[[],[[]],[],[],[[]]]
 => [[[[[.,.],[.,.]],.],.],[.,.]]
 => [1,3,2,4,5,7,6] => ([(3,6),(4,5)],7)
 => ? = 0
[[],[[]],[],[[]],[]]
 => [[[[[.,.],[.,.]],.],[.,.]],.]
 => [1,3,2,4,6,5,7] => ([(3,6),(4,5)],7)
 => ? = 0
[[],[[]],[],[[],[]]]
 => [[[[.,.],[.,.]],.],[[.,.],.]]
 => [1,3,2,4,6,7,5] => ([(2,3),(4,6),(5,6)],7)
 => ? = 0
[[],[[]],[],[[[]]]]
 => [[[[.,.],[.,.]],.],[.,[.,.]]]
 => [1,3,2,4,7,6,5] => ([(2,3),(4,5),(4,6),(5,6)],7)
 => ? = 0
[[],[[]],[[]],[],[]]
 => [[[[[.,.],[.,.]],[.,.]],.],.]
 => [1,3,2,5,4,6,7] => ([(3,6),(4,5)],7)
 => ? = 0
[[],[[]],[[]],[[]]]
 => [[[[.,.],[.,.]],[.,.]],[.,.]]
 => [1,3,2,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
 => ? = 0
[[],[[]],[[],[]],[]]
 => [[[[.,.],[.,.]],[[.,.],.]],.]
 => [1,3,2,5,6,4,7] => ([(2,3),(4,6),(5,6)],7)
 => ? = 0
Description
The minimal degree of a vertex of a graph.
The following 1 statistic also match your data. Click on any of them to see the details.
St000261The edge connectivity of a graph.