Identifier
Values
[1,0] => [1,0] => 10 => 11 => 1
[1,0,1,0] => [1,1,0,0] => 1100 => 1001 => 2
[1,1,0,0] => [1,0,1,0] => 1010 => 1111 => 1
[1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 101101 => 3
[1,0,1,1,0,0] => [1,1,0,0,1,0] => 110010 => 100111 => 2
[1,1,0,0,1,0] => [1,0,1,1,0,0] => 101100 => 111001 => 2
[1,1,0,1,0,0] => [1,1,0,1,0,0] => 110100 => 100001 => 2
[1,1,1,0,0,0] => [1,0,1,0,1,0] => 101010 => 111111 => 1
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 11110000 => 10100101 => 4
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 11100010 => 10110111 => 3
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 10011001 => 2
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => 11100100 => 10110001 => 3
[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 11001010 => 10011111 => 2
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 10111000 => 11101101 => 3
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 11100111 => 2
[1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 11011000 => 10001101 => 3
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => 10111101 => 3
[1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 10000111 => 2
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 10101100 => 11111001 => 2
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 11100001 => 2
[1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => 10000001 => 2
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => 11111111 => 1
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Description
The length of the longest alternating subword.
This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
Lalanne-Kreweras involution
Description
The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.
Map
alternating inverse
Description
Sends a binary word $w_1\cdots w_m$ to the binary word $v_1 \cdots v_m$ with $v_i = w_i$ if $i$ is odd and $v_i = 1 - w_i$ if $i$ is even.
This map is used in [1], see Definitions 3.2 and 5.1.