- FindStat

Identifier

Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St000392: Binary words ⟶ ℤ

Values

[1,0] => 10 => 11 => 01 => 1
[1,0,1,0] => 1010 => 1111 => 0111 => 3
[1,1,0,0] => 1100 => 1001 => 0001 => 1
[1,0,1,0,1,0] => 101010 => 111111 => 011111 => 5
[1,0,1,1,0,0] => 101100 => 111001 => 011001 => 2
[1,1,0,0,1,0] => 110010 => 100111 => 000111 => 3
[1,1,0,1,0,0] => 110100 => 100001 => 000001 => 1
[1,1,1,0,0,0] => 111000 => 101101 => 001101 => 2
[1,0,1,0,1,0,1,0] => 10101010 => 11111111 => 01111111 => 7
[1,0,1,0,1,1,0,0] => 10101100 => 11111001 => 01111001 => 4
[1,0,1,1,0,0,1,0] => 10110010 => 11100111 => 01100111 => 3
[1,0,1,1,0,1,0,0] => 10110100 => 11100001 => 01100001 => 2
[1,0,1,1,1,0,0,0] => 10111000 => 11101101 => 01101101 => 2
[1,1,0,0,1,0,1,0] => 11001010 => 10011111 => 00011111 => 5
[1,1,0,0,1,1,0,0] => 11001100 => 10011001 => 00011001 => 2
[1,1,0,1,0,0,1,0] => 11010010 => 10000111 => 00000111 => 3
[1,1,0,1,0,1,0,0] => 11010100 => 10000001 => 00000001 => 1
[1,1,0,1,1,0,0,0] => 11011000 => 10001101 => 00001101 => 2
[1,1,1,0,0,0,1,0] => 11100010 => 10110111 => 00110111 => 3
[1,1,1,0,0,1,0,0] => 11100100 => 10110001 => 00110001 => 2
[1,1,1,0,1,0,0,0] => 11101000 => 10111101 => 00111101 => 4
[1,1,1,1,0,0,0,0] => 11110000 => 10100101 => 00100101 => 1
[1,0,1,0,1,0,1,0,1,0] => 1010101010 => 1111111111 => 0111111111 => 9
[1,0,1,0,1,0,1,1,0,0] => 1010101100 => 1111111001 => 0111111001 => 6
[1,0,1,0,1,1,0,1,0,0] => 1010110100 => 1111100001 => 0111100001 => 4
[1,0,1,1,0,0,1,1,0,0] => 1011001100 => 1110011001 => 0110011001 => 2
[1,0,1,1,0,1,0,1,0,0] => 1011010100 => 1110000001 => 0110000001 => 2
[1,0,1,1,1,0,0,1,0,0] => 1011100100 => 1110110001 => 0110110001 => 2
[1,0,1,1,1,1,0,0,0,0] => 1011110000 => 1110100101 => 0110100101 => 2
[1,1,0,0,1,0,1,0,1,0] => 1100101010 => 1001111111 => 0001111111 => 7
[1,1,0,0,1,0,1,1,0,0] => 1100101100 => 1001111001 => 0001111001 => 4
[1,1,0,0,1,1,0,0,1,0] => 1100110010 => 1001100111 => 0001100111 => 3
[1,1,0,0,1,1,0,1,0,0] => 1100110100 => 1001100001 => 0001100001 => 2
[1,1,0,0,1,1,1,0,0,0] => 1100111000 => 1001101101 => 0001101101 => 2
[1,1,0,1,0,0,1,0,1,0] => 1101001010 => 1000011111 => 0000011111 => 5
[1,1,0,1,0,0,1,1,0,0] => 1101001100 => 1000011001 => 0000011001 => 2
[1,1,0,1,0,1,0,0,1,0] => 1101010010 => 1000000111 => 0000000111 => 3
[1,1,0,1,0,1,0,1,0,0] => 1101010100 => 1000000001 => 0000000001 => 1
[1,1,0,1,0,1,1,0,0,0] => 1101011000 => 1000001101 => 0000001101 => 2
[1,1,0,1,1,0,0,0,1,0] => 1101100010 => 1000110111 => 0000110111 => 3
[1,1,0,1,1,0,0,1,0,0] => 1101100100 => 1000110001 => 0000110001 => 2
[1,1,0,1,1,0,1,0,0,0] => 1101101000 => 1000111101 => 0000111101 => 4
[1,1,0,1,1,1,0,0,0,0] => 1101110000 => 1000100101 => 0000100101 => 1
[1,1,1,0,0,1,0,0,1,0] => 1110010010 => 1011000111 => 0011000111 => 3
[1,1,1,0,0,1,1,0,0,0] => 1110011000 => 1011001101 => 0011001101 => 2
[1,1,1,0,1,0,1,0,0,0] => 1110101000 => 1011111101 => 0011111101 => 6
[1,1,1,1,0,0,0,0,1,0] => 1111000010 => 1010010111 => 0010010111 => 3
[1,1,1,1,0,0,0,1,0,0] => 1111000100 => 1010010001 => 0010010001 => 1
[1,1,1,1,0,0,1,0,0,0] => 1111001000 => 1010011101 => 0010011101 => 3
[1,1,1,1,0,1,0,0,0,0] => 1111010000 => 1010000101 => 0010000101 => 1
[1,1,1,1,1,0,0,0,0,0] => 1111100000 => 1010110101 => 0010110101 => 2

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$F_{1} = q$

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

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

$F_{4} = 2\ q + 5\ q^{2} + 3\ q^{3} + 2\ q^{4} + q^{5} + q^{7}$

Description

The length of the longest run of ones in a binary word.

Map

twist

Description

Return the binary word with first letter inverted.

Map

to binary word

Description

Return the Dyck word as binary word.

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.