Your data matches 22 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000381
(load all 7 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => 1
[[],[]]
 => [1,0,1,0]
 => [1,1] => 1
[[[]]]
 => [1,1,0,0]
 => [2] => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,2] => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [2,1] => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [3] => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => 3
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [4] => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => 4
Description
The largest part of an integer composition.
Matching statistic: St000392
(load all 6 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 10 => 1
[[],[]]
 => [1,0,1,0]
 => 1010 => 1
[[[]]]
 => [1,1,0,0]
 => 1100 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => 101010 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 101100 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => 110010 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => 110100 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => 111000 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 3
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 4
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
(load all 5 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => 10 => 1
[[],[]]
 => [1,0,1,0]
 => 1010 => 1
[[[]]]
 => [1,1,0,0]
 => 1100 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => 101010 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 101100 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => 110010 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => 110100 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => 111000 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 3
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 11101000 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 11110000 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 4
Description
The length of a longest cyclic run of ones of a binary word. Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St000147
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => [1]
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,1] => [1,1]
 => 1
[[[]]]
 => [1,1,0,0]
 => [2] => [2]
 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,2] => [2,1]
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => [2,1]
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [2,1] => [2,1]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [3] => [3]
 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1]
 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => [2,1,1]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => [3,1]
 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => [2,1,1]
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => [2,2]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => [2,1,1]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => [3,1]
 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => [2,1,1]
 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => [2,2]
 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => [3,1]
 => 3
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => [3,1]
 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [4] => [4]
 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1,1]
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [2,1,1,1]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1]
 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,1,1,1]
 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [2,1,1,1]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [2,1,1,1]
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [2,2,1]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [3,1,1]
 => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [3,1,1]
 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1]
 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [2,2,1]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2]
 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => [3,1,1]
 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => [3,1,1]
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [4,1]
 => 4
Description
The largest part of an integer partition.
Matching statistic: St000013
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => [1,0]
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,1] => [1,0,1,0]
 => 1
[[[]]]
 => [1,1,0,0]
 => [2] => [1,1,0,0]
 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => [1,0,1,0,1,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,2] => [1,0,1,1,0,0]
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => [1,1,0,0,1,0]
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
[[],[[[[],[[[]]]]]]]
 => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
[[[[[],[[[]]]]]],[]]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4
[[[[[[],[[]]]]]],[]]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 5
Description
The height of a Dyck path. The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000444
(load all 11 compositions to match this statistic)
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => ? = 1
[[],[]]
 => [1,0,1,0]
 => 1
[[[]]]
 => [1,1,0,0]
 => 2
[[],[],[]]
 => [1,0,1,0,1,0]
 => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => 2
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => 3
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => 3
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[[[[[],[[[]]]]]],[]]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 4
[[[[[[[]]],[]]]],[]]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 6
[[[[[[],[],[]]]]],[]]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => ? = 5
[[[[[[],[[]]]]]],[]]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => ? = 5
[[[[[[[]],[]]]]],[]]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ? = 6
[[[[[[[],[]]]]]],[]]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 6
[[[[[[[[]]]]]]],[]]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000442
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => [1,0]
 => ? = 1 - 1
[[],[]]
 => [1,0,1,0]
 => [1,1] => [1,0,1,0]
 => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => [2] => [1,1,0,0]
 => 1 = 2 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => 2 = 3 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 3 = 4 - 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 2 - 1
