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Identifier

Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St001313: Binary words ⟶ ℤ

Values

0 => 0 => 1

1 => 1 => 1

00 => 00 => 1

01 => 10 => 1

10 => 10 => 1

11 => 11 => 1

000 => 000 => 1

001 => 100 => 1

010 => 100 => 1

011 => 110 => 1

100 => 100 => 1

101 => 110 => 1

110 => 110 => 1

111 => 111 => 1

0000 => 0000 => 1

0001 => 1000 => 1

0010 => 1000 => 1

0011 => 1010 => 2

0100 => 1000 => 1

0101 => 1100 => 1

0110 => 1010 => 2

0111 => 1110 => 1

1000 => 1000 => 1

1001 => 1010 => 2

1010 => 1100 => 1

1011 => 1110 => 1

1100 => 1010 => 2

1101 => 1110 => 1

1110 => 1110 => 1

1111 => 1111 => 1

00000 => 00000 => 1

00001 => 10000 => 1

00010 => 10000 => 1

00011 => 10010 => 3

00100 => 10000 => 1

00101 => 11000 => 1

00110 => 10010 => 3

00111 => 10110 => 3

01000 => 10000 => 1

01001 => 11000 => 1

01010 => 11000 => 1

01011 => 11100 => 1

01100 => 10010 => 3

01101 => 11100 => 1

01110 => 10110 => 3

01111 => 11110 => 1

10000 => 10000 => 1

10001 => 10010 => 3

10010 => 11000 => 1

10011 => 10110 => 3

10100 => 11000 => 1

10101 => 11100 => 1

10110 => 11100 => 1

10111 => 11110 => 1

11000 => 10010 => 3

11001 => 10110 => 3

11010 => 11100 => 1

11011 => 11110 => 1

11100 => 10110 => 3

11101 => 11110 => 1

11110 => 11110 => 1

11111 => 11111 => 1

000000 => 000000 => 1

000001 => 100000 => 1

000010 => 100000 => 1

000011 => 100010 => 4

000100 => 100000 => 1

000101 => 101000 => 2

000110 => 100010 => 4

000111 => 100110 => 6

001000 => 100000 => 1

001001 => 110000 => 1

001010 => 101000 => 2

001011 => 101100 => 3

001100 => 100010 => 4

001101 => 110010 => 3

001110 => 100110 => 6

001111 => 101110 => 4

010000 => 100000 => 1

010001 => 101000 => 2

010010 => 110000 => 1

010011 => 110010 => 3

010100 => 101000 => 2

010101 => 111000 => 1

010110 => 101100 => 3

010111 => 111010 => 2

011000 => 100010 => 4

011001 => 101100 => 3

011010 => 110010 => 3

011011 => 111100 => 1

011100 => 100110 => 6

011101 => 111010 => 2

011110 => 101110 => 4

011111 => 111110 => 1

100000 => 100000 => 1

100001 => 100010 => 4

100010 => 101000 => 2

100011 => 100110 => 6

100100 => 110000 => 1

100101 => 101100 => 3

100110 => 110010 => 3

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Description

The number of Dyck paths above the lattice path given by a binary word.
One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$.
See St001312Number of parabolic noncrossing partitions indexed by the composition. for this statistic on compositions treated as bounce paths.

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.