- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000008: Integer compositions ⟶ ℤ

Values

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

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

00 => [2] => [1] => 0

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

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

11 => [2] => [1] => 0

000 => [3] => [1] => 0

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

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

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

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

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

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

111 => [3] => [1] => 0

0000 => [4] => [1] => 0

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

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

0011 => [2,2] => [2] => 0

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

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

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

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

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

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

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

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

1100 => [2,2] => [2] => 0

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

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

1111 => [4] => [1] => 0

00000 => [5] => [1] => 0

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

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

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

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

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

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

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

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

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

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

01011 => [1,1,1,2] => [3,1] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

11010 => [2,1,1,1] => [1,3] => 1

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

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

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

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

11111 => [5] => [1] => 0

000000 => [6] => [1] => 0

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

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

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

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

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

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

000111 => [3,3] => [2] => 0

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

001001 => [2,1,2,1] => [1,1,1,1] => 6

001010 => [2,1,1,1,1] => [1,4] => 1

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

001100 => [2,2,2] => [3] => 0

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

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

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

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

010001 => [1,1,3,1] => [2,1,1] => 5

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

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

010100 => [1,1,1,1,2] => [4,1] => 4

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

010110 => [1,1,1,2,1] => [3,1,1] => 7

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

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

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

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

011011 => [1,2,1,2] => [1,1,1,1] => 6

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

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

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

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

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

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

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

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

100100 => [1,2,1,2] => [1,1,1,1] => 6

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

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

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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

delta morphism

Description

Apply the delta morphism to an integer composition.
The delta morphism of a composition $C$ is the compositions composed of the lengths of consecutive runs of the same integer in $C$.

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 $w$.