- FindStat

Identifier

Mp00040: Integer compositions —to partition⟶ Integer partitions
St000320: Integer partitions ⟶ ℤ (values match St000319The spin of an integer partition.)

Values

[1] => [1] => 0

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

[2] => [2] => 1

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

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

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

[3] => [3] => 2

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

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

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

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

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

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

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

[4] => [4] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5] => [5] => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[6] => [6] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1023 entries. <<<

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Description

The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.

Map

to partition

Description

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