[[[[[],[[[]]]]]],[]]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4 - 1
[[[[[[[]]],[]]]],[]]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 6 - 1
[[[[[[],[],[]]]]],[]]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 5 - 1
[[[[[[],[[]]]]]],[]]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 5 - 1
[[[[[[[]],[]]]]],[]]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 6 - 1
[[[[[[[],[]]]]]],[]]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 6 - 1
[[[[[[[[]]]]]]],[]]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7 - 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St001062
Mapping
Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [[]]
 => [1,0]
 => {{1}}
 => ? = 1
[[],[]]
 => [[],[]]
 => [1,0,1,0]
 => {{1},{2}}
 => 1
[[[]]]
 => [[[]]]
 => [1,1,0,0]
 => {{1,2}}
 => 2
[[],[],[]]
 => [[],[],[]]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[[],[[]]]
 => [[[]],[]]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 2
[[[]],[]]
 => [[],[[]]]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 2
[[[],[]]]
 => [[[],[]]]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 2
[[[[]]]]
 => [[[[]]]]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 3
[[],[],[],[]]
 => [[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 1
[[],[],[[]]]
 => [[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 2
[[],[[]],[]]
 => [[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 2
[[],[[],[]]]
 => [[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 2
[[],[[[]]]]
 => [[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 3
[[[]],[],[]]
 => [[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 2
[[[]],[[]]]
 => [[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
[[[],[]],[]]
 => [[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 2
[[[[]]],[]]
 => [[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 3
[[[],[],[]]]
 => [[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 2
[[[],[[]]]]
 => [[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 2
[[[[]],[]]]
 => [[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 3
[[[[],[]]]]
 => [[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 3
[[[[[]]]]]
 => [[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 4
[[],[],[],[],[]]
 => [[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 1
[[],[],[],[[]]]
 => [[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 2
[[],[],[[]],[]]
 => [[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 2
[[],[],[[],[]]]
 => [[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 2
[[],[],[[[]]]]
 => [[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 3
[[],[[]],[],[]]
 => [[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 2
[[],[[]],[[]]]
 => [[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 2
[[],[[],[]],[]]
 => [[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 2
[[],[[[]]],[]]
 => [[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 3
[[],[[],[],[]]]
 => [[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 2
[[],[[],[[]]]]
 => [[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 2
[[],[[[]],[]]]
 => [[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 3
[[],[[[],[]]]]
 => [[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => {{1,2,4},{3},{5}}
 => 3
[[],[[[[]]]]]
 => [[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 4
[[[]],[],[],[]]
 => [[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 2
[[[]],[],[[]]]
 => [[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2
[[[]],[[]],[]]
 => [[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 2
[[[]],[[],[]]]
 => [[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 2
[[[]],[[[]]]]
 => [[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 3
[[[],[]],[],[]]
 => [[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 2
[[[[]]],[],[]]
 => [[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 3
[[[],[]],[[]]]
 => [[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 2
[[[[]]],[[]]]
 => [[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 3
[[[],[],[]],[]]
 => [[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 2
[[[],[[]]],[]]
 => [[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 2
[[[[]],[]],[]]
 => [[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => 3
[[[[],[]]],[]]
 => [[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => 3
[[[[[]]]],[]]
 => [[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 4
[[[],[],[],[]]]
 => [[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 2
[[],[[[[],[[[]]]]]]]
 => [[[[[[[]]],[]]]],[]]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => {{1,2,3,7},{4,5,6},{8}}
 => ? = 4
[[],[[[[[[]]],[]]]]]
 => [[[[[],[[[]]]]]],[]]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => {{1,2,3,5,6,7},{4},{8}}
 => ? = 6
[[],[[[[[],[],[]]]]]]
 => [[[[[[],[],[]]]]],[]]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => {{1,2,3,4,7},{5},{6},{8}}
 => ? = 5
[[],[[[[[],[[]]]]]]]
 => [[[[[[[]],[]]]]],[]]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => {{1,2,3,4,7},{5,6},{8}}
 => ? = 5
[[],[[[[[[]],[]]]]]]
 => [[[[[[],[[]]]]]],[]]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => {{1,2,3,4,6,7},{5},{8}}
 => ? = 6
[[],[[[[[[],[]]]]]]]
 => [[[[[[[],[]]]]]],[]]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => {{1,2,3,4,5,7},{6},{8}}
 => ? = 6
[[[[[],[[[]]]]]],[]]
 => [[],[[[[[[]]],[]]]]]
 => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => {{1},{2,3,4,8},{5,6,7}}
 => ? = 4
[[[[[[],[],[]]]]],[]]
 => [[],[[[[[],[],[]]]]]]
 => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => {{1},{2,3,4,5,8},{6},{7}}
 => ? = 5
[[[[[[],[[]]]]]],[]]
 => [[],[[[[[[]],[]]]]]]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => {{1},{2,3,4,5,8},{6,7}}
 => ? = 5
Description
The maximal size of a block of a set partition.
Matching statistic: St000308
(load all 5 compositions to match this statistic)
Mapping
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 92%●distinct values known / distinct values provided: 86%
Values
[[]]
 => [.,.]
 => [1] => 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => 1
[[[]]]
 => [[.,.],.]
 => [1,2] => 2
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => 2
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => 2
[[[],[]]]
 => [[.,.],[.,.]]
