- FindStat

Identifier

Mp00040: Integer compositions —to partition⟶ Integer partitions
St000278: Integer partitions ⟶ ℤ

Values

[1] => [1] => 1

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

[2] => [2] => 1

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

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

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

[3] => [3] => 1

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

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

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

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

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

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

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

[4] => [4] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5] => [5] => 1

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

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

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

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

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

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

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

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

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

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

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

[1,2,3] => [3,2,1] => 6

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

[1,3,2] => [3,2,1] => 6

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

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

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

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

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

[2,1,3] => [3,2,1] => 6

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

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

[2,3,1] => [3,2,1] => 6

[2,4] => [4,2] => 2

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

[3,1,2] => [3,2,1] => 6

[3,2,1] => [3,2,1] => 6

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

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

[4,2] => [4,2] => 2

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

[6] => [6] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1884 entries. <<<

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$F_{1} = q$

$F_{2} = 2\ q$

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

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

$F_{5} = 2\ q + 4\ q^{2} + 6\ q^{3} + 4\ q^{4}$

$F_{6} = 4\ q + 4\ q^{2} + 3\ q^{3} + 4\ q^{4} + 5\ q^{5} + 12\ q^{6}$

$F_{7} = 2\ q + 6\ q^{2} + 9\ q^{3} + 8\ q^{4} + 5\ q^{5} + 12\ q^{6} + 10\ q^{10} + 12\ q^{12}$

$F_{8} = 4\ q + 6\ q^{2} + 9\ q^{3} + 4\ q^{4} + 5\ q^{5} + 24\ q^{6} + 7\ q^{7} + 10\ q^{10} + 24\ q^{12} + 15\ q^{15} + 20\ q^{20}$

$F_{9} = 3\ q + 8\ q^{2} + 9\ q^{3} + 8\ q^{4} + 10\ q^{5} + 24\ q^{6} + 7\ q^{7} + 8\ q^{8} + 10\ q^{10} + 48\ q^{12} + 40\ q^{20} + 21\ q^{21} + 60\ q^{30}$

$F_{10} = 4\ q + 8\ q^{2} + 12\ q^{3} + 12\ q^{4} + 5\ q^{5} + 42\ q^{6} + 7\ q^{7} + 8\ q^{8} + 9\ q^{9} + 36\ q^{12} + 30\ q^{15} + 60\ q^{20} + 24\ q^{24} + 28\ q^{28} + 90\ q^{30} + 35\ q^{35} + 42\ q^{42} + 60\ q^{60}$

Description

The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions.
This is the multinomial of the multiplicities of the parts, see [1].
This is the same as

$m_\lambda(x_1,\dotsc,x_k)$ evaluated at

$x_1=\dotsb=x_k=1$ ,
where

$k$ is the number of parts of

$\lambda$ .
An explicit formula is

$\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$
where

$m_i(\lambda)$ is the number of parts of

$\lambda$ equal to

$i$ .

Map

to partition

Description

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