- FindStat

Identifier

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St000983: Binary words ⟶ ℤ

Values

[1,0] => [1,0] => 10 => 01 => 2
[1,0,1,0] => [1,1,0,0] => 1100 => 1011 => 3
[1,1,0,0] => [1,0,1,0] => 1010 => 1001 => 2
[1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 010111 => 4
[1,0,1,1,0,0] => [1,1,0,0,1,0] => 110010 => 011011 => 3
[1,1,0,0,1,0] => [1,0,1,1,0,0] => 101100 => 010001 => 3
[1,1,0,1,0,0] => [1,1,0,1,0,0] => 110100 => 010011 => 3
[1,1,1,0,0,0] => [1,0,1,0,1,0] => 101010 => 011001 => 2
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 11110000 => 10101111 => 5
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 11100010 => 10010111 => 4
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 10111011 => 3
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => 11100100 => 10110111 => 3
[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 11001010 => 10011011 => 3
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 10111000 => 10100001 => 4
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 10010001 => 3
[1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 11011000 => 10100011 => 4
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => 10100111 => 4
[1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 10010011 => 3
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 10101100 => 10111001 => 3
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 10110001 => 3
[1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => 10110011 => 3
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => 10011001 => 2

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$F_{1} = q^{2}$

$F_{2} = q^{2} + q^{3}$

$F_{3} = q^{2} + 3\ q^{3} + q^{4}$

$F_{4} = q^{2} + 8\ q^{3} + 4\ q^{4} + q^{5}$

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

zeros to flag zeros

Description

Return a binary word of the same length, such that the number of occurrences of

$10$ in the word obtained by prepending the reverse of the complement equals the number of

$0$ s in the original word.
For example, the image of the word

$w=1\dots1$ is

$1\dots1$ , because

$w$ has no zeros, and

$1\dots1$ is the only word such that prepending the reverse of its complement has no occurrence of the factor

$10$ .
On the other hand,

$0\dots0$ must be mapped to

$10\dots10$ if the length is even, and

$010\dots10$ if it is odd.

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.