Identifier
-
Mp00172:
Integer compositions
—rotate back to front⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000278: Integer partitions ⟶ ℤ
Values
[1] => [1] => [[1],[]] => [] => 1
[1,1] => [1,1] => [[1,1],[]] => [] => 1
[2] => [2] => [[2],[]] => [] => 1
[1,1,1] => [1,1,1] => [[1,1,1],[]] => [] => 1
[1,2] => [2,1] => [[2,2],[1]] => [1] => 1
[2,1] => [1,2] => [[2,1],[]] => [] => 1
[3] => [3] => [[3],[]] => [] => 1
[1,1,1,1] => [1,1,1,1] => [[1,1,1,1],[]] => [] => 1
[1,1,2] => [2,1,1] => [[2,2,2],[1,1]] => [1,1] => 1
[1,2,1] => [1,1,2] => [[2,1,1],[]] => [] => 1
[1,3] => [3,1] => [[3,3],[2]] => [2] => 1
[2,1,1] => [1,2,1] => [[2,2,1],[1]] => [1] => 1
[2,2] => [2,2] => [[3,2],[1]] => [1] => 1
[3,1] => [1,3] => [[3,1],[]] => [] => 1
[4] => [4] => [[4],[]] => [] => 1
[1,1,1,1,1] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => [] => 1
[1,1,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => 1
[1,1,2,1] => [1,1,1,2] => [[2,1,1,1],[]] => [] => 1
[1,1,3] => [3,1,1] => [[3,3,3],[2,2]] => [2,2] => 1
[1,2,1,1] => [1,1,2,1] => [[2,2,1,1],[1]] => [1] => 1
[1,2,2] => [2,1,2] => [[3,2,2],[1,1]] => [1,1] => 1
[1,3,1] => [1,1,3] => [[3,1,1],[]] => [] => 1
[1,4] => [4,1] => [[4,4],[3]] => [3] => 1
[2,1,1,1] => [1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => 1
[2,1,2] => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 2
[2,2,1] => [1,2,2] => [[3,2,1],[1]] => [1] => 1
[2,3] => [3,2] => [[4,3],[2]] => [2] => 1
[3,1,1] => [1,3,1] => [[3,3,1],[2]] => [2] => 1
[3,2] => [2,3] => [[4,2],[1]] => [1] => 1
[4,1] => [1,4] => [[4,1],[]] => [] => 1
[5] => [5] => [[5],[]] => [] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => [] => 1
[1,1,1,1,2] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 1
[1,1,1,2,1] => [1,1,1,1,2] => [[2,1,1,1,1],[]] => [] => 1
[1,1,1,3] => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 1
[1,1,2,1,1] => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => [1] => 1
[1,1,2,2] => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => 1
[1,1,3,1] => [1,1,1,3] => [[3,1,1,1],[]] => [] => 1
[1,1,4] => [4,1,1] => [[4,4,4],[3,3]] => [3,3] => 1
[1,2,1,1,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => 1
[1,2,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 3
[1,2,2,1] => [1,1,2,2] => [[3,2,1,1],[1]] => [1] => 1
[1,2,3] => [3,1,2] => [[4,3,3],[2,2]] => [2,2] => 1
[1,3,1,1] => [1,1,3,1] => [[3,3,1,1],[2]] => [2] => 1
[1,3,2] => [2,1,3] => [[4,2,2],[1,1]] => [1,1] => 1
[1,4,1] => [1,1,4] => [[4,1,1],[]] => [] => 1
[1,5] => [5,1] => [[5,5],[4]] => [4] => 1
[2,1,1,1,1] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => 1
[2,1,1,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => 3
[2,1,2,1] => [1,2,1,2] => [[3,2,2,1],[1,1]] => [1,1] => 1
[2,1,3] => [3,2,1] => [[4,4,3],[3,2]] => [3,2] => 2
[2,2,1,1] => [1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => 2
[2,2,2] => [2,2,2] => [[4,3,2],[2,1]] => [2,1] => 2
[2,3,1] => [1,2,3] => [[4,2,1],[1]] => [1] => 1
[2,4] => [4,2] => [[5,4],[3]] => [3] => 1
[3,1,1,1] => [1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => 1
[3,1,2] => [2,3,1] => [[4,4,2],[3,1]] => [3,1] => 2
[3,2,1] => [1,3,2] => [[4,3,1],[2]] => [2] => 1
[3,3] => [3,3] => [[5,3],[2]] => [2] => 1
[4,1,1] => [1,4,1] => [[4,4,1],[3]] => [3] => 1
[4,2] => [2,4] => [[5,2],[1]] => [1] => 1
[5,1] => [1,5] => [[5,1],[]] => [] => 1
[6] => [6] => [[6],[]] => [] => 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => [] => 1
[1,1,1,1,1,2] => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => 1
