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
to binary word
Description
Return the Dyck word as binary word.
Map
bounce path
Description
The bounce path determined by an integer composition.
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 $w$.
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