- FindStat

Identifier

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤ

Values

{{1}} => [1] => 1 => 0

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

{{1},{2}} => [1,1] => 11 => 0

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

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

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

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

{{1},{2},{3}} => [1,1,1] => 111 => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1},{2},{3},{4}} => [1,1,1,1] => 1111 => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1},{2},{3},{4},{5}} => [1,1,1,1,1] => 11111 => 0

{{1,2,3,4,5,6}} => [6] => 100000 => 1

{{1,2,3,4,5},{6}} => [5,1] => 100001 => 1

{{1,2,3,4,6},{5}} => [5,1] => 100001 => 1

{{1,2,3,4},{5,6}} => [4,2] => 100010 => 2

{{1,2,3,4},{5},{6}} => [4,1,1] => 100011 => 1

{{1,2,3,5,6},{4}} => [5,1] => 100001 => 1

{{1,2,3,5},{4,6}} => [4,2] => 100010 => 2

{{1,2,3,5},{4},{6}} => [4,1,1] => 100011 => 1

{{1,2,3,6},{4,5}} => [4,2] => 100010 => 2

{{1,2,3},{4,5,6}} => [3,3] => 100100 => 2

{{1,2,3},{4,5},{6}} => [3,2,1] => 100101 => 2

{{1,2,3,6},{4},{5}} => [4,1,1] => 100011 => 1

{{1,2,3},{4,6},{5}} => [3,2,1