- FindStat

Identifier

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤ

Values

[[]] => [1,0] => [1,0] => 10 => 1

[[],[]] => [1,0,1,0] => [1,1,0,0] => 1100 => 2

[[[]]] => [1,1,0,0] => [1,0,1,0] => 1010 => 1

[[],[],[]] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 3

[[],[[]]] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => 110100 => 2

[[[]],[]] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => 110010 => 2

[[[],[]]] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 101100 => 2

[[[[]]]] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 101010 => 1

[[],[],[],[]] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 11110000 => 4

[[],[],[[]]] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => 3

[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 11100100 => 3

[[],[[],[]]] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 2

[[],[[[]]]] => [1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => 2

[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 3

[[[]],[[]]] => [1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 2

[[[],[]],[]] => [1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 2

[[[[]]],[]] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => 11001010 => 2

[[[],[],[]]] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 3

[[[],[[]]]] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 2

[[[[]],[]]] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 2

[[[[],[]]]] => [1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 10101100 => 2

[[[[[]]]]] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => 1

[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1111100000 => 5

[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1111010000 => 4

[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0] => 1111001000 => 4

[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => 1110110000 => 3

[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1110101000 => 3

[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 1111000100 => 4

[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0] => 1110100100 => 3

[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 1110011000 => 3

[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1110010100 => 3

[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1101110000 => 3

[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => 1101101000 => 2

[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 1101100100 => 2

[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1101011000 => 2

[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 1101010100 => 2

[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 1111000010 => 4

[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 1110100010 => 3

[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => 1110010010 => 3

[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 1101100010 => 2

[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => 1101010010 => 2

[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1110001100 => 3

[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => 1110001010 => 3

[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => 1101001100 => 2

[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 1101001010 => 2

[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1100111000 => 3

[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 1100110100 => 2

[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 1100110010 => 2

[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 1100101100 => 2

[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 1100101010 => 2

[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1011110000 => 4

[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => 1011101000 => 3

[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 1011100100 => 3

[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => 1011011000 => 2

[[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 2

[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 1011100010 => 3

[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 1011010010 => 2

[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 1011001100 => 2

[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => 1011001010 => 2

[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 1010111000 => 3

[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 1010110100 => 2

[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 1010110010 => 2

[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 1010101100 => 2

[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 1010101010 => 1

[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 111111000000 => 6

[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 111110100000 => 5

[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 111110010000 => 5

[[],[],[],[[],[]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 111101100000 => 4

[[],[],[],[[[]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => 111101010000 => 4

[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 111110001000 => 5

[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 111101001000 => 4

[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 111100110000 => 4

[[],[],[[[]]],[]] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 111100101000 => 4

[[],[],[[],[],[]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 111011100000 => 3

[[],[],[[],[[]]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => 111011010000 => 3

[[],[],[[[]],[]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => 111011001000 => 3

[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => 111010110000 => 3

[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => 111010101000 => 3

[[],[[]],[],[],[]] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 111110000100 => 5

[[],[[]],[],[[]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => 111101000100 => 4

[[],[[]],[[]],[]] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => 111100100100 => 4

[[],[[]],[[],[]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 111011000100 => 3

[[],[[]],[[[]]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => 111010100100 => 3

[[],[[],[]],[],[]] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 111100011000 => 4

[[],[[[]]],[],[]] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 111100010100 => 4

[[],[[],[]],[[]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => 111010011000 => 3

[[],[[[]]],[[]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 111010010100 => 3

[[],[[],[],[]],[]] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 111001110000 => 3

[[],[[],[[]]],[]] => [1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => 111001101000 => 3

[[],[[[]],[]],[]] => [1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => 111001100100 => 3

[[],[[[],[]]],[]] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => 111001011000 => 3

[[],[[[[]]]],[]] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 111001010100 => 3

[[],[[],[],[],[]]] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 110111100000 => 4

[[],[[],[],[[]]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => 110111010000 => 3

[[],[[],[[]],[]]] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 110111001000 => 3

[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => 110110110000 => 2

[[],[[],[[[]]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => 110110101000 => 2

