- FindStat

Identifier

Mp00104: Binary words —reverse⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000982: Binary words ⟶ ℤ

Values

0 => 0 => 1 => 1 => 1

1 => 1 => 0 => 0 => 1

00 => 00 => 11 => 10 => 1

01 => 10 => 01 => 00 => 2

10 => 01 => 10 => 11 => 2

11 => 11 => 00 => 01 => 1

000 => 000 => 111 => 101 => 1

001 => 100 => 011 => 001 => 2

010 => 010 => 101 => 111 => 3

011 => 110 => 001 => 011 => 2

100 => 001 => 110 => 100 => 2

101 => 101 => 010 => 000 => 3

110 => 011 => 100 => 110 => 2

111 => 111 => 000 => 010 => 1

0000 => 0000 => 1111 => 1010 => 1

0001 => 1000 => 0111 => 0010 => 2

0010 => 0100 => 1011 => 1110 => 3

0011 => 1100 => 0011 => 0110 => 2

0100 => 0010 => 1101 => 1000 => 3

0101 => 1010 => 0101 => 0000 => 4

0110 => 0110 => 1001 => 1100 => 2

0111 => 1110 => 0001 => 0100 => 2

1000 => 0001 => 1110 => 1011 => 2

1001 => 1001 => 0110 => 0011 => 2

1010 => 0101 => 1010 => 1111 => 4

1011 => 1101 => 0010 => 0111 => 3

1100 => 0011 => 1100 => 1001 => 2

1101 => 1011 => 0100 => 0001 => 3

1110 => 0111 => 1000 => 1101 => 2

1111 => 1111 => 0000 => 0101 => 1

00000 => 00000 => 11111 => 10101 => 1

00001 => 10000 => 01111 => 00101 => 2

00010 => 01000 => 10111 => 11101 => 3

00011 => 11000 => 00111 => 01101 => 2

00100 => 00100 => 11011 => 10001 => 3

00101 => 10100 => 01011 => 00001 => 4

00110 => 01100 => 10011 => 11001 => 2

00111 => 11100 => 00011 => 01001 => 2

01000 => 00010 => 11101 => 10111 => 3

01001 => 10010 => 01101 => 00111 => 3

01010 => 01010 => 10101 => 11111 => 5

01011 => 11010 => 00101 => 01111 => 4

01100 => 00110 => 11001 => 10011 => 2

01101 => 10110 => 01001 => 00011 => 3

01110 => 01110 => 10001 => 11011 => 2

01111 => 11110 => 00001 => 01011 => 2

10000 => 00001 => 11110 => 10100 => 2

10001 => 10001 => 01110 => 00100 => 2

10010 => 01001 => 10110 => 11100 => 3

10011 => 11001 => 00110 => 01100 => 2

10100 => 00101 => 11010 => 10000 => 4

10101 => 10101 => 01010 => 00000 => 5

10110 => 01101 => 10010 => 11000 => 3

10111 => 11101 => 00010 => 01000 => 3

11000 => 00011 => 11100 => 10110 => 2

11001 => 10011 => 01100 => 00110 => 2

11010 => 01011 => 10100 => 11110 => 4

11011 => 11011 => 00100 => 01110 => 3

11100 => 00111 => 11000 => 10010 => 2

11101 => 10111 => 01000 => 00010 => 3

11110 => 01111 => 10000 => 11010 => 2

11111 => 11111 => 00000 => 01010 => 1

000000 => 000000 => 111111 => 101010 => 1

000001 => 100000 => 011111 => 001010 => 2

000010 => 010000 => 101111 => 111010 => 3

000011 => 110000 => 001111 => 011010 => 2

000100 => 001000 => 110111 => 100010 => 3

000101 => 101000 => 010111 => 000010 => 4

000110 => 011000 => 100111 => 110010 => 2

000111 => 111000 => 000111 => 010010 => 2

001000 => 000100 => 111011 => 101110 => 3

001001 => 100100 => 011011 => 001110 => 3

001010 => 010100 => 101011 => 111110 => 5

001011 => 110100 => 001011 => 011110 => 4

001100 => 001100 => 110011 => 100110 => 2

001101 => 101100 => 010011 => 000110 => 3

001110 => 011100 => 100011 => 110110 => 2

001111 => 111100 => 000011 => 010110 => 2

010000 => 000010 => 111101 => 101000 => 3

010001 => 100010 => 011101 => 001000 => 3

010010 => 010010 => 101101 => 111000 => 3

010011 => 110010 => 001101 => 011000 => 3

010100 => 001010 => 110101 => 100000 => 5

010101 => 101010 => 010101 => 000000 => 6

010110 => 011010 => 100101 => 110000 => 4

010111 => 111010 => 000101 => 010000 => 4

011000 => 000110 => 111001 => 101100 => 2

011001 => 100110 => 011001 => 001100 => 2

011010 => 010110 => 101001 => 111100 => 4

011011 => 110110 => 001001 => 011100 => 3

011100 => 001110 => 110001 => 100100 => 2

011101 => 101110 => 010001 => 000100 => 3

011110 => 011110 => 100001 => 110100 => 2

011111 => 111110 => 000001 => 010100 => 2

100000 => 000001 => 111110 => 101011 => 2

100001 => 100001 => 011110 => 001011 => 2

100010 => 010001 => 101110 => 111011 => 3

100011 => 110001 => 001110 => 011011 => 2

100100 => 001001 => 110110 => 100011 => 3

100101 => 101001 => 010110 => 000011 => 4

100110 => 011001 => 100110 => 110011 => 2

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$F_{1} = 2\ q$

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

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

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

$F_{5} = 2\ q + 14\ q^{2} + 10\ q^{3} + 4\ q^{4} + 2\ q^{5}$

$F_{6} = 2\ q + 24\ q^{2} + 22\ q^{3} + 10\ q^{4} + 4\ q^{5} + 2\ q^{6}$

$F_{7} = 2\ q + 40\ q^{2} + 46\ q^{3} + 24\ q^{4} + 10\ q^{5} + 4\ q^{6} + 2\ q^{7}$

$F_{8} = 2\ q + 66\ q^{2} + 94\ q^{3} + 54\ q^{4} + 24\ q^{5} + 10\ q^{6} + 4\ q^{7} + 2\ q^{8}$

$F_{9} = 2\ q + 108\ q^{2} + 188\ q^{3} + 118\ q^{4} + 56\ q^{5} + 24\ q^{6} + 10\ q^{7} + 4\ q^{8} + 2\ q^{9}$

Description

The length of the longest constant subword.

Map

reverse

Description

Return the reversal of a 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.

Map

complement

Description

Send a binary word to the word obtained by interchanging the two letters.