- FindStat

Identifier

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000913: Integer partitions ⟶ ℤ

Values

[1] => [1] => [1] => 1

[1,1] => [2] => [2] => 1

[2] => [1,1] => [1,1] => 1

[1,1,1] => [3] => [3] => 1

[1,2] => [2,1] => [2,1] => 1

[2,1] => [1,2] => [2,1] => 1

[3] => [1,1,1] => [1,1,1] => 1

[1,1,1,1] => [4] => [4] => 2

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

[1,2,1] => [2,2] => [2,2] => 1

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

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

[2,2] => [1,2,1] => [2,1,1] => 1

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

[4] => [1,1,1,1] => [1,1,1,1] => 1

[1,1,1,1,1] => [5] => [5] => 4

[1,1,1,2] => [4,1] => [4,1] => 2

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

[1,1,3] => [3,1,1] => [3,1,1] => 1

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

[1,2,2] => [2,2,1] => [2,2,1] => 1

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

[1,4] => [2,1,1,1] => [2,1,1,1] => 1

[2,1,1,1] => [1,4] => [4,1] => 2

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

[2,2,1] => [1,2,2] => [2,2,1] => 1

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

[3,1,1] => [1,1,3] => [3,1,1] => 1

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

[4,1] => [1,1,1,2] => [2,1,1,1] => 1

[5] => [1,1,1,1,1] => [1,1,1,1,1] => 1

[1,1,1,1,1,1] => [6] => [6] => 11

[1,1,1,1,2] => [5,1] => [5,1] => 4

[1,1,1,2,1] => [4,2] => [4,2] => 5

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

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

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

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

[1,1,4] => [3,1,1,1] => [3,1,1,1] => 1

[1,2,1,1,1] => [2,4] => [4,2] => 5

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

[1,2,2,1] => [2,2,2] => [2,2,2] => 1

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

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

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

[1,4,1] => [2,1,1,2] => [2,2,1,1] => 1

[1,5] => [2,1,1,1,1] => [2,1,1,1,1] => 1

[2,1,1,1,1] => [1,5] => [5,1] => 4

[2,1,1,2] => [1,4,1] => [4,1,1] => 2

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

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

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

[2,2,2] => [1,2,2,1] => [2,2,1,1] => 1

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

[2,4] => [1,2,1,1,1] => [2,1,1,1,1] => 1

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

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

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

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

[4,1,1] => [1,1,1,3] => [3,1,1,1] => 1

[4,2] => [1,1,1,2,1] => [2,1,1,1,1] => 1

[5,1] => [1,1,1,1,2] => [2,1,1,1,1] => 1

[6] => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 1

[1,1,1,1,1,1,1] => [7] => [7] => 33

[1,1,1,1,1,2] => [6,1] => [6,1] => 11

[1,1,1,1,2,1] => [5,2] => [5,2] => 12

[1,1,1,1,3] => [5,1,1] => [5,1,1] => 4

[1,1,1,2,1,1] => [4,3] => [4,3] => 10

[1,1,1,2,2] => [4,2,1] => [4,2,1] => 5

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

[1,1,1,4] => [4,1,1,1] => [4,1,1,1] => 2

[1,1,2,1,1,1] => [3,4] => [4,3] => 10

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

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

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

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

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

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

[1,1,5] => [3,1,1,1,1] => [3,1,1,1,1] => 1

[1,2,1,1,1,1] => [2,5] => [5,2] => 12

[1,2,1,1,2] => [2,4,1] => [4,2,1] => 5

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

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

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

[1,2,2,2] => [2,2,2,1] => [2,2,2,1] => 1

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

[1,2,4] => [2,2,1,1,1] => [2,2,1,1,1] => 1

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

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

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

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

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

[1,4,2] => [2,1,1,2,1] => [2,2,1,1,1] => 1

[1,5,1] => [2,1,1,1,2] => [2,2,1,1,1] => 1

[1,6] => [2,1,1,1,1,1] => [2,1,1,1,1,1] => 1

[2,1,1,1,1,1] => [1,6] => [6,1] => 11

[2,1,1,1,2] => [1,5,1] => [5,1,1] => 4

[2,1,1,2,1] => [1,4,2] => [4,2,1] => 5

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

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

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

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Description

The number of ways to refine the partition into singletons.
For example there is only one way to refine $[2,2]$: $[2,2] > [2,1,1] > [1,1,1,1]$. However, there are two ways to refine $[3,2]$: $[3,2] > [2,2,1] > [2,1,1,1] > [1,1,1,1,1$ and $[3,2] > [3,1,1] > [2,1,1,1] > [1,1,1,1,1]$.
In other words, this is the number of saturated chains in the refinement order from the bottom element to the given partition.
The sequence of values on the partitions with only one part is A002846.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

Map

complement

Description

The complement of a composition.
The complement of a composition $I$ is defined as follows:
If $I$ is the empty composition, then the complement is also the empty composition. Otherwise, let $S$ be the descent set corresponding to $I=(i_1,\dots,i_k)$, that is, the subset
$$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$$
of $\{ 1, 2, \ldots, |I|-1 \}$. Then, the complement of $I$ is the composition of the same size as $I$, whose descent set is $\{ 1, 2, \ldots, |I|-1 \} \setminus S$.
The complement of a composition $I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to $I$.