- FindStat

Identifier

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000382: Integer compositions ⟶ ℤ

Values

{{1}} => [1] => [1] => 1

{{1,2}} => [2] => [1,1] => 1

{{1},{2}} => [1,1] => [2] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5,6}} => [6] => [1,1,1,1,1,1] => 1

{{1,2,3,4,5},{6}} => [5,1] => [1,1,1,1,2] => 1

{{1,2,3,4,6},{5}} => [5,1] => [1,1,1,1,2] => 1

{{1,2,3,4},{5,6}} => [4,2] => [1,1,1,2,1] => 1

{{1,2,3,4},{5},{6}} => [4,1,1] => [1,1,1,3] => 1

{{1,2,3,5,6},{4}} => [5,1] => [1,1,1,1,2] => 1

{{1,2,3,5},{4,6}} => [4,2] => [1,1,1,2,1] => 1

{{1,2,3,5},{4},{6}} => [4,1,1] => [1,1,1,3] => 1

{{1,2,3,6},{4,5}} => [4,2] => [1,1,1,2,1] => 1

{{1,2,3},{4,5,6}} => [3,3] => [1,1,2,1,1] => 1

{{1,2,3},{4,5},{6}} => [3,2,1] => [1,1,2,2] => 1

{{1,2,3,6},{4},{5}} => [4,1,1] => [1,1,1,3] => 1

{{1,2,3},{4,6},{5}} => [3,2,1] => [1,1,2,2] => 1

{{1,2,3},{4},{5,6}} => [3,1,2] => [1,1,3,1] => 1

{{1,2,3},{4},{5},{6}} => [3,1,1,1] => [1,1,4] => 1

{{1,2,4,5,6},{3}} => [5,1] => [1,1,1,1,2] => 1

{{1,2,4,5},{3,6}} => [4,2] => [1,1,1,2,1] => 1

{{1,2,4,5},{3},{6}} => [4,1,1] => [1,1,1,3] => 1

{{1,2,4,6},{3,5}} => [4,2] => [1,1,1,2,1] => 1

{{1,2,4},{3,5,6}} => [3,3] => [1,1,2,1,1] => 1

{{1,2,4},{3,5},{6}} => [3,2,1] => [1,1,2,2] => 1

{{1,2,4,6},{3},{5}} => [4,1,1] => [1,1,1,3] => 1

{{1,2,4},{3,6},{5}} => [3,2,1] => [1,1,2,2] => 1

{{1,2,4},{3},{5,6}} => [3,1,2] => [1,1,3,1] => 1

{{1,2,4},{3},{5},{6}} => [3,1,1,1] => [1,1,4] => 1

{{1,2,5,6},{3,4}} => [4,2] => [1,1,1,2,1] => 1

>>> Load all 4154 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = 3\ q + q^{2} + q^{3}$

$F_{4} = 10\ q + 3\ q^{2} + q^{3} + q^{4}$

$F_{5} = 37\ q + 10\ q^{2} + 3\ q^{3} + q^{4} + q^{5}$

$F_{6} = 151\ q + 37\ q^{2} + 10\ q^{3} + 3\ q^{4} + q^{5} + q^{6}$

$F_{7} = 674\ q + 151\ q^{2} + 37\ q^{3} + 10\ q^{4} + 3\ q^{5} + q^{6} + q^{7}$

Description

The first part of an integer composition.

Map

to composition

Description

The integer composition of block sizes of a set partition.
For a set partition of

$\{1,2,\dots,n\}$ , this is the integer composition of

$n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.

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

$I$ .