Identifier
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Mp00132:
Dyck paths
—switch returns and last double rise⟶
Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000390: Binary words ⟶ ℤ
Values
[1,0] => [1,0] => 10 => 00 => 0
[1,0,1,0] => [1,0,1,0] => 1010 => 0010 => 1
[1,1,0,0] => [1,1,0,0] => 1100 => 0100 => 1
[1,0,1,0,1,0] => [1,0,1,0,1,0] => 101010 => 001010 => 2
[1,0,1,1,0,0] => [1,1,0,1,0,0] => 110100 => 010100 => 2
[1,1,0,0,1,0] => [1,1,0,0,1,0] => 110010 => 010010 => 2
[1,1,0,1,0,0] => [1,0,1,1,0,0] => 101100 => 001100 => 1
[1,1,1,0,0,0] => [1,1,1,0,0,0] => 111000 => 011000 => 1
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => 10101010 => 00101010 => 3
[1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => 01010100 => 3
[1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,0] => 11010010 => 01010010 => 3
[1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 00110100 => 2
[1,0,1,1,1,0,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => 01101000 => 2
[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 11001010 => 01001010 => 3
[1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0] => 11100100 => 01100100 => 2
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 10110010 => 00110010 => 2
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => 10101100 => 00101100 => 2
[1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 01011000 => 2
[1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 01100010 => 2
[1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => 11001100 => 01001100 => 2
[1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 00111000 => 1
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 11110000 => 01110000 => 1
[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => 1010101010 => 0010101010 => 4
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => 1010101100 => 0010101100 => 3
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Description
The number of runs of ones in a binary word.
Map
twist
Description
Return the binary word with first letter inverted.
Map
switch returns and last double rise
Description
An alternative to the Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.
Map
to binary word
Description
Return the Dyck word as binary word.
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