Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000320: Integer partitions ⟶ ℤ (values match St000319The spin 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 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 $4+3+1+0 = 8$.
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 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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