- FindStat

Identifier

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤ

Values

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

{{1,2}} => [2] => [1,1] => 11 => 2

{{1},{2}} => [1,1] => [2] => 10 => 1

{{1,2,3}} => [3] => [1,1,1] => 111 => 3

{{1,2},{3}} => [2,1] => [1,2] => 110 => 2

{{1,3},{2}} => [2,1] => [1,2] => 110 => 2

{{1},{2,3}} => [1,2] => [2,1] => 101 => 1

{{1},{2},{3}} => [1,1,1] => [3] => 100 => 1

{{1,2,3,4}} => [4] => [1,1,1,1] => 1111 => 4

{{1,2,3},{4}} => [3,1] => [1,1,2] => 1110 => 3

{{1,2,4},{3}} => [3,1] => [1,1,2] => 1110 => 3

{{1,2},{3,4}} => [2,2] => [1,2,1] => 1101 => 2

{{1,2},{3},{4}} => [2,1,1] => [1,3] => 1100 => 2

{{1,3,4},{2}} => [3,1] => [1,1,2] => 1110 => 3

{{1,3},{2,4}} => [2,2] => [1,2,1] => 1101 => 2

{{1,3},{2},{4}} => [2,1,1] => [1,3] => 1100 => 2

{{1,4},{2,3}} => [2,2] => [1,2,1] => 1101 => 2

{{1},{2,3,4}} => [1,3] => [2,1,1] => 1011 => 1

{{1},{2,3},{4}} => [1,2,1] => [2,2] => 1010 => 1

{{1,4},{2},{3}} => [2,1,1] => [1,3] => 1100 => 2

{{1},{2,4},{3}} => [1,2,1] => [2,2] => 1010 => 1

{{1},{2},{3,4}} => [1,1,2] => [3,1] => 1001 => 1

{{1},{2},{3},{4}} => [1,1,1,1] => [4] => 1000 => 1

{{1,2,3,4,5}} => [5] => [1,1,1,1,1] => 11111 => 5

{{1,2,3,4},{5}} => [4,1] => [1,1,1,2] => 11110 => 4

{{1,2,3,5},{4}} => [4,1] => [1,1,1,2] => 11110 => 4

{{1,2,3},{4,5}} => [3,2] => [1,1,2,1] => 11101 => 3

{{1,2,3},{4},{5}} => [3,1,1] => [1,1,3] => 11100 => 3

{{1,2,4,5},{3}} => [4,1] => [1,1,1,2] => 11110 => 4

{{1,2,4},{3,5}} => [3,2] => [1,1,2,1] => 11101 => 3

{{1,2,4},{3},{5}} => [3,1,1] => [1,1,3] => 11100 => 3

{{1,2,5},{3,4}} => [3,2] => [1,1,2,1] => 11101 => 3

{{1,2},{3,4,5}} => [2,3] => [1,2,1,1] => 11011 => 2

{{1,2},{3,4},{5}} => [2,2,1] => [1,2,2] => 11010 => 2

{{1,2,5},{3},{4}} => [3,1,1] => [1,1,3] => 11100 => 3

{{1,2},{3,5},{4}} => [2,2,1] => [1,2,2] => 11010 => 2

{{1,2},{3},{4,5}} => [2,1,2] => [1,3,1] => 11001 => 2

{{1,2},{3},{4},{5}} => [2,1,1,1] => [1,4] => 11000 => 2

{{1,3,4,5},{2}} => [4,1] => [1,1,1,2] => 11110 => 4

{{1,3,4},{2,5}} => [3,2] => [1,1,2,1] => 11101 => 3

{{1,3,4},{2},{5}} => [3,1,1] => [1,1,3] => 11100 => 3

{{1,3,5},{2,4}} => [3,2] => [1,1,2,1] => 11101 => 3

{{1,3},{2,4,5}} => [2,3] => [1,2,1,1] => 11011 => 2

{{1,3},{2,4},{5}} => [2,2,1] => [1,2,2] => 11010 => 2

{{1,3,5},{2},{4}} => [3,1,1] => [1,1,3] => 11100 => 3

{{1,3},{2,5},{4}} => [2,2,1] => [1,2,2] => 11010 => 2

{{1,3},{2},{4,5}} => [2,1,2] => [1,3,1] => 11001 => 2

{{1,3},{2},{4},{5}} => [2,1,1,1] => [1,4] => 11000 => 2

{{1,4,5},{2,3}} => [3,2] => [1,1,2,1] => 11101 => 3

{{1,4},{2,3,5}} => [2,3] => [1,2,1,1] => 11011 => 2

{{1,4},{2,3},{5}} => [2,2,1] => [1,2,2] => 11010 => 2

{{1,5},{2,3,4}} => [2,3] => [1,2,1,1] => 11011 => 2

{{1},{2,3,4,5}} => [1,4] => [2,1,1,1] => 10111 => 1

{{1},{2,3,4},{5}} => [1,3,1] => [2,1,2] => 10110 => 1

{{1,5},{2,3},{4}} => [2,2,1] => [1,2,2] => 11010 => 2

{{1},{2,3,5},{4}} => [1,3,1] => [2,1,2] => 10110 => 1

{{1},{2,3},{4,5}} => [1,2,2] => [2,2,1] => 10101 => 1

{{1},{2,3},{4},{5}} => [1,2,1,1] => [2,3] => 10100 => 1

