- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤ (values match St000439The position of the first down step of a Dyck path.)

Values

0 => [2] => [1,1,0,0] => 2

1 => [1,1] => [1,0,1,0] => 1

00 => [3] => [1,1,1,0,0,0] => 3

01 => [2,1] => [1,1,0,0,1,0] => 2

10 => [1,2] => [1,0,1,1,0,0] => 1

11 => [1,1,1] => [1,0,1,0,1,0] => 1

000 => [4] => [1,1,1,1,0,0,0,0] => 4

001 => [3,1] => [1,1,1,0,0,0,1,0] => 3

010 => [2,2] => [1,1,0,0,1,1,0,0] => 2

011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 2

100 => [1,3] => [1,0,1,1,1,0,0,0] => 1

101 => [1,2,1] => [1,0,1,1,0,0,1,0] => 1

110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 1

111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 1

0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => 5

0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4

0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3

0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 3

0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2

0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 2

0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2

0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 2

1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1

1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 1

1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 1

1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 1

1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 1

1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 1

1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 1

1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 1

00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 6

00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5

00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4

00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 4

00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3

00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3

00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 3

00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 3

01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2

01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 2

01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 2

01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 2

01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 2

01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 2

01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 2

01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 2

10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1

10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 1

10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 1

10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 1

10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 1

10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 1

10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 1

10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 1

11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 1

11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 1

11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 1

11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 1

11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1

11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 1

11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1

11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1

000000 => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 7

000001 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 6

000010 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 5

000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 5

000100 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 4

000101 => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 4

000110 => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => 4

000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => 4

001000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 3

001001 => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 3

001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 3

001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => 3

001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => 3

001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => 3

001110 => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => 3

001111 => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 3

010000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 2

010001 => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 2

010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 2

010011 => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => 2

010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 2

010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 2

010110 => [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => 2

010111 => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => 2

011000 => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => 2

011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 2

011010 => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 2

011011 => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 2

011100 => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 2

011101 => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 2

011110 => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 2

011111 => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 2

100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1

100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 1

100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 1

100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 1

100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 1

100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 1

100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 1

>>> Load all 127 entries. <<<

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$F_{1} = q + q^{2}$

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

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

$F_{4} = 8\ q + 4\ q^{2} + 2\ q^{3} + q^{4} + q^{5}$

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

$F_{6} = 32\ q + 16\ q^{2} + 8\ q^{3} + 4\ q^{4} + 2\ q^{5} + q^{6} + q^{7}$

Description

The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of

$D$ .

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending

$1$ to a binary word

$w$ , the

$i$ -th part of the composition equals

$1$ plus the number of zeros after the

$i$ -th

$1$ in

$w$ .
This map is not surjective, since the empty composition does not have a preimage.