Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000319: Integer partitions ⟶ ℤ (values match St000320The dinv adjustment of an integer partition.)
Values
[2,1] => [[2,2],[1]] => [1] => 0
[1,2,1] => [[2,2,1],[1]] => [1] => 0
[2,1,1] => [[2,2,2],[1,1]] => [1,1] => 0
[2,2] => [[3,2],[1]] => [1] => 0
[3,1] => [[3,3],[2]] => [2] => 1
[1,1,2,1] => [[2,2,1,1],[1]] => [1] => 0
[1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => 0
[1,2,2] => [[3,2,1],[1]] => [1] => 0
[1,3,1] => [[3,3,1],[2]] => [2] => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => 0
[2,1,2] => [[3,2,2],[1,1]] => [1,1] => 0
[2,2,1] => [[3,3,2],[2,1]] => [2,1] => 1
[2,3] => [[4,2],[1]] => [1] => 0
[3,1,1] => [[3,3,3],[2,2]] => [2,2] => 1
[3,2] => [[4,3],[2]] => [2] => 1
[4,1] => [[4,4],[3]] => [3] => 2
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [1] => 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => 0
[1,1,2,2] => [[3,2,1,1],[1]] => [1] => 0
[1,1,3,1] => [[3,3,1,1],[2]] => [2] => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => 0
[1,2,1,2] => [[3,2,2,1],[1,1]] => [1,1] => 0
[1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => 1
[1,2,3] => [[4,2,1],[1]] => [1] => 0
[1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => 1
[1,3,2] => [[4,3,1],[2]] => [2] => 1
[1,4,1] => [[4,4,1],[3]] => [3] => 2
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 1
[2,1,3] => [[4,2,2],[1,1]] => [1,1] => 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => 1
[2,2,2] => [[4,3,2],[2,1]] => [2,1] => 1
[2,3,1] => [[4,4,2],[3,1]] => [3,1] => 2
[2,4] => [[5,2],[1]] => [1] => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 1
[3,1,2] => [[4,3,3],[2,2]] => [2,2] => 1
[3,2,1] => [[4,4,3],[3,2]] => [3,2] => 2
[3,3] => [[5,3],[2]] => [2] => 1
[4,1,1] => [[4,4,4],[3,3]] => [3,3] => 3
[4,2] => [[5,4],[3]] => [3] => 2
[5,1] => [[5,5],[4]] => [4] => 3
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [1] => 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [1] => 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => [2] => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => 0
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [1,1] => 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => 1
[1,1,2,3] => [[4,2,1,1],[1]] => [1] => 0
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => 1
[1,1,3,2] => [[4,3,1,1],[2]] => [2] => 1
[1,1,4,1] => [[4,4,1,1],[3]] => [3] => 2
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => 0
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => 0
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => 1
[1,2,1,3] => [[4,2,2,1],[1,1]] => [1,1] => 0
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => 1
[1,2,2,2] => [[4,3,2,1],[2,1]] => [2,1] => 1
[1,2,3,1] => [[4,4,2,1],[3,1]] => [3,1] => 2
[1,2,4] => [[5,2,1],[1]] => [1] => 0
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => 1
[1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => 1
[1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => 2
[1,3,3] => [[5,3,1],[2]] => [2] => 1
[1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => 3
[1,4,2] => [[5,4,1],[3]] => [3] => 2
[1,5,1] => [[5,5,1],[4]] => [4] => 3
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => 0
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 0
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => 0
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => 1
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => 2
[2,1,4] => [[5,2,2],[1,1]] => [1,1] => 0
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => [3,2,1] => 2
[2,2,3] => [[5,3,2],[2,1]] => [2,1] => 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => 3
[2,3,2] => [[5,4,2],[3,1]] => [3,1] => 2
[2,4,1] => [[5,5,2],[4,1]] => [4,1] => 3
[2,5] => [[6,2],[1]] => [1] => 0
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => 1
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => 1
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => 2
[3,1,3] => [[5,3,3],[2,2]] => [2,2] => 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => 3
[3,2,2] => [[5,4,3],[3,2]] => [3,2] => 2
[3,3,1] => [[5,5,3],[4,2]] => [4,2] => 3
[3,4] => [[6,3],[2]] => [2] => 1
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => 3
[4,1,2] => [[5,4,4],[3,3]] => [3,3] => 3
[4,2,1] => [[5,5,4],[4,3]] => [4,3] => 4
[4,3] => [[6,4],[3]] => [3] => 2
[5,1,1] => [[5,5,5],[4,4]] => [4,4] => 5
[5,2] => [[6,5],[4]] => [4] => 3
[6,1] => [[6,6],[5]] => [5] => 4
[1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]] => [1] => 0
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => [1,1] => 0
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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 $4+3+1 = 8$.
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
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
inner shape
Description
The inner shape of a skew partition.
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