 => [3,1,2] => 2
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => 3
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => 2
[[],[[]],[]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => 2
[[],[[],[]]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => 2
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 3
[[[]],[],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 2
[[[]],[[]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 2
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 2
[[[[]]],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 3
[[[],[],[]]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => 2
[[[],[[]]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 2
[[[[]],[]]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 3
[[[[],[]]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 3
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => 4
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => 2
[[],[],[[]],[]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => 2
[[],[],[[],[]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => 2
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => 3
[[],[[]],[],[]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => 2
[[],[[]],[[]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => 2
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => 2
[[],[[[]]],[]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => 3
[[],[[],[],[]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => 2
[[],[[],[[]]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => 2
[[],[[[]],[]]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => 3
[[],[[[],[]]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => 3
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 4
[[[]],[],[],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => 2
[[[]],[],[[]]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => 2
[[[]],[[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => 2
[[[]],[[],[]]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => 2
[[[]],[[[]]]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => 3
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => 3
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => 2
[[[[]]],[[]]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => 3
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => 2
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => 2
[[[[]],[]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => 3
[[[[],[]]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => 3
[[[[[]]]],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => 4
[[],[[],[[[[]]]]]]
 => [.,[[.,.],[[[[.,.],.],.],.]]]
 => [4,5,6,7,2,3,1] => ? = 4
[[],[[[],[[[]]]]]]
 => [.,[[[.,.],[[[.,.],.],.]],.]]
 => [4,5,6,2,3,7,1] => ? = 3
[[],[[[[]],[[]]]]]
 => [.,[[[[.,.],.],[[.,.],.]],.]]
 => [5,6,2,3,4,7,1] => ? = 4
[[],[[[[],[[]]]]]]
 => [.,[[[[.,.],[[.,.],.]],.],.]]
 => [4,5,2,3,6,7,1] => ? = 4
[[],[[[[],[[[]]]]]]]
 => [.,[[[[.,.],[[[.,.],.],.]],.],.]]
 => [4,5,6,2,3,7,8,1] => ? = 4
[[],[[[[[[]]],[]]]]]
 => [.,[[[[[[.,.],.],.],[.,.]],.],.]]
 => [6,2,3,4,5,7,8,1] => ? = 6
[[],[[[[[],[],[]]]]]]
 => [.,[[[[[.,.],[.,[.,.]]],.],.],.]]
 => [5,4,2,3,6,7,8,1] => ? = 5
[[],[[[[[],[[]]]]]]]
 => [.,[[[[[.,.],[[.,.],.]],.],.],.]]
 => [4,5,2,3,6,7,8,1] => ? = 5
[[],[[[[[[]],[]]]]]]
 => [.,[[[[[[.,.],.],[.,.]],.],.],.]]
 => [5,2,3,4,6,7,8,1] => ? = 6
[[],[[[[[[],[]]]]]]]
 => [.,[[[[[[.,.],[.,.]],.],.],.],.]]
 => [4,2,3,5,6,7,8,1] => ? = 6
[[],[[[[[[[]]]]]]]]
 => [.,[[[[[[[.,.],.],.],.],.],.],.]]
 => [2,3,4,5,6,7,8,1] => ? = 7
[[[[[],[[[]]]]]],[]]
 => [[[[.,[[[.,.],.],.]],[.,.]],.],.]
 => [6,2,3,4,1,5,7,8] => ? = 4
[[[[[[[]]],[]]]],[]]
 => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
 => [6,2,1,3,4,5,7,8] => ? = 6
[[[[[[],[],[]]]]],[]]
 => [[[[[.,[.,[.,.]]],[.,.]],.],.],.]
 => [5,3,2,1,4,6,7,8] => ? = 5
[[[[[[],[[]]]]]],[]]
 => [[[[[.,[[.,.],.]],[.,.]],.],.],.]
 => [5,2,3,1,4,6,7,8] => ? = 5
[[[[[[[]],[]]]]],[]]
 => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
 => [5,2,1,3,4,6,7,8] => ? = 6
[[[[[[[],[]]]]]],[]]
 => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
 => [4,2,1,3,5,6,7,8] => ? = 6
[[[[[[[[]]]]]]],[]]
 => [[[[[[[.,[.,.]],.],.],.],.],.],.]