[[],[[[]],[],[]]] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 110111000100 => 3

[[],[[[]],[[]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => 110110100100 => 2

[[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => 110110011000 => 2

[[],[[[[]]],[]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 110110010100 => 2

>>> Load all 236 entries. <<<

[[],[[[],[],[]]]] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 110101110000 => 3

[[],[[[],[[]]]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => 110101101000 => 2

[[],[[[[]],[]]]] => [1,0,1,1,1,1,0,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => 110101100100 => 2

[[],[[[[],[]]]]] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 110101011000 => 2

[[],[[[[[]]]]]] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 110101010100 => 2

[[[]],[],[],[],[]] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 111110000010 => 5

[[[]],[],[],[[]]] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => 111101000010 => 4

[[[]],[],[[]],[]] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => 111100100010 => 4

[[[]],[],[[],[]]] => [1,1,0,0,1,0,1,1,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => 111011000010 => 3

[[[]],[],[[[]]]] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => 111010100010 => 3

[[[]],[[]],[],[]] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => 111100010010 => 4

[[[]],[[]],[[]]] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => 111010010010 => 3

[[[]],[[],[]],[]] => [1,1,0,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => 111001100010 => 3

[[[]],[[[]]],[]] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => 111001010010 => 3

[[[]],[[],[],[]]] => [1,1,0,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 110111000010 => 3

[[[]],[[],[[]]]] => [1,1,0,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => 110110100010 => 2

[[[]],[[[]],[]]] => [1,1,0,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => 110110010010 => 2

[[[]],[[[],[]]]] => [1,1,0,0,1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => 110101100010 => 2

[[[]],[[[[]]]]] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 110101010010 => 2

[[[],[]],[],[],[]] => [1,1,0,1,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 111100001100 => 4

[[[[]]],[],[],[]] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 111100001010 => 4

[[[],[]],[],[[]]] => [1,1,0,1,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,1,0,0] => 111010001100 => 3

[[[[]]],[],[[]]] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => 111010001010 => 3

[[[],[]],[[]],[]] => [1,1,0,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,1,0,0] => 111001001100 => 3

[[[[]]],[[]],[]] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => 111001001010 => 3

[[[],[]],[[],[]]] => [1,1,0,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => 110110001100 => 2

[[[],[]],[[[]]]] => [1,1,0,1,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => 110101001100 => 2

[[[[]]],[[],[]]] => [1,1,1,0,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => 110110001010 => 2

[[[[]]],[[[]]]] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => 110101001010 => 2

[[[],[],[]],[],[]] => [1,1,0,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => 111000111000 => 3

[[[],[[]]],[],[]] => [1,1,0,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,1,0,0] => 111000110100 => 3

[[[[]],[]],[],[]] => [1,1,1,0,0,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 111000110010 => 3

[[[[],[]]],[],[]] => [1,1,1,0,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => 111000101100 => 3

[[[[[]]]],[],[]] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 111000101010 => 3

[[[],[],[]],[[]]] => [1,1,0,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => 110100111000 => 3

[[[],[[]]],[[]]] => [1,1,0,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => 110100110100 => 2

[[[[]],[]],[[]]] => [1,1,1,0,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => 110100110010 => 2

[[[[],[]]],[[]]] => [1,1,1,0,1,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => 110100101100 => 2

[[[[[]]]],[[]]] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 110100101010 => 2

[[[],[],[],[]],[]] => [1,1,0,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 110011110000 => 4

[[[],[],[[]]],[]] => [1,1,0,1,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => 110011101000 => 3

[[[],[[]],[]],[]] => [1,1,0,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => 110011100100 => 3

[[[],[[],[]]],[]] => [1,1,0,1,1,0,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => 110011011000 => 2

[[[],[[[]]]],[]] => [1,1,0,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,1,0,0] => 110011010100 => 2

[[[[]],[],[]],[]] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 110011100010 => 3

[[[[]],[[]]],[]] => [1,1,1,0,0,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => 110011010010 => 2

[[[[],[]],[]],[]] => [1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 110011001100 => 2

