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Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
St000767: 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] => 2

0011 => [2,2] => 1

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

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] => 2

1100 => [2,2] => 1

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

1110 => [3,1] => 2

1111 => [4] => 1

00000 => [5] => 1

00001 => [4,1] => 2

00010 => [3,1,1] => 2

00011 => [3,2] => 2

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

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

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

00111 => [2,3] => 2

01000 => [1,1,3] => 2

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

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

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

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

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

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

01111 => [1,4] => 2

10000 => [1,4] => 2

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

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

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

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

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

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

10111 => [1,1,3] => 2

11000 => [2,3] => 2

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

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

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

11100 => [3,2] => 2

11101 => [3,1,1] => 2

11110 => [4,1] => 2

11111 => [5] => 1

000000 => [6] => 1

000001 => [5,1] => 2

000010 => [4,1,1] => 2

000011 => [4,2] => 2

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

000101 => [3,1,1,1] => 2

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

000111 => [3,3] => 1

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

001001 => [2,1,2,1] => 4

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

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

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

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

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

001111 => [2,4] => 2

010000 => [1,1,4] => 2

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

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

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

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

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

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

010111 => [1,1,1,3] => 2

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

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

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

011011 => [1,2,1,2] => 4

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

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

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

011111 => [1,5] => 2

100000 => [1,5] => 2

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

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

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

100100 => [1,2,1,2] => 4

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

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

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Description

The number of runs in an integer composition.
Writing the composition as $c_1^{e_1} \dots c_\ell^{e_\ell}$, where $c_i \neq c_{i+1}$ for all $i$, the number of runs is $\ell$, see [def.2.8, 1].
It turns out that the total number of runs in all compositions of $n$ equals the total number of odd parts in all these compositions, see [1].

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 $w$.