- FindStat

Identifier

Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000047: Integer compositions ⟶ ℤ

Values

[1] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,2,1] => [2,2,1] => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,1,1,1] => [4,2] => 10

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

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

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

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

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

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

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

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

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

[2,1,2,1] => [2,3,1] => 15

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

[2,2,1,1] => [3,2,1] => 20

[2,2,2] => [1,2,2,1] => 8

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

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

[3,1,1,1] => [4,1,1] => 10

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

[3,2,1] => [2,2,1,1] => 15

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

[4,1,1] => [3,1,1,1] => 10

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

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

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

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

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

[1,1,1,1,2,1] => [2,5] => 6

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

[1,1,1,2,1,1] => [3,4] => 15

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

[1,1,1,3,1] => [2,1,4] => 6

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

[1,1,2,1,1,1] => [4,3] => 20

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

[1,1,2,2,1] => [2,2,3] => 24

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

[1,1,3,1,1] => [3,1,3] => 15

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

[1,1,4,1] => [2,1,1,3] => 6

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

[1,2,1,1,1,1] => [5,2] => 15

[1,2,1,1,2] => [1,4,2] => 10

[1,2,1,2,1] => [2,3,2] => 36

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

[1,2,2,1,1] => [3,2,2] => 45

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

[1,2,3,1] => [2,1,2,2] => 18

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

[1,3,1,1,1] => [4,1,2] => 20

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

[1,3,2,1] => [2,2,1,2] => 24

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

[1,4,1,1] => [3,1,1,2] => 15

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

[1,5,1] => [2,1,1,1,2] => 6

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

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

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

[2,1,1,2,1] => [2,4,1] => 24

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

[2,1,2,1,1] => [3,3,1] => 45

[2,1,2,2] => [1,2,3,1] => 15

>>> Load all 435 entries. <<<

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$F_{1} = q$

$F_{2} = 2\ q$

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

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

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

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

$F_{7} = 7\ q + q^{2} + 3\ q^{3} + 4\ q^{4} + 5\ q^{5} + 8\ q^{6} + q^{8} + 6\ q^{10} + q^{12} + 9\ q^{15} + 3\ q^{18} + 4\ q^{20} + 4\ q^{24} + q^{30} + 2\ q^{36} + q^{40} + 3\ q^{45} + q^{48}$

Description

The number of standard immaculate tableaux of a given shape.
See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.

Map

conjugate

Description

The conjugate of a composition.
The conjugate of a composition

$C$ is defined as the complement (Mp00039complement) of the reversal (Mp00038reverse) of

$C$ .
Equivalently, the ribbon shape corresponding to the conjugate of

$C$ is the conjugate of the ribbon shape of

$C$ .