- FindStat

Identifier

Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000876: Binary words ⟶ ℤ

Values

0 => 0 => 1

1 => 1 => 1

00 => 00 => 2

01 => 10 => 1

10 => 10 => 1

11 => 11 => 2

000 => 000 => 3

001 => 100 => 1

010 => 100 => 1

011 => 110 => 1

100 => 100 => 1

101 => 110 => 1

110 => 110 => 1

111 => 111 => 3

0000 => 0000 => 4

0001 => 1000 => 2

0010 => 1000 => 2

0011 => 1010 => 1

0100 => 1000 => 2

0101 => 1100 => 1

0110 => 1010 => 1

0111 => 1110 => 2

1000 => 1000 => 2

1001 => 1010 => 1

1010 => 1100 => 1

1011 => 1110 => 2

1100 => 1010 => 1

1101 => 1110 => 2

1110 => 1110 => 2

1111 => 1111 => 4

00000 => 00000 => 5

00001 => 10000 => 3

00010 => 10000 => 3

00011 => 10010 => 2

00100 => 10000 => 3

00101 => 11000 => 1

00110 => 10010 => 2

00111 => 10110 => 2

01000 => 10000 => 3

01001 => 11000 => 1

01010 => 11000 => 1

01011 => 11100 => 1

01100 => 10010 => 2

01101 => 11100 => 1

01110 => 10110 => 2

01111 => 11110 => 3

10000 => 10000 => 3

10001 => 10010 => 2

10010 => 11000 => 1

10011 => 10110 => 2

10100 => 11000 => 1

10101 => 11100 => 1

10110 => 11100 => 1

10111 => 11110 => 3

11000 => 10010 => 2

11001 => 10110 => 2

11010 => 11100 => 1

11011 => 11110 => 3

11100 => 10110 => 2

11101 => 11110 => 3

11110 => 11110 => 3

11111 => 11111 => 5

000000 => 000000 => 6

000001 => 100000 => 4

000010 => 100000 => 4

000011 => 100010 => 3

000100 => 100000 => 4

000101 => 101000 => 2

000110 => 100010 => 3

000111 => 100110 => 2

001000 => 100000 => 4

001001 => 110000 => 2

001010 => 101000 => 2

001011 => 101100 => 1

001100 => 100010 => 3

001101 => 110010 => 1

001110 => 100110 => 2

001111 => 101110 => 3

010000 => 100000 => 4

010001 => 101000 => 2

010010 => 110000 => 2

010011 => 110010 => 1

010100 => 101000 => 2

010101 => 111000 => 1

010110 => 101100 => 1

010111 => 111010 => 2

011000 => 100010 => 3

011001 => 101100 => 1

011010 => 110010 => 1

011011 => 111100 => 2

011100 => 100110 => 2

011101 => 111010 => 2

011110 => 101110 => 3

011111 => 111110 => 4

100000 => 100000 => 4

100001 => 100010 => 3

100010 => 101000 => 2

100011 => 100110 => 2

100100 => 110000 => 2

100101 => 101100 => 1

100110 => 110010 => 1

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$F_{1} = 2\ q$

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

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

$F_{4} = 6\ q + 8\ q^{2} + 2\ q^{4}$

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

$F_{6} = 14\ q + 24\ q^{2} + 12\ q^{3} + 12\ q^{4} + 2\ q^{6}$

$F_{7} = 28\ q + 28\ q^{2} + 42\ q^{3} + 14\ q^{4} + 14\ q^{5} + 2\ q^{7}$

$F_{8} = 46\ q + 64\ q^{2} + 48\ q^{3} + 64\ q^{4} + 16\ q^{5} + 16\ q^{6} + 2\ q^{8}$

$F_{9} = 72\ q + 108\ q^{2} + 150\ q^{3} + 54\ q^{4} + 90\ q^{5} + 18\ q^{6} + 18\ q^{7} + 2\ q^{9}$

Description

The number of factors in the Catalan decomposition of a binary word.
Every binary word can be written in a unique way as

$(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$ , where

$\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the number of factors in the Catalan factorisation, that is,

$\ell + m$ if the middle Dyck word is empty and

$\ell + 1 + m$ otherwise.

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.