- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
St000820: Integer compositions ⟶ ℤ

Values

0 => [1] => 1

1 => [1] => 1

00 => [2] => 1

01 => [1,1] => 1

10 => [1,1] => 1

11 => [2] => 1

000 => [3] => 1

001 => [2,1] => 2

010 => [1,1,1] => 1

011 => [1,2] => 2

100 => [1,2] => 2

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

110 => [2,1] => 2

111 => [3] => 1

0000 => [4] => 1

0001 => [3,1] => 2

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

0011 => [2,2] => 1

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

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

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

0111 => [1,3] => 2

1000 => [1,3] => 2

1001 => [1,2,1] => 3

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

1011 => [1,1,2] => 3

1100 => [2,2] => 1

1101 => [2,1,1] => 3

1110 => [3,1] => 2

1111 => [4] => 1

00000 => [5] => 1

00001 => [4,1] => 2

00010 => [3,1,1] => 3

00011 => [3,2] => 2

00100 => [2,1,2] => 3

00101 => [2,1,1,1] => 4

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

00111 => [2,3] => 2

01000 => [1,1,3] => 3

01001 => [1,1,2,1] => 4

01010 => [1,1,1,1,1] => 1

01011 => [1,1,1,2] => 4

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

01101 => [1,2,1,1] => 4

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

01111 => [1,4] => 2

10000 => [1,4] => 2

10001 => [1,3,1] => 3

10010 => [1,2,1,1] => 4

10011 => [1,2,2] => 3

10100 => [1,1,1,2] => 4

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

10110 => [1,1,2,1] => 4

10111 => [1,1,3] => 3

11000 => [2,3] => 2

11001 => [2,2,1] => 3

11010 => [2,1,1,1] => 4

11011 => [2,1,2] => 3

11100 => [3,2] => 2

11101 => [3,1,1] => 3

11110 => [4,1] => 2

11111 => [5] => 1

000000 => [6] => 1

000001 => [5,1] => 2

000010 => [4,1,1] => 3

000011 => [4,2] => 2

000100 => [3,1,2] => 3

000101 => [3,1,1,1] => 4

000110 => [3,2,1] => 3

000111 => [3,3] => 1

001000 => [2,1,3] => 3

001001 => [2,1,2,1] => 2

001010 => [2,1,1,1,1] => 5

001011 => [2,1,1,2] => 4

001100 => [2,2,2] => 1

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

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

001111 => [2,4] => 2

010000 => [1,1,4] => 3

010001 => [1,1,3,1] => 4

010010 => [1,1,2,1,1] => 5

010011 => [1,1,2,2] => 4

010100 => [1,1,1,1,2] => 5

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

010110 => [1,1,1,2,1] => 5

010111 => [1,1,1,3] => 4

011000 => [1,2,3] => 3

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

011010 => [1,2,1,1,1] => 5

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

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

011101 => [1,3,1,1] => 4

011110 => [1,4,1] => 3

011111 => [1,5] => 2

100000 => [1,5] => 2

100001 => [1,4,1] => 3

100010 => [1,3,1,1] => 4

100011 => [1,3,2] => 3

100100 => [1,2,1,2] => 2

100101 => [1,2,1,1,1] => 5

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

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$F_{1} = 2\ q$

$F_{2} = 4\ q$

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

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

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

$F_{6} = 8\ q + 12\ q^{2} + 18\ q^{3} + 16\ q^{4} + 10\ q^{5}$

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

$F_{8} = 8\ q + 16\ q^{2} + 48\ q^{3} + 64\ q^{4} + 70\ q^{5} + 36\ q^{6} + 14\ q^{7}$

$F_{9} = 6\ q + 20\ q^{2} + 54\ q^{3} + 112\ q^{4} + 140\ q^{5} + 108\ q^{6} + 56\ q^{7} + 16\ q^{8}$

Description

The number of compositions obtained by rotating the composition.

Map

delta morphism

Description

Applies the delta morphism to a binary word.
The delta morphism of a finite word

$w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in

$w$ .