- FindStat

Identifier

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤ

Values

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

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

[2] => [1] => 1 => 1

[1,1,1] => [3] => 100 => 1

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

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

[3] => [1] => 1 => 1

[1,1,1,1] => [4] => 1000 => 1

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

[1,2,1] => [1,1,1] => 111 => 3

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

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

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

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

[4] => [1] => 1 => 1

[1,1,1,1,1] => [5] => 10000 => 1

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

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

[1,1,3] => [2,1] => 101 => 2

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

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

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

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

[2,1,1,1] => [1,3] => 1100 => 2

[2,1,2] => [1,1,1] => 111 => 3

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

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

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

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

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

[5] => [1] => 1 => 1

[1,1,1,1,1,1] => [6] => 100000 => 1

[1,1,1,1,2] => [4,1] => 10001 => 2

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

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

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

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

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

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

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

[1,2,1,2] => [1,1,1,1] => 1111 => 4

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

[1,2,3] => [1,1,1] => 111 => 3

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

[1,3,2] => [1,1,1] => 111 => 3

[1,4,1] => [1,1,1] => 111 => 3

[1,5] => [1,1] => 11 => 2

[2,1,1,1,1] => [1,4] => 11000 => 2

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

[2,1,2,1] => [1,1,1,1] => 1111 => 4

[2,1,3] => [1,1,1] => 111 => 3

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

[2,2,2] => [3] => 100 => 1

[2,3,1] => [1,1,1] => 111 => 3

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

[3,1,1,1] => [1,3] => 1100 => 2

[3,1,2] => [1,1,1] => 111 => 3

[3,2,1] => [1,1,1] => 111 => 3

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

[4,1,1] => [1,2] => 110 => 2

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

[5,1] => [1,1] => 11 => 2

[6] => [1] => 1 => 1

[1,1,1,1,1,1,1] => [7] => 1000000 => 1

[1,1,1,1,1,2] => [5,1] => 100001 => 2

[1,1,1,1,2,1] => [4,1,1] => 100011 => 3

[1,1,1,1,3] => [4,1] => 10001 => 2

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

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

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

[1,1,1,4] => [3,1] => 1001 => 2

[1,1,2,1,1,1] => [2,1,3] => 101100 => 3

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

[1,1,2,2,1] => [2,2,1] => 10101 => 3

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

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

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

[1,1,4,1] => [2,1,1] => 1011 => 3

[1,1,5] => [2,1] => 101 => 2

[1,2,1,1,1,1] => [1,1,4] => 111000 => 3

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

[1,2,1,2,1] => [1,1,1,1,1] => 11111 => 5

[1,2,1,3] => [1,1,1,1] => 1111 => 4

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

[1,2,2,2] => [1,3] => 1100 => 2

[1,2,3,1] => [1,1,1,1] => 1111 => 4

[1,2,4] => [1,1,1] => 111 => 3

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

[1,3,1,2] => [1,1,1,1] => 1111 => 4

[1,3,2,1] => [1,1,1,1] => 1111 => 4

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

[1,4,1,1] => [1,1,2] => 1110 => 3

[1,4,2] => [1,1,1] => 111 => 3

[1,5,1] => [1,1,1] => 111 => 3

[1,6] => [1,1] => 11 => 2

[2,1,1,1,1,1] => [1,5] => 110000 => 2

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

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

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

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

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

>>> Load all 2151 entries. <<<

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$F_{1} = q$

$F_{2} = 2\ q$

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

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

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

$F_{6} = 4\ q + 12\ q^{2} + 14\ q^{3} + 2\ q^{4}$

$F_{7} = 2\ q + 22\ q^{2} + 29\ q^{3} + 10\ q^{4} + q^{5}$

$F_{8} = 4\ q + 26\ q^{2} + 56\ q^{3} + 36\ q^{4} + 6\ q^{5}$

$F_{9} = 3\ q + 34\ q^{2} + 100\ q^{3} + 86\ q^{4} + 31\ q^{5} + 2\ q^{6}$

Description

The number of ones in a binary word.
This is also known as the Hamming weight of the word.

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

to binary word

Description

Return the composition as a binary word, treating ones as separators.
Encoding a positive integer

$i$ as the word

$10\dots 0$ consisting of a one followed by

$i-1$ zeros, the binary word of a composition

$(i_1,\dots,i_k)$ is the concatenation of of words for

$i_1,\dots,i_k$ .
The image of this map contains precisely the words which do not begin with a

$0$ .