 => [2,1,3,4,5,6,7,8] => ? = 7
Description
The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.
Matching statistic: St000328
(load all 6 compositions to match this statistic)
Mapping
Mp00328: Ordered trees —DeBruijn-Morselt plane tree automorphism⟶ Ordered trees
St000328: Ordered trees ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 86%
Values
[[]]
 => [[]]
 => 1
[[],[]]
 => [[[]]]
 => 1
[[[]]]
 => [[],[]]
 => 2
[[],[],[]]
 => [[[[]]]]
 => 1
[[],[[]]]
 => [[[],[]]]
 => 2
[[[]],[]]
 => [[],[[]]]
 => 2
[[[],[]]]
 => [[[]],[]]
 => 2
[[[[]]]]
 => [[],[],[]]
 => 3
[[],[],[],[]]
 => [[[[[]]]]]
 => 1
[[],[],[[]]]
 => [[[[],[]]]]
 => 2
[[],[[]],[]]
 => [[[],[[]]]]
 => 2
[[],[[],[]]]
 => [[[[]],[]]]
 => 2
[[],[[[]]]]
 => [[[],[],[]]]
 => 3
[[[]],[],[]]
 => [[],[[[]]]]
 => 2
[[[]],[[]]]
 => [[],[[],[]]]
 => 2
[[[],[]],[]]
 => [[[]],[[]]]
 => 2
[[[[]]],[]]
 => [[],[],[[]]]
 => 3
[[[],[],[]]]
 => [[[[]]],[]]
 => 2
[[[],[[]]]]
 => [[[],[]],[]]
 => 2
[[[[]],[]]]
 => [[],[[]],[]]
 => 3
[[[[],[]]]]
 => [[[]],[],[]]
 => 3
[[[[[]]]]]
 => [[],[],[],[]]
 => 4
[[],[],[],[],[]]
 => [[[[[[]]]]]]
 => 1
[[],[],[],[[]]]
 => [[[[[],[]]]]]
 => 2
[[],[],[[]],[]]
 => [[[[],[[]]]]]
 => 2
[[],[],[[],[]]]
 => [[[[[]],[]]]]
 => 2
[[],[],[[[]]]]
 => [[[[],[],[]]]]
 => 3
[[],[[]],[],[]]
 => [[[],[[[]]]]]
 => 2
[[],[[]],[[]]]
 => [[[],[[],[]]]]
 => 2
[[],[[],[]],[]]
 => [[[[]],[[]]]]
 => 2
[[],[[[]]],[]]
 => [[[],[],[[]]]]
 => 3
[[],[[],[],[]]]
 => [[[[[]]],[]]]
 => 2
[[],[[],[[]]]]
 => [[[[],[]],[]]]
 => 2
[[],[[[]],[]]]
 => [[[],[[]],[]]]
 => 3
[[],[[[],[]]]]
 => [[[[]],[],[]]]
 => 3
[[],[[[[]]]]]
 => [[[],[],[],[]]]
 => 4
[[[]],[],[],[]]
 => [[],[[[[]]]]]
 => 2
[[[]],[],[[]]]
 => [[],[[[],[]]]]
 => 2
[[[]],[[]],[]]
 => [[],[[],[[]]]]
 => 2
[[[]],[[],[]]]
 => [[],[[[]],[]]]
 => 2
[[[]],[[[]]]]
 => [[],[[],[],[]]]
 => 3
[[[],[]],[],[]]
 => [[[]],[[[]]]]
 => 2
[[[[]]],[],[]]
 => [[],[],[[[]]]]
 => 3
[[[],[]],[[]]]
 => [[[]],[[],[]]]
 => 2
[[[[]]],[[]]]
 => [[],[],[[],[]]]
 => 3
[[[],[],[]],[]]
 => [[[[]]],[[]]]
 => 2
[[[],[[]]],[]]
 => [[[],[]],[[]]]
 => 2
[[[[]],[]],[]]
 => [[],[[]],[[]]]
 => 3
[[[[],[]]],[]]
 => [[[]],[],[[]]]
 => 3
[[[[[]]]],[]]
 => [[],[],[],[[]]]
 => 4
[[],[[],[[[[]]]]]]
 => [[[[],[],[],[]],[]]]
 => ? = 4
[[],[[[[[]]]],[]]]
 => [[[],[],[],[[]],[]]]
 => ? = 5
[[],[[[],[[],[]]]]]
 => [[[[[]],[]],[],[]]]
 => ? = 3
[[],[[[],[[[]]]]]]
 => [[[[],[],[]],[],[]]]
