- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤ (values match St000319The spin of an integer partition.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 3512 entries. <<<

search for individual values

searching the database for the individual values of this statistic

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

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.

Map

first row removal

Description

Removes the first entry of an integer partition

Map

to partition

Description

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