- FindStat

Identifier

Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤ

Values

[1] => [1] => [1] => [] => 0

[2] => [1,1] => [2] => [] => 0

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

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

[2,1] => [1,1,1] => [3] => [] => 0

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

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

[3,1] => [1,1,1,1] => [4] => [] => 0

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

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

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

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

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

[3,2] => [1,1,1,1,1] => [5] => [] => 0

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

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

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

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

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

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

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

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

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

[3,2,1] => [1,1,1,1,1,1] => [6] => [] => 0

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

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

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

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

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

[7] => [4,3] => [2,2,2,1] => [2,2,1] => 5

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

[5,2] => [3,2,1,1] => [4,2,1] => [2,1] => 3

[5,1,1] => [4,2,1] => [3,2,1,1] => [2,1,1] => 4

[4,3] => [2,2,1,1,1] => [5,2] => [2] => 2

[4,2,1] => [1,1,1,1,1,1,1] => [7] => [] => 0

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

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

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

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

[3,1,1,1,1] => [5,1,1] => [3,1,1,1,1] => [1,1,1,1] => 4

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

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

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

[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1] => [1,1,1,1,1,1] => 6

[8] => [4,4] => [2,2,2,2] => [2,2,2] => 6

[7,1] => [4,3,1] => [3,2,2,1] => [2,2,1] => 5

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

[6,1,1] => [4,2,2] => [3,3,1,1] => [3,1,1] => 5

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

[5,2,1] => [4,1,1,1,1] => [5,1,1,1] => [1,1,1] => 3

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

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

[4,3,1] => [1,1,1,1,1,1,1,1] => [8] => [] => 0

[4,2,2] => [2,1,1,1,1,1,1] => [7,1] => [1] => 1

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

[4,1,1,1,1] => [5,2,1] => [3,2,1,1,1] => [2,1,1,1] => 5

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

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

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

[3,2,1,1,1] => [5,1,1,1] => [4,1,1,1,1] => [1,1,1,1] => 4

[3,1,1,1,1,1] => [6,1,1] => [3,1,1,1,1,1] => [1,1,1,1,1] => 5

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

[2,2,2,1,1] => [5,3] => [2,2,2,1,1] => [2,2,1,1] => 6

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

[2,1,1,1,1,1,1] => [7,1] => [2,1,1,1,1,1,1] => [1,1,1,1,1,1] => 6

[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 7

[9] => [5,4] => [2,2,2,2,1] => [2,2,2,1] => 7

[8,1] => [4,4,1] => [3,2,2,2] => [2,2,2] => 6

[7,2] => [4,3,1,1] => [4,2,2,1] => [2,2,1] => 5

[7,1,1] => [4,3,2] => [3,3,2,1] => [3,2,1] => 6

[6,3] => [4,2,2,1] => [4,3,1,1] => [3,1,1] => 5

[6,2,1] => [4,2,1,1,1] => [5,2,1,1] => [2,1,1] => 4

[6,1,1,1] => [5,2,2] => [3,3,1,1,1] => [3,1,1,1] => 6

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

[5,3,1] => [3,1,1,1,1,1,1] => [7,1,1] => [1,1] => 2

[5,2,2] => [3,2,1,1,1,1] => [6,2,1] => [2,1] => 3

[5,2,1,1] => [3,3,1,1,1] => [5,2,2] => [2,2] => 4

[5,1,1,1,1] => [3,2,2,2] => [4,4,1] => [4,1] => 5

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

[4,3,2] => [1,1,1,1,1,1,1,1,1] => [9] => [] => 0

[4,3,1,1] => [2,1,1,1,1,1,1,1] => [8,1] => [1] => 1

[4,2,2,1] => [2,2,1,1,1,1,1] => [7,2] => [2] => 2

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

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

[3,3,3] => [3,2,2,1,1] => [5,3,1] => [3,1] => 4

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

[3,3,1,1,1] => [5,2,1,1] => [4,2,1,1,1] => [2,1,1,1] => 5

[3,2,2,2] => [3,3,2,1] => [4,3,2] => [3,2] => 5

[3,2,2,1,1] => [5,3,1] => [3,2,2,1,1] => [2,2,1,1] => 6

[3,2,1,1,1,1] => [6,1,1,1] => [4,1,1,1,1,1] => [1,1,1,1,1] => 5

[3,1,1,1,1,1,1] => [7,1,1] => [3,1,1,1,1,1,1] => [1,1,1,1,1,1] => 6

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

[2,2,2,1,1,1] => [6,3] => [2,2,2,1,1,1] => [2,2,1,1,1] => 7

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

[2,1,1,1,1,1,1,1] => [8,1] => [2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 7

[1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 8

[10] => [5,5] => [2,2,2,2,2] => [2,2,2,2] => 8

[9,1] => [5,4,1] => [3,2,2,2,1] => [2,2,2,1] => 7

[8,2] => [4,4,1,1] => [4,2,2,2] => [2,2,2] => 6

[8,1,1] => [4,4,2] => [3,3,2,2] => [3,2,2] => 7

[7,3] => [4,3,2,1] => [4,3,2,1] => [3,2,1] => 6

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$F_{1} = 1$

$F_{2} = 1 + q$

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

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

$F_{5} = 1 + q + 2\ q^{2} + 2\ q^{3} + q^{4}$

$F_{6} = 1 + q + 2\ q^{2} + 3\ q^{3} + 3\ q^{4} + q^{5}$

$F_{7} = 1 + q + 2\ q^{2} + 3\ q^{3} + 4\ q^{4} + 3\ q^{5} + q^{6}$

$F_{8} = 1 + q + 2\ q^{2} + 3\ q^{3} + 5\ q^{4} + 5\ q^{5} + 4\ q^{6} + q^{7}$

$F_{9} = 1 + q + 2\ q^{2} + 3\ q^{3} + 5\ q^{4} + 6\ q^{5} + 7\ q^{6} + 4\ q^{7} + q^{8}$

$F_{10} = 1 + q + 2\ q^{2} + 3\ q^{3} + 5\ q^{4} + 7\ q^{5} + 9\ q^{6} + 8\ q^{7} + 5\ q^{8} + q^{9}$

$F_{11} = 1 + q + 2\ q^{2} + 3\ q^{3} + 5\ q^{4} + 7\ q^{5} + 10\ q^{6} + 11\ q^{7} + 10\ q^{8} + 5\ q^{9} + q^{10}$

Description

The size of a partition.
This statistic is the constant statistic of the level sets.

Map

first row removal

Description

Removes the first entry of an integer partition

Map

Loehr-Warrington inverse

Description

Return a partition whose length is the diagonal inversion number of the preimage.

Map

conjugate

Description

Return the conjugate partition of the partition.
The conjugate partition of the partition

$\lambda$ of

$n$ is the partition

$\lambda^*$ whose Ferrers diagram is obtained from the diagram of

$\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.