- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000008: Integer compositions ⟶ ℤ

Values

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

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

00 => [2] => [1,1] => 1

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

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

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

000 => [3] => [1,1,1] => 3

001 => [2,1] => [1,2] => 1

010 => [1,1,1] => [3] => 0

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

100 => [1,2] => [2,1] => 2

101 => [1,1,1] => [3] => 0

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

111 => [3] => [1,1,1] => 3

0000 => [4] => [1,1,1,1] => 6

0001 => [3,1] => [1,1,2] => 3

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

0011 => [2,2] => [1,2,1] => 4

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

0101 => [1,1,1,1] => [4] => 0

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

0111 => [1,3] => [2,1,1] => 5

1000 => [1,3] => [2,1,1] => 5

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

1010 => [1,1,1,1] => [4] => 0

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

1100 => [2,2] => [1,2,1] => 4

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

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

1111 => [4] => [1,1,1,1] => 6

00000 => [5] => [1,1,1,1,1] => 10

00001 => [4,1] => [1,1,1,2] => 6

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

00011 => [3,2] => [1,1,2,1] => 7

00100 => [2,1,2] => [1,3,1] => 5

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

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

00111 => [2,3] => [1,2,1,1] => 8

01000 => [1,1,3] => [3,1,1] => 7

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

01010 => [1,1,1,1,1] => [5] => 0

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

01100 => [1,2,2] => [2,2,1] => 6

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

01110 => [1,3,1] => [2,1,2] => 5

01111 => [1,4] => [2,1,1,1] => 9

10000 => [1,4] => [2,1,1,1] => 9

10001 => [1,3,1] => [2,1,2] => 5

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

10011 => [1,2,2] => [2,2,1] => 6

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

10101 => [1,1,1,1,1] => [5] => 0

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

10111 => [1,1,3] => [3,1,1] => 7

11000 => [2,3] => [1,2,1,1] => 8

11001 => [2,2,1] => [1,2,2] => 4

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

11011 => [2,1,2] => [1,3,1] => 5

11100 => [3,2] => [1,1,2,1] => 7

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

11110 => [4,1] => [1,1,1,2] => 6

11111 => [5] => [1,1,1,1,1] => 10

000000 => [6] => [1,1,1,1,1,1] => 15

000001 => [5,1] => [1,1,1,1,2] => 10

000010 => [4,1,1] => [1,1,1,3] => 6

000011 => [4,2] => [1,1,1,2,1] => 11

000100 => [3,1,2] => [1,1,3,1] => 8

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

000110 => [3,2,1] => [1,1,2,2] => 7

000111 => [3,3] => [1,1,2,1,1] => 12

001000 => [2,1,3] => [1,3,1,1] => 10

001001 => [2,1,2,1] => [1,3,2] => 5

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

001011 => [2,1,1,2] => [1,4,1] => 6

001100 => [2,2,2] => [1,2,2,1] => 9

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

001110 => [2,3,1] => [1,2,1,2] => 8

001111 => [2,4] => [1,2,1,1,1] => 13

010000 => [1,1,4] => [3,1,1,1] => 12

010001 => [1,1,3,1] => [3,1,2] => 7

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

010011 => [1,1,2,2] => [3,2,1] => 8

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

010101 => [1,1,1,1,1,1] => [6] => 0

010110 => [1,1,1,2,1] => [4,2] => 4

010111 => [1,1,1,3] => [4,1,1] => 9

011000 => [1,2,3] => [2,2,1,1] => 11

011001 => [1,2,2,1] => [2,2,2] => 6

011010 => [1,2,1,1,1] => [2,4] => 2

011011 => [1,2,1,2] => [2,3,1] => 7

011100 => [1,3,2] => [2,1,2,1] => 10

011101 => [1,3,1,1] => [2,1,3] => 5

011110 => [1,4,1] => [2,1,1,2] => 9

011111 => [1,5] => [2,1,1,1,1] => 14

100000 => [1,5] => [2,1,1,1,1] => 14

100001 => [1,4,1] => [2,1,1,2] => 9

100010 => [1,3,1,1] => [2,1,3] => 5

100011 => [1,3,2] => [2,1,2,1] => 10

100100 => [1,2,1,2] => [2,3,1] => 7

100101 => [1,2,1,1,1] => [2,4] => 2

100110 => [1,2,2,1] => [2,2,2] => 6

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Description

The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see Permutations/Descents-Major.

Map

complement

Description

The complement of a composition.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.

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 $w$.