- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤ

Values

0 => [2] => 2

1 => [1,1] => 2

00 => [3] => 2

01 => [2,1] => 3

10 => [1,2] => 3

11 => [1,1,1] => 2

000 => [4] => 2

001 => [3,1] => 3

010 => [2,2] => 4

011 => [2,1,1] => 3

100 => [1,3] => 3

101 => [1,2,1] => 4

110 => [1,1,2] => 3

111 => [1,1,1,1] => 2

0000 => [5] => 2

0001 => [4,1] => 3

0010 => [3,2] => 4

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

0100 => [2,3] => 4

0101 => [2,2,1] => 5

0110 => [2,1,2] => 4

0111 => [2,1,1,1] => 3

1000 => [1,4] => 3

1001 => [1,3,1] => 4

1010 => [1,2,2] => 5

1011 => [1,2,1,1] => 4

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

1101 => [1,1,2,1] => 4

1110 => [1,1,1,2] => 3

1111 => [1,1,1,1,1] => 2

00000 => [6] => 2

00001 => [5,1] => 3

00010 => [4,2] => 4

00011 => [4,1,1] => 3

00100 => [3,3] => 4

00101 => [3,2,1] => 5

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

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

01000 => [2,4] => 4

01001 => [2,3,1] => 5

01010 => [2,2,2] => 6

01011 => [2,2,1,1] => 5

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

01101 => [2,1,2,1] => 5

01110 => [2,1,1,2] => 4

01111 => [2,1,1,1,1] => 3

10000 => [1,5] => 3

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

10010 => [1,3,2] => 5

10011 => [1,3,1,1] => 4

10100 => [1,2,3] => 5

10101 => [1,2,2,1] => 6

10110 => [1,2,1,2] => 5

10111 => [1,2,1,1,1] => 4

11000 => [1,1,4] => 3

11001 => [1,1,3,1] => 4

11010 => [1,1,2,2] => 5

11011 => [1,1,2,1,1] => 4

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

11101 => [1,1,1,2,1] => 4

11110 => [1,1,1,1,2] => 3

11111 => [1,1,1,1,1,1] => 2

000000 => [7] => 2

000001 => [6,1] => 3

000010 => [5,2] => 4

000011 => [5,1,1] => 3

000100 => [4,3] => 4

000101 => [4,2,1] => 5

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

000111 => [4,1,1,1] => 3

001000 => [3,4] => 4

001001 => [3,3,1] => 5

001010 => [3,2,2] => 6

001011 => [3,2,1,1] => 5

001100 => [3,1,3] => 4

001101 => [3,1,2,1] => 5

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

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

010000 => [2,5] => 4

010001 => [2,4,1] => 5

010010 => [2,3,2] => 6

010011 => [2,3,1,1] => 5

010100 => [2,2,3] => 6

010101 => [2,2,2,1] => 7

010110 => [2,2,1,2] => 6

010111 => [2,2,1,1,1] => 5

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

011001 => [2,1,3,1] => 5

011010 => [2,1,2,2] => 6

011011 => [2,1,2,1,1] => 5

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

011101 => [2,1,1,2,1] => 5

011110 => [2,1,1,1,2] => 4

011111 => [2,1,1,1,1,1] => 3

100000 => [1,6] => 3

100001 => [1,5,1] => 4

100010 => [1,4,2] => 5

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

100100 => [1,3,3] => 5

100101 => [1,3,2,1] => 6

100110 => [1,3,1,2] => 5

>>> Load all 1023 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

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

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

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

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

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

$F_{6} = 2\ q^{2} + 10\ q^{3} + 20\ q^{4} + 20\ q^{5} + 10\ q^{6} + 2\ q^{7}$

$F_{7} = 2\ q^{2} + 12\ q^{3} + 30\ q^{4} + 40\ q^{5} + 30\ q^{6} + 12\ q^{7} + 2\ q^{8}$

$F_{8} = 2\ q^{2} + 14\ q^{3} + 42\ q^{4} + 70\ q^{5} + 70\ q^{6} + 42\ q^{7} + 14\ q^{8} + 2\ q^{9}$

$F_{9} = 2\ q^{2} + 16\ q^{3} + 56\ q^{4} + 112\ q^{5} + 140\ q^{6} + 112\ q^{7} + 56\ q^{8} + 16\ q^{9} + 2\ q^{10}$

Description

The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition

$c=(c_1,\dots,c_n)$ with

$c_i$ cells in the

$i$ -th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.

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.