- FindStat

Identifier

Mp00079: Set partitions —shape⟶ Integer partitions
St000319: Integer partitions ⟶ ℤ (values match St000320The dinv adjustment of an integer partition.)

Values

{{1}} => [1] => 0

{{1,2}} => [2] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1},{2},{3},{4}} => [1,1,1,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1},{2},{3},{4},{5}} => [1,1,1,1,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.

Map

shape

Description

Sends a set partition to the integer partition obtained by the sizes of the blocks.