Identifier
-
Mp00093:
Dyck paths
—to binary word⟶
Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤ
Values
[1,0] => 10 => [1,1] => [1,0,1,0] => 2
[1,0,1,0] => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 4
[1,1,0,0] => 1100 => [2,2] => [1,1,0,0,1,1,0,0] => 0
[1,0,1,0,1,0] => 101010 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,1,0,0] => 101100 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 2
[1,1,0,0,1,0] => 110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2
[1,1,0,1,0,0] => 110100 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 2
[1,1,1,0,0,0] => 111000 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 0
[1,0,1,0,1,0,1,0] => 10101010 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 8
[1,0,1,0,1,1,0,0] => 10101100 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,1,0,0,1,0] => 10110010 => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0] => 4
[1,0,1,1,0,1,0,0] => 10110100 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 4
[1,0,1,1,1,0,0,0] => 10111000 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 2
[1,1,0,0,1,0,1,0] => 11001010 => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => 4
[1,1,0,0,1,1,0,0] => 11001100 => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => 0
[1,1,0,1,0,0,1,0] => 11010010 => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0] => 4
[1,1,0,1,0,1,0,0] => 11010100 => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => 4
[1,1,0,1,1,0,0,0] => 11011000 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0] => 1
[1,1,1,0,0,0,1,0] => 11100010 => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0] => 2
[1,1,1,0,0,1,0,0] => 11100100 => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0] => 1
[1,1,1,0,1,0,0,0] => 11101000 => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0] => 2
[1,1,1,1,0,0,0,0] => 11110000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => 0
[1,0,1,0,1,0,1,0,1,0] => 1010101010 => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 10
[1,0,1,0,1,0,1,1,0,0] => 1010101100 => [1,1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 6
[1,0,1,1,0,0,1,1,0,0] => 1011001100 => [1,1,2,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => 2
[1,1,0,1,0,1,0,1,0,0] => 1101010100 => [2,1,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 6
[1,1,0,1,1,1,0,0,0,0] => 1101110000 => [2,1,3,4] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 1
[1,1,1,0,1,1,0,0,0,0] => 1110110000 => [3,1,2,4] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 1
[1,1,1,1,0,0,0,1,0,0] => 1111000100 => [4,3,1,2] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,1,0,0] => 1
[1,1,1,1,0,0,1,0,0,0] => 1111001000 => [4,2,1,3] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,0] => 1
[1,1,1,1,1,0,0,0,0,0] => 1111100000 => [5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => 101010101010 => [1,1,1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 12
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Description
The number of rises of length 1 of a Dyck path.
Map
delta morphism
Description
Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
bounce path
Description
The bounce path determined by an integer composition.
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