[[[[[]]],[]],[]] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 110011001010 => 2

[[[[],[],[]]],[]] => [1,1,1,0,1,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => 110010111000 => 3

[[[[],[[]]]],[]] => [1,1,1,0,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,1,0,0] => 110010110100 => 2

[[[[[]],[]]],[]] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => 110010110010 => 2

[[[[[],[]]]],[]] => [1,1,1,1,0,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => 110010101100 => 2

[[[[[[]]]]],[]] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => 110010101010 => 2

[[[],[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 101111100000 => 5

[[[],[],[],[[]]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,1,0,1,0,0,0,0] => 101111010000 => 4

[[[],[],[[]],[]]] => [1,1,0,1,0,1,1,0,0,1,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 101111001000 => 4

[[[],[],[[],[]]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => [1,0,1,1,1,0,1,1,0,0,0,0] => 101110110000 => 3

[[[],[],[[[]]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,1,0,0,0] => 101110101000 => 3

[[[],[[]],[],[]]] => [1,1,0,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,1,0,0] => 101111000100 => 4

[[[],[[]],[[]]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,1,0,0,1,0,0] => 101110100100 => 3

[[[],[[],[]],[]]] => [1,1,0,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,1,0,0,0] => 101110011000 => 3

[[[],[[[]]],[]]] => [1,1,0,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,1,0,0] => 101110010100 => 3

[[[],[[],[],[]]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,1,1,0,0,0,0] => 101101110000 => 3

[[[],[[],[[]]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,1,1,0,1,0,0,0] => 101101101000 => 2

[[[],[[[]],[]]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,1,0,0] => 101101100100 => 2

[[[],[[[],[]]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => [1,0,1,1,0,1,0,1,1,0,0,0] => 101101011000 => 2

[[[],[[[[]]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,0,1,1,0,1,0,1,0,1,0,0] => 101101010100 => 2

[[[[]],[],[],[]]] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 101111000010 => 4

[[[[]],[],[[]]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => 101110100010 => 3

[[[[]],[[]],[]]] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => 101110010010 => 3

[[[[]],[[],[]]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => 101101100010 => 2

[[[[]],[[[]]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => 101101010010 => 2

[[[[],[]],[],[]]] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 101110001100 => 3

[[[[[]]],[],[]]] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => 101110001010 => 3

[[[[],[]],[[]]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,1,0,0] => 101101001100 => 2

[[[[[]]],[[]]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => 101101001010 => 2

[[[[],[],[]],[]]] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 101100111000 => 3

[[[[],[[]]],[]]] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => 101100110100 => 2

[[[[[]],[]],[]]] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 101100110010 => 2

[[[[[],[]]],[]]] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 101100101100 => 2

[[[[[[]]]],[]]] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 101100101010 => 2

[[[[],[],[],[]]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 101011110000 => 4

[[[[],[],[[]]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,1,0,1,0,0,0] => 101011101000 => 3

[[[[],[[]],[]]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,1,0,0] => 101011100100 => 3

[[[[],[[],[]]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,1,1,0,0,0] => 101011011000 => 2

[[[[],[[[]]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,0,1,0,1,1,0,1,0,1,0,0] => 101011010100 => 2

[[[[[]],[],[]]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 101011100010 => 3

[[[[[]],[[]]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => 101011010010 => 2

[[[[[],[]],[]]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 101011001100 => 2

[[[[[[]]],[]]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 101011001010 => 2

[[[[[],[],[]]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 101010111000 => 3

[[[[[],[[]]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 2

[[[[[[]],[]]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 101010110010 => 2

[[[[[[],[]]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => 2

[[[[[[[]]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 101010101010 => 1

[[[]],[],[],[],[],[]] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 11111100000010 => 6

[[[]],[],[],[],[[]]] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => 11111010000010 => 5

[[[]],[],[],[[]],[]] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0] => 11111001000010 => 5

[[[]],[],[],[[],[]]] => [1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => 11110110000010 => 4

[[[]],[],[],[[[]]]] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0] => 11110101000010 => 4

[[[]],[],[[]],[],[]] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0] => 11111000100010 => 5