[1,1,1,1,2,1] => [1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => [] => 1
[1,1,1,1,3] => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => 1
[1,1,1,2,1,1] => [1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [1] => 1
[1,1,1,2,2] => [2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 1
[1,1,1,3,1] => [1,1,1,1,3] => [[3,1,1,1,1],[]] => [] => 1
[1,1,1,4] => [4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => 1
[1,1,2,1,1,1] => [1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => 1
[1,1,2,1,2] => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => 4
[1,1,2,2,1] => [1,1,1,2,2] => [[3,2,1,1,1],[1]] => [1] => 1
[1,1,2,3] => [3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => 1
[1,1,3,1,1] => [1,1,1,3,1] => [[3,3,1,1,1],[2]] => [2] => 1
[1,1,3,2] => [2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => 1
[1,1,4,1] => [1,1,1,4] => [[4,1,1,1],[]] => [] => 1
[1,1,5] => [5,1,1] => [[5,5,5],[4,4]] => [4,4] => 1
[1,2,1,1,1,1] => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => 1
[1,2,1,1,2] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => 6
[1,2,1,2,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [1,1] => 1
[1,2,1,3] => [3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => 3
[1,2,2,1,1] => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => 2
[1,2,2,2] => [2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => 3
[1,2,3,1] => [1,1,2,3] => [[4,2,1,1],[1]] => [1] => 1
[1,2,4] => [4,1,2] => [[5,4,4],[3,3]] => [3,3] => 1
[1,3,1,1,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => 1
[1,3,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => 3
[1,3,2,1] => [1,1,3,2] => [[4,3,1,1],[2]] => [2] => 1
[1,3,3] => [3,1,3] => [[5,3,3],[2,2]] => [2,2] => 1
[1,4,1,1] => [1,1,4,1] => [[4,4,1,1],[3]] => [3] => 1
[1,4,2] => [2,1,4] => [[5,2,2],[1,1]] => [1,1] => 1
[1,5,1] => [1,1,5] => [[5,1,1],[]] => [] => 1
[1,6] => [6,1] => [[6,6],[5]] => [5] => 1
[2,1,1,1,1,1] => [1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => 1
[2,1,1,1,2] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => 4
[2,1,1,2,1] => [1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => 1
[2,1,1,3] => [3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => 3
[2,1,2,1,1] => [1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => 3
[2,1,2,2] => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => 3
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Description
The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions.
This is the multinomial of the multiplicities of the parts, see [1].
This is the same as $m_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=\dotsb=x_k=1$,
where $k$ is the number of parts of $\lambda$.
An explicit formula is $\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$
where $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$.
This is the multinomial of the multiplicities of the parts, see [1].
This is the same as $m_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=\dotsb=x_k=1$,
where $k$ is the number of parts of $\lambda$.
An explicit formula is $\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$
where $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$.
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
rotate back to front
Description
The back to front rotation of an integer composition.
Map
inner shape
Description
The inner shape of a skew partition.
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