- FindStat

Identifier

Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤ

Values

0 => 0 => 0 => 0

1 => 1 => 1 => 0

00 => 00 => 00 => 0

01 => 10 => 10 => 1

10 => 10 => 10 => 1

11 => 11 => 11 => 0

000 => 000 => 000 => 0

001 => 100 => 010 => 2

010 => 100 => 010 => 2

011 => 110 => 110 => 2

100 => 100 => 010 => 2

101 => 110 => 110 => 2

110 => 110 => 110 => 2

111 => 111 => 111 => 0

0000 => 0000 => 0000 => 0

0001 => 1000 => 0010 => 3

0010 => 1000 => 0010 => 3

0011 => 1010 => 0110 => 3

0100 => 1000 => 0010 => 3

0101 => 1100 => 1010 => 4

0110 => 1010 => 0110 => 3

0111 => 1110 => 1110 => 3

1000 => 1000 => 0010 => 3

1001 => 1010 => 0110 => 3

1010 => 1100 => 1010 => 4

1011 => 1110 => 1110 => 3

1100 => 1010 => 0110 => 3

1101 => 1110 => 1110 => 3

1110 => 1110 => 1110 => 3

1111 => 1111 => 1111 => 0

00000 => 00000 => 00000 => 0

00001 => 10000 => 00010 => 4

00010 => 10000 => 00010 => 4

00011 => 10010 => 00110 => 4

00100 => 10000 => 00010 => 4

00101 => 11000 => 01010 => 6

00110 => 10010 => 00110 => 4

00111 => 10110 => 01110 => 4

01000 => 10000 => 00010 => 4

01001 => 11000 => 01010 => 6

01010 => 11000 => 01010 => 6

01011 => 11100 => 11010 => 6

01100 => 10010 => 00110 => 4

01101 => 11100 => 11010 => 6

01110 => 10110 => 01110 => 4

01111 => 11110 => 11110 => 4

10000 => 10000 => 00010 => 4

10001 => 10010 => 00110 => 4

10010 => 11000 => 01010 => 6

10011 => 10110 => 01110 => 4

10100 => 11000 => 01010 => 6

10101 => 11100 => 11010 => 6

10110 => 11100 => 11010 => 6

10111 => 11110 => 11110 => 4

11000 => 10010 => 00110 => 4

11001 => 10110 => 01110 => 4

11010 => 11100 => 11010 => 6

11011 => 11110 => 11110 => 4

11100 => 10110 => 01110 => 4

11101 => 11110 => 11110 => 4

11110 => 11110 => 11110 => 4

11111 => 11111 => 11111 => 0

000000 => 000000 => 000000 => 0

000001 => 100000 => 000010 => 5

000010 => 100000 => 000010 => 5

000011 => 100010 => 000110 => 5

000100 => 100000 => 000010 => 5

000101 => 101000 => 010010 => 7

000110 => 100010 => 000110 => 5

000111 => 100110 => 001110 => 5

001000 => 100000 => 000010 => 5

001001 => 110000 => 001010 => 8

001010 => 101000 => 010010 => 7

001011 => 101100 => 110010 => 7

001100 => 100010 => 000110 => 5

001101 => 110010 => 010110 => 7

001110 => 100110 => 001110 => 5

001111 => 101110 => 011110 => 5

010000 => 100000 => 000010 => 5

010001 => 101000 => 010010 => 7

010010 => 110000 => 001010 => 8

010011 => 110010 => 010110 => 7

010100 => 101000 => 010010 => 7

010101 => 111000 => 101010 => 9

010110 => 101100 => 110010 => 7

010111 => 111010 => 110110 => 7

011000 => 100010 => 000110 => 5

011001 => 101100 => 110010 => 7

011010 => 110010 => 010110 => 7

011011 => 111100 => 111010 => 8

011100 => 100110 => 001110 => 5

011101 => 111010 => 110110 => 7

011110 => 101110 => 011110 => 5

011111 => 111110 => 111110 => 5

100000 => 100000 => 000010 => 5

100001 => 100010 => 000110 => 5

100010 => 101000 => 010010 => 7

100011 => 100110 => 001110 => 5

100100 => 110000 => 001010 => 8

100101 => 101100 => 110010 => 7

100110 => 110010 => 010110 => 7

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$F_{1} = 2$

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

$F_{3} = 2 + 6\ q^{2}$

$F_{4} = 2 + 12\ q^{3} + 2\ q^{4}$

$F_{5} = 2 + 20\ q^{4} + 10\ q^{6}$

$F_{6} = 2 + 30\ q^{5} + 24\ q^{7} + 6\ q^{8} + 2\ q^{9}$

$F_{7} = 2 + 42\ q^{6} + 42\ q^{8} + 28\ q^{10} + 14\ q^{12}$

$F_{8} = 2 + 56\ q^{7} + 64\ q^{9} + 64\ q^{11} + 12\ q^{12} + 40\ q^{13} + 16\ q^{15} + 2\ q^{16}$

$F_{9} = 2 + 72\ q^{8} + 90\ q^{10} + 108\ q^{12} + 126\ q^{14} + 72\ q^{16} + 24\ q^{18} + 18\ q^{20}$

Description

The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length

$n$ with

$a$ zeros, the generating function for the major index is the

$q$ -binomial coefficient

$\binom{n}{a}_q$ .

Map

inverse Foata bijection

Description

The inverse of Foata's bijection.
See Mp00096Foata bijection.

Map

Burrows-Wheeler

Description

The Burrows-Wheeler transform of a binary word.
The Burrows-Wheeler transform of a finite word

$w$ is obtained from

$w$ by first listing the conjugates of

$w$ in lexicographic order and then concatenating the final letters of the conjugates in this order.