[[[]],[],[[]],[[]]] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0,1,0] => 11110100100010 => 4

[[[]],[],[[],[]],[]] => [1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0] => 11110011000010 => 4

[[[]],[],[[],[],[]]] => [1,1,0,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0] => 11101110000010 => 3

[[[]],[],[[],[[]]]] => [1,1,0,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0,1,0] => 11101101000010 => 3

[[[]],[[]],[],[],[]] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0] => 11111000010010 => 5

[[[]],[[],[],[],[]]] => [1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0] => 11011110000010 => 4

[[[],[]],[],[],[],[]] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 11111000001100 => 5

[[[],[],[],[],[]],[]] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 11001111100000 => 5

[[[],[],[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 10111111000000 => 6

[[[],[],[],[],[[]]]] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => 10111110100000 => 5

[[[],[],[],[[]],[]]] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 10111110010000 => 5

[[[],[],[],[[],[]]]] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0] => 10111101100000 => 4

[[[],[],[],[[[]]]]] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 10111101010000 => 4

[[[],[],[[]],[],[]]] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => 10111110001000 => 5

[[[],[],[[]],[[]]]] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,1,0,1,0,0,1,0,0,0] => 10111101001000 => 4

[[[],[],[[],[]],[]]] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0] => 10111100110000 => 4

[[[],[],[[],[],[]]]] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => 10111011100000 => 3

[[[],[],[[],[[]]]]] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => 10111011010000 => 3

[[[],[[]],[],[],[]]] => [1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 10111110000100 => 5

[[[],[[],[],[],[]]]] => [1,1,0,1,1,0,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0] => 10110111100000 => 4

[[[]],[],[],[],[],[],[]] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => 1111111000000010 => 7

[[[]],[],[],[],[],[[]]] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => 1111110100000010 => 6

[[[]],[],[],[],[[]],[]] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => 1111110010000010 => 6

[[[]],[],[],[],[[],[]]] => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0] => 1111101100000010 => 5

[[[]],[],[],[],[[[]]]] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0] => 1111101010000010 => 5

[[[]],[],[],[[]],[],[]] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => 1111110001000010 => 6

[[[]],[],[],[[],[],[]]] => [1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0] => 1111011100000010 => 4

[[[],[],[],[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 1011111110000000 => 7

[[[],[],[],[],[],[[]]]] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 1011111101000000 => 6

[[[],[],[],[],[[]],[]]] => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0] => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1011111100100000 => 6

[[[],[],[],[],[[],[]]]] => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 1011111011000000 => 5

[[[],[],[],[],[[[]]]]] => [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => 1011111010100000 => 5

[[[],[],[],[[]],[],[]]] => [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1011111100010000 => 6

[[[],[],[],[[],[],[]]]] => [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0] => [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 1011110111000000 => 4

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$F_{2} = q$

$F_{3} = q + q^{2}$

$F_{4} = q + 3\ q^{2} + q^{3}$

$F_{5} = q + 8\ q^{2} + 4\ q^{3} + q^{4}$

$F_{6} = q + 20\ q^{2} + 15\ q^{3} + 5\ q^{4} + q^{5}$

$F_{7} = q + 50\ q^{2} + 53\ q^{3} + 21\ q^{4} + 6\ q^{5} + q^{6}$

Description

The length of the longest run of ones in a binary word.

Map

to binary word

Description

Return the Dyck word as binary word.

Map

to Dyck path

Description

Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.

Map

decomposition reverse

Description

This map is recursively defined as follows.
The unique empty path of semilength

$0$ is sent to itself.
Let

$D$ be a Dyck path of semilength

$n > 0$ and decompose it into

$1 D_1 0 D_2$ with Dyck paths

$D_1, D_2$ of respective semilengths

$n_1$ and

$n_2$ such that

$n_1$ is minimal. One then has

$n_1+n_2 = n-1$ .
Now let

$\tilde D_1$ and

$\tilde D_2$ be the recursively defined respective images of

$D_1$ and

$D_2$ under this map. The image of

$D$ is then defined as

$1 \tilde D_2 0 \tilde D_1$ .