- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
St000808: Integer compositions ⟶ ℤ

Values

0 => [2] => 2

1 => [1,1] => 1

00 => [3] => 3

01 => [2,1] => 2

10 => [1,2] => 2

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

000 => [4] => 4

001 => [3,1] => 3

010 => [2,2] => 2

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

100 => [1,3] => 3

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

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

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

0000 => [5] => 5

0001 => [4,1] => 4

0010 => [3,2] => 3

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

0100 => [2,3] => 3

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

0110 => [2,1,2] => 3

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

1000 => [1,4] => 4

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

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

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

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

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

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

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

00000 => [6] => 6

00001 => [5,1] => 5

00010 => [4,2] => 4

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

00100 => [3,3] => 3

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

00110 => [3,1,2] => 4

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

01000 => [2,4] => 4

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

01010 => [2,2,2] => 2

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

01100 => [2,1,3] => 4

01101 => [2,1,2,1] => 3

01110 => [2,1,1,2] => 3

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

10000 => [1,5] => 5

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

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

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

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

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

10110 => [1,2,1,2] => 3

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

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

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

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

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

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

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

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

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

000000 => [7] => 7

000001 => [6,1] => 6

000010 => [5,2] => 5

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

000100 => [4,3] => 4

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

000110 => [4,1,2] => 5

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

001000 => [3,4] => 4

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

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

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

001100 => [3,1,3] => 5

001101 => [3,1,2,1] => 4

001110 => [3,1,1,2] => 4

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

010000 => [2,5] => 5

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

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

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

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

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

010110 => [2,2,1,2] => 3

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

011000 => [2,1,4] => 5

011001 => [2,1,3,1] => 4

011010 => [2,1,2,2] => 3

011011 => [2,1,2,1,1] => 3

011100 => [2,1,1,3] => 4

011101 => [2,1,1,2,1] => 3

011110 => [2,1,1,1,2] => 3

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

100000 => [1,6] => 6

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

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

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

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

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

100110 => [1,3,1,2] => 4

>>> Load all 511 entries. <<<

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

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

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

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

$F_{5} = q + 9\ q^{2} + 12\ q^{3} + 7\ q^{4} + 2\ q^{5} + q^{6}$

$F_{6} = q + 12\ q^{2} + 24\ q^{3} + 16\ q^{4} + 8\ q^{5} + 2\ q^{6} + q^{7}$

$F_{7} = q + 16\ q^{2} + 41\ q^{3} + 39\ q^{4} + 19\ q^{5} + 9\ q^{6} + 2\ q^{7} + q^{8}$

$F_{8} = q + 20\ q^{2} + 69\ q^{3} + 78\ q^{4} + 53\ q^{5} + 22\ q^{6} + 10\ q^{7} + 2\ q^{8} + q^{9}$

Description

The number of up steps of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.

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.