Identifier
-
Mp00044:
Integer partitions
—conjugate⟶
Integer partitions
St000160: Integer partitions ⟶ ℤ
Values
[1] => [1] => 1
[2] => [1,1] => 2
[1,1] => [2] => 1
[3] => [1,1,1] => 3
[2,1] => [2,1] => 1
[1,1,1] => [3] => 1
[4] => [1,1,1,1] => 4
[3,1] => [2,1,1] => 2
[2,2] => [2,2] => 2
[2,1,1] => [3,1] => 1
[1,1,1,1] => [4] => 1
[5] => [1,1,1,1,1] => 5
[4,1] => [2,1,1,1] => 3
[3,2] => [2,2,1] => 1
[3,1,1] => [3,1,1] => 2
[2,2,1] => [3,2] => 1
[2,1,1,1] => [4,1] => 1
[1,1,1,1,1] => [5] => 1
[6] => [1,1,1,1,1,1] => 6
[5,1] => [2,1,1,1,1] => 4
[4,2] => [2,2,1,1] => 2
[4,1,1] => [3,1,1,1] => 3
[3,3] => [2,2,2] => 3
[3,2,1] => [3,2,1] => 1
[3,1,1,1] => [4,1,1] => 2
[2,2,2] => [3,3] => 2
[2,2,1,1] => [4,2] => 1
[2,1,1,1,1] => [5,1] => 1
[1,1,1,1,1,1] => [6] => 1
[7] => [1,1,1,1,1,1,1] => 7
[6,1] => [2,1,1,1,1,1] => 5
[5,2] => [2,2,1,1,1] => 3
[5,1,1] => [3,1,1,1,1] => 4
[4,3] => [2,2,2,1] => 1
[4,2,1] => [3,2,1,1] => 2
[4,1,1,1] => [4,1,1,1] => 3
[3,3,1] => [3,2,2] => 2
[3,2,2] => [3,3,1] => 1
[3,2,1,1] => [4,2,1] => 1
[3,1,1,1,1] => [5,1,1] => 2
[2,2,2,1] => [4,3] => 1
[2,2,1,1,1] => [5,2] => 1
[2,1,1,1,1,1] => [6,1] => 1
[1,1,1,1,1,1,1] => [7] => 1
[8] => [1,1,1,1,1,1,1,1] => 8
[7,1] => [2,1,1,1,1,1,1] => 6
[6,2] => [2,2,1,1,1,1] => 4
[6,1,1] => [3,1,1,1,1,1] => 5
[5,3] => [2,2,2,1,1] => 2
[5,2,1] => [3,2,1,1,1] => 3
[5,1,1,1] => [4,1,1,1,1] => 4
[4,4] => [2,2,2,2] => 4
[4,3,1] => [3,2,2,1] => 1
[4,2,2] => [3,3,1,1] => 2
[4,2,1,1] => [4,2,1,1] => 2
[4,1,1,1,1] => [5,1,1,1] => 3
[3,3,2] => [3,3,2] => 1
[3,3,1,1] => [4,2,2] => 2
[3,2,2,1] => [4,3,1] => 1
[3,2,1,1,1] => [5,2,1] => 1
[3,1,1,1,1,1] => [6,1,1] => 2
[2,2,2,2] => [4,4] => 2
[2,2,2,1,1] => [5,3] => 1
[2,2,1,1,1,1] => [6,2] => 1
[2,1,1,1,1,1,1] => [7,1] => 1
[1,1,1,1,1,1,1,1] => [8] => 1
[9] => [1,1,1,1,1,1,1,1,1] => 9
[8,1] => [2,1,1,1,1,1,1,1] => 7
[7,2] => [2,2,1,1,1,1,1] => 5
[7,1,1] => [3,1,1,1,1,1,1] => 6
[6,3] => [2,2,2,1,1,1] => 3
[6,2,1] => [3,2,1,1,1,1] => 4
[6,1,1,1] => [4,1,1,1,1,1] => 5
[5,4] => [2,2,2,2,1] => 1
[5,3,1] => [3,2,2,1,1] => 2
[5,2,2] => [3,3,1,1,1] => 3
[5,2,1,1] => [4,2,1,1,1] => 3
[5,1,1,1,1] => [5,1,1,1,1] => 4
[4,4,1] => [3,2,2,2] => 3
[4,3,2] => [3,3,2,1] => 1
[4,3,1,1] => [4,2,2,1] => 1
[4,2,2,1] => [4,3,1,1] => 2
[4,2,1,1,1] => [5,2,1,1] => 2
[4,1,1,1,1,1] => [6,1,1,1] => 3
[3,3,3] => [3,3,3] => 3
[3,3,2,1] => [4,3,2] => 1
[3,3,1,1,1] => [5,2,2] => 2
[3,2,2,2] => [4,4,1] => 1
[3,2,2,1,1] => [5,3,1] => 1
[3,2,1,1,1,1] => [6,2,1] => 1
[3,1,1,1,1,1,1] => [7,1,1] => 2
[2,2,2,2,1] => [5,4] => 1
[2,2,2,1,1,1] => [6,3] => 1
[2,2,1,1,1,1,1] => [7,2] => 1
[2,1,1,1,1,1,1,1] => [8,1] => 1
[1,1,1,1,1,1,1,1,1] => [9] => 1
[10] => [1,1,1,1,1,1,1,1,1,1] => 10
[9,1] => [2,1,1,1,1,1,1,1,1] => 8
[8,2] => [2,2,1,1,1,1,1,1] => 6
[8,1,1] => [3,1,1,1,1,1,1,1] => 7
[7,3] => [2,2,2,1,1,1,1] => 4
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Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
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