- FindStat

Identifier

Mp00040: Integer compositions —to partition⟶ Integer partitions
St001961: Integer partitions ⟶ ℤ

Values

[1] => [1] => 0

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

[2] => [2] => 0

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

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

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

[3] => [3] => 0

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

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

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

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

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

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

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

[4] => [4] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5] => [5] => 0

[1,1,1,1,1,1] => [1,1,1,1,1,1] => 15

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[6] => [6] => 0

[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 21

[1,1,1,1,1,2] => [2,1,1,1,1,1] => 15

[1,1,1,1,2,1] => [2,1,1,1,1,1] => 15

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

[1,1,1,2,1,1] => [2,1,1,1,1,1] => 15

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

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

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

[1,1,2,1,1,1] => [2,1,1,1,1,1] => 15

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

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

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

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

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

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

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

[1,2,1,1,1,1] => [2,1,1,1,1,1] => 15

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,1,1,1,1,1] => [2,1,1,1,1,1] => 15

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

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

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

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

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

>>> Load all 2280 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + q^{3}$

$F_{4} = 1 + 2\ q + q^{2} + 3\ q^{3} + q^{6}$

$F_{5} = 1 + 4\ q + 3\ q^{3} + 3\ q^{4} + 4\ q^{6} + q^{10}$

$F_{6} = 1 + 2\ q + 2\ q^{2} + 10\ q^{3} + 5\ q^{6} + 6\ q^{7} + 5\ q^{10} + q^{15}$

$F_{7} = 1 + 6\ q + 3\ q^{3} + 9\ q^{4} + 3\ q^{5} + 16\ q^{6} + 4\ q^{9} + 5\ q^{10} + 10\ q^{11} + 6\ q^{15} + q^{21}$

$F_{8} = 1 + 4\ q + 2\ q^{2} + 15\ q^{3} + q^{4} + 3\ q^{5} + 7\ q^{6} + 24\ q^{7} + 6\ q^{8} + 25\ q^{10} + q^{12} + 10\ q^{13} + 6\ q^{15} + 15\ q^{16} + 7\ q^{21} + q^{28}$

$F_{9} = 1 + 6\ q + 11\ q^{3} + 15\ q^{4} + 31\ q^{6} + 12\ q^{8} + 17\ q^{9} + 5\ q^{10} + 50\ q^{11} + 10\ q^{12} + 36\ q^{15} + 5\ q^{16} + 20\ q^{18} + 7\ q^{21} + 21\ q^{22} + 8\ q^{28} + q^{36}$

$F_{10} = 1 + 4\ q + 4\ q^{2} + 21\ q^{3} + 10\ q^{5} + 19\ q^{6} + 48\ q^{7} + 3\ q^{8} + 12\ q^{9} + 45\ q^{10} + 38\ q^{12} + 50\ q^{13} + 6\ q^{15} + 90\ q^{16} + 15\ q^{17} + q^{20} + 64\ q^{21} + 35\ q^{24} + 8\ q^{28} + 28\ q^{29} + 9\ q^{36} + q^{45}$

Description

The sum of the greatest common divisors of all pairs of parts.

Map

to partition

Description

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