Processing math: 100%

Identifier
Values
[1,0] => [1,0] => 10 => 11 => 1
[1,0,1,0] => [1,1,0,0] => 1100 => 0111 => 1
[1,1,0,0] => [1,0,1,0] => 1010 => 1101 => 2
[1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 001111 => 1
[1,0,1,1,0,0] => [1,1,0,0,1,0] => 110010 => 011101 => 2
[1,1,0,0,1,0] => [1,0,1,1,0,0] => 101100 => 110011 => 2
[1,1,0,1,0,0] => [1,0,1,0,1,0] => 101010 => 110101 => 3
[1,1,1,0,0,0] => [1,1,0,1,0,0] => 110100 => 111001 => 2
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 11110000 => 00011111 => 1
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 11100010 => 00111101 => 2
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 01110011 => 2
[1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => 11001010 => 01110101 => 3
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 11100100 => 01111001 => 2
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 10111000 => 11000111 => 2
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 11001101 => 3
[1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => 10101100 => 11010011 => 3
[1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => 11010101 => 4
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 11011001 => 3
[1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 11011000 => 11100011 => 2
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 11100101 => 3
[1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => 11101001 => 3
[1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => 11110001 => 2
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1111100000 => 0000111111 => 1
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1111000010 => 0001111101 => 2
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 1110010100 => 0111101001 => 3
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => 1111001000 => 0111110001 => 2
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Description
The number of runs of ones in a binary word.
Map
peaks-to-valleys
Description
Return the path that has a valley wherever the original path has a peak of height at least one.
More precisely, the height of a valley in the image is the height of the corresponding peak minus 2.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
path rowmotion
Description
Return the rowmotion of the binary word, regarded as a lattice path.
Consider the binary word of length n as a lattice path with n steps, where a 1 corresponds to an up step and a 0 corresponds to a down step.
This map returns the path whose peaks are the valleys of the original path with an up step appended.