{{1,4,5},{2},{3}} => [3,1,1] => [1,1,3] => 11100 => 3

{{1,4},{2,5},{3}} => [2,2,1] => [1,2,2] => 11010 => 2

{{1,4},{2},{3,5}} => [2,1,2] => [1,3,1] => 11001 => 2

{{1,4},{2},{3},{5}} => [2,1,1,1] => [1,4] => 11000 => 2

{{1,5},{2,4},{3}} => [2,2,1] => [1,2,2] => 11010 => 2

{{1},{2,4,5},{3}} => [1,3,1] => [2,1,2] => 10110 => 1

{{1},{2,4},{3,5}} => [1,2,2] => [2,2,1] => 10101 => 1

{{1},{2,4},{3},{5}} => [1,2,1,1] => [2,3] => 10100 => 1

{{1,5},{2},{3,4}} => [2,1,2] => [1,3,1] => 11001 => 2

{{1},{2,5},{3,4}} => [1,2,2] => [2,2,1] => 10101 => 1

{{1},{2},{3,4,5}} => [1,1,3] => [3,1,1] => 10011 => 1

{{1},{2},{3,4},{5}} => [1,1,2,1] => [3,2] => 10010 => 1

{{1,5},{2},{3},{4}} => [2,1,1,1] => [1,4] => 11000 => 2

{{1},{2,5},{3},{4}} => [1,2,1,1] => [2,3] => 10100 => 1

{{1},{2},{3,5},{4}} => [1,1,2,1] => [3,2] => 10010 => 1

{{1},{2},{3},{4,5}} => [1,1,1,2] => [4,1] => 10001 => 1

{{1},{2},{3},{4},{5}} => [1,1,1,1,1] => [5] => 10000 => 1

{{1,2,3,4,5,6}} => [6] => [1,1,1,1,1,1] => 111111 => 6

{{1,2,3,4,5},{6}} => [5,1] => [1,1,1,1,2] => 111110 => 5

{{1,2,3,4,6},{5}} => [5,1] => [1,1,1,1,2] => 111110 => 5

{{1,2,3,4},{5,6}} => [4,2] => [1,1,1,2,1] => 111101 => 4

{{1,2,3,4},{5},{6}} => [4,1,1] => [1,1,1,3] => 111100 => 4

{{1,2,3,5,6},{4}} => [5,1] => [1,1,1,1,2] => 111110 => 5

{{1,2,3,5},{4,6}} => [4,2] => [1,1,1,2,1] => 111101 => 4

{{1,2,3,5},{4},{6}} => [4,1,1] => [1,1,1,3] => 111100 => 4

{{1,2,3,6},{4,5}} => [4,2] => [1,1,1,2,1] => 111101 => 4

{{1,2,3},{4,5,6}} => [3,3] => [1,1,2,1,1] => 111011 => 3

{{1,2,3},{4,5},{6}} => [3,2,1] => [1,1,2,2] => 111010 => 3

{{1,2,3,6},{4},{5}} => [4,1,1] => [1,1,1,3] => 111100 => 4

{{1,2,3},{4,6},{5}} => [3,2,1] => [1,1,2,2] => 111010 => 3

{{1,2,3},{4},{5,6}} => [3,1,2] => [1,1,3,1] => 111001 => 3

{{1,2,3},{4},{5},{6}} => [3,1,1,1] => [1,1,4] => 111000 => 3

{{1,2,4,5,6},{3}} => [5,1] => [1,1,1,1,2] => 111110 => 5

{{1,2,4,5},{3,6}} => [4,2] => [1,1,1,2,1] => 111101 => 4

{{1,2,4,5},{3},{6}} => [4,1,1] => [1,1,1,3] => 111100 => 4

{{1,2,4,6},{3,5}} => [4,2] => [1,1,1,2,1] => 111101 => 4

{{1,2,4},{3,5,6}} => [3,3] => [1,1,2,1,1] => 111011 => 3

{{1,2,4},{3,5},{6}} => [3,2,1] => [1,1,2,2] => 111010 => 3

{{1,2,4,6},{3},{5}} => [4,1,1] => [1,1,1,3] => 111100 => 4

{{1,2,4},{3,6},{5}} => [3,2,1] => [1,1,2,2] => 111010 => 3

{{1,2,4},{3},{5,6}} => [3,1,2] => [1,1,3,1] => 111001 => 3

{{1,2,4},{3},{5},{6}} => [3,1,1,1] => [1,1,4] => 111000 => 3

{{1,2,5,6},{3,4}} => [4,2] => [1,1,1,2,1] => 111101 => 4

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Description

The number of leading ones in a binary word.

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 $I$.

Map

to binary word

Description

Return the composition as a binary word, treating ones as separators.
Encoding a positive integer $i$ as the word $10\dots 0$ consisting of a one followed by $i-1$ zeros, the binary word of a composition $(i_1,\dots,i_k)$ is the concatenation of of words for $i_1,\dots,i_k$.
The image of this map contains precisely the words which do not begin with a $0$.