 => ? = 3
[[],[[[[]],[[]]]]]
 => [[[],[[],[]],[],[]]]
 => ? = 4
[[],[[[[],[]],[]]]]
 => [[[[]],[[]],[],[]]]
 => ? = 4
[[],[[[[[]]],[]]]]
 => [[[],[],[[]],[],[]]]
 => ? = 5
[[],[[[[],[],[]]]]]
 => [[[[[]]],[],[],[]]]
 => ? = 4
[[],[[[[],[[]]]]]]
 => [[[[],[]],[],[],[]]]
 => ? = 4
[[],[[[[[]],[]]]]]
 => [[[],[[]],[],[],[]]]
 => ? = 5
[[],[[[[[],[]]]]]]
 => [[[[]],[],[],[],[]]]
 => ? = 5
[[],[[[[[[]]]]]]]
 => [[[],[],[],[],[],[]]]
 => ? = 6
[[[]],[[[[[]]]]]]
 => [[],[[],[],[],[],[]]]
 => ? = 5
[[[[[[]]]]],[[]]]
 => [[],[],[],[],[[],[]]]
 => ? = 5
[[[],[[[[]]]]],[]]
 => [[[],[],[],[]],[[]]]
 => ? = 4
[[[[[[]]]],[]],[]]
 => [[],[],[],[[]],[[]]]
 => ? = 5
[[[[],[[],[]]]],[]]
 => [[[[]],[]],[],[[]]]
 => ? = 3
[[[[],[[[]]]]],[]]
 => [[[],[],[]],[],[[]]]
 => ? = 3
[[[[[]],[[]]]],[]]
 => [[],[[],[]],[],[[]]]
 => ? = 4
[[[[[],[]],[]]],[]]
 => [[[]],[[]],[],[[]]]
 => ? = 4
[[[[[[]]],[]]],[]]
 => [[],[],[[]],[],[[]]]
 => ? = 5
[[[[[],[],[]]]],[]]
 => [[[[]]],[],[],[[]]]
 => ? = 4
[[[[[],[[]]]]],[]]
 => [[[],[]],[],[],[[]]]
 => ? = 4
[[[[[[]],[]]]],[]]
 => [[],[[]],[],[],[[]]]
 => ? = 5
[[[[[[],[]]]]],[]]
 => [[[]],[],[],[],[[]]]
 => ? = 5
[[[[[[[]]]]]],[]]
 => [[],[],[],[],[],[[]]]
 => ? = 6
[[],[[[[],[[[]]]]]]]
 => [[[[],[],[]],[],[],[]]]
 => ? = 4
[[],[[[[[[]]],[]]]]]
 => [[[],[],[[]],[],[],[]]]
 => ? = 6
[[],[[[[[],[],[]]]]]]
 => [[[[[]]],[],[],[],[]]]
 => ? = 5
[[],[[[[[],[[]]]]]]]
 => [[[[],[]],[],[],[],[]]]
 => ? = 5
[[],[[[[[[]],[]]]]]]
 => [[[],[[]],[],[],[],[]]]
 => ? = 6
[[],[[[[[[],[]]]]]]]
 => [[[[]],[],[],[],[],[]]]
 => ? = 6
[[],[[[[[[[]]]]]]]]
 => [[[],[],[],[],[],[],[]]]
 => ? = 7
[[[[[],[[[]]]]]],[]]
 => [[[],[],[]],[],[],[[]]]
 => ? = 4
[[[[[[[]]],[]]]],[]]
 => [[],[],[[]],[],[],[[]]]
 => ? = 6
[[[[[[],[],[]]]]],[]]
 => [[[[]]],[],[],[],[[]]]
 => ? = 5
[[[[[[],[[]]]]]],[]]
 => [[[],[]],[],[],[],[[]]]
 => ? = 5
[[[[[[[]],[]]]]],[]]
 => [[],[[]],[],[],[],[[]]]
 => ? = 6
[[[[[[[],[]]]]]],[]]
 => [[[]],[],[],[],[],[[]]]
 => ? = 6
[[[[[[[[]]]]]]],[]]
 => [[],[],[],[],[],[],[[]]]
 => ? = 7
Description
The maximum number of child nodes in a tree.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000846The maximal number of elements covering an element of a poset. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St000845The maximal number of elements covered by an element in a poset. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2.