- FindStat

Identifier

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1] => [1,1,1,1,1] => 111110 => [5,1] => 5

[1,1,1,2] => [2,1,1,1] => 101110 => [1,1,3,1] => 1

[1,1,2,1] => [2,1,1,1] => 101110 => [1,1,3,1] => 1

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

[1,2,1,1] => [2,1,1,1] => 101110 => [1,1,3,1] => 1

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

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

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

[2,1,1,1] => [2,1,1,1] => 101110 => [1,1,3,1] => 1

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

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

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

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

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

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

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

[1,1,1,1,1,1] => [1,1,1,1,1,1] => 1111110 => [6,1] => 6

[1,1,1,1,2] => [2,1,1,1,1] => 1011110 => [1,1,4,1] => 1

[1,1,1,2,1] => [2,1,1,1,1] => 1011110 => [1,1,4,1] => 1

[1,1,1,3] => [3,1,1,1] => 1001110 => [1,2,3,1] => 1

[1,1,2,1,1] => [2,1,1,1,1] => 1011110 => [1,1,4,1] => 1

[1,1,2,2] => [2,2,1,1] => 110110 => [2,1,2,1] => 2

[1,1,3,1] => [3,1,1,1] => 1001110 => [1,2,3,1] => 1

[1,1,4] => [4,1,1] => 1000110 => [1,3,2,1] => 1

[1,2,1,1,1] => [2,1,1,1,1] => 1011110 => [1,1,4,1] => 1

[1,2,1,2] => [2,2,1,1] => 110110 => [2,1,2,1] => 2

[1,2,2,1] => [2,2,1,1] => 110110 => [2,1,2,1] => 2

[1,2,3] => [3,2,1] => 101010 => [1,1,1,1,1,1] => 1

[1,3,1,1] => [3,1,1,1] => 1001110 => [1,2,3,1] => 1

[1,3,2] => [3,2,1] => 101010 => [1,1,1,1,1,1] => 1

[1,4,1] => [4,1,1] => 1000110 => [1,3,2,1] => 1

[1,5] => [5,1] => 1000010 => [1,4,1,1] => 1

[2,1,1,1,1] => [2,1,1,1,1] => 1011110 => [1,1,4,1] => 1

[2,1,1,2] => [2,2,1,1] => 110110 => [2,1,2,1] => 2

[2,1,2,1] => [2,2,1,1] => 110110 => [2,1,2,1] => 2

[2,1,3] => [3,2,1] => 101010 => [1,1,1,1,1,1] => 1

[2,2,1,1] => [2,2,1,1] => 110110 => [2,1,2,1] => 2

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

[2,3,1] => [3,2,1] => 101010 => [1,1,1,1,1,1] => 1

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

[3,1,1,1] => [3,1,1,1] => 1001110 => [1,2,3,1] => 1

[3,1,2] => [3,2,1] => 101010 => [1,1,1,1,1,1] => 1

[3,2,1] => [3,2,1] => 101010 => [1,1,1,1,1,1] => 1

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

[4,1,1] => [4,1,1] => 1000110 => [1,3,2,1] => 1

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

[5,1] => [5,1] => 1000010 => [1,4,1,1] => 1

[6] => [6] => 1000000 => [1,6] => 1

[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 11111110 => [7,1] => 7

[1,1,1,1,1,2] => [2,1,1,1,1,1] => 10111110 => [1,1,5,1] => 1

[1,1,1,1,2,1] => [2,1,1,1,1,1] => 10111110 => [1,1,5,1] => 1

[1,1,1,1,3] => [3,1,1,1,1] => 10011110 => [1,2,4,1] => 1

[1,1,1,2,1,1] => [2,1,1,1,1,1] => 10111110 => [1,1,5,1] => 1

[1,1,1,2,2] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 2

[1,1,1,3,1] => [3,1,1,1,1] => 10011110 => [1,2,4,1] => 1

[1,1,1,4] => [4,1,1,1] => 10001110 => [1,3,3,1] => 1

[1,1,2,1,1,1] => [2,1,1,1,1,1] => 10111110 => [1,1,5,1] => 1

[1,1,2,1,2] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 2

[1,1,2,2,1] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 2

[1,1,2,3] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1

[1,1,3,1,1] => [3,1,1,1,1] => 10011110 => [1,2,4,1] => 1

[1,1,3,2] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1

[1,1,4,1] => [4,1,1,1] => 10001110 => [1,3,3,1] => 1

[1,1,5] => [5,1,1] => 10000110 => [1,4,2,1] => 1

[1,2,1,1,1,1] => [2,1,1,1,1,1] => 10111110 => [1,1,5,1] => 1

[1,2,1,1,2] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 2

[1,2,1,2,1] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 2

[1,2,1,3] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1

[1,2,2,1,1] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 2

[1,2,2,2] => [2,2,2,1] => 111010 => [3,1,1,1] => 3

[1,2,3,1] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1

[1,2,4] => [4,2,1] => 1001010 => [1,2,1,1,1,1] => 1

[1,3,1,1,1] => [3,1,1,1,1] => 10011110 => [1,2,4,1] => 1

[1,3,1,2] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1

[1,3,2,1] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1

[1,3,3] => [3,3,1] => 110010 => [2,2,1,1] => 2

[1,4,1,1] => [4,1,1,1] => 10001110 => [1,3,3,1] => 1

[1,4,2] => [4,2,1] => 1001010 => [1,2,1,1,1,1] => 1

[1,5,1] => [5,1,1] => 10000110 => [1,4,2,1] => 1

[1,6] => [6,1] => 10000010 => [1,5,1,1] => 1

[2,1,1,1,1,1] => [2,1,1,1,1,1] => 10111110 => [1,1,5,1] => 1

[2,1,1,1,2] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 2

[2,1,1,2,1] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 2

[2,1,1,3] => [3,2,1,1] => 1010110 => [1,1,1,1,2,1] => 1

[2,1,2,1,1] => [2,2,1,1,1] => 1101110 => [2,1,3,1] => 2

[2,1,2,2] => [2,2,2,1] => 111010 => [3,1,1,1] => 3

>>> Load all 1299 entries. <<<

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$F_{1} = q$

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

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

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

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

$F_{6} = 23\ q + 7\ q^{2} + q^{3} + q^{6}$

$F_{7} = 46\ q + 13\ q^{2} + 4\ q^{3} + q^{7}$

$F_{8} = 91\ q + 25\ q^{2} + 10\ q^{3} + q^{4} + q^{8}$

$F_{9} = 183\ q + 46\ q^{2} + 21\ q^{3} + 5\ q^{4} + q^{9}$

$F_{10} = 367\ q + 89\ q^{2} + 39\ q^{3} + 15\ q^{4} + q^{5} + q^{10}$

Description

The first part of an integer composition.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

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

$w$ .

Map

to binary word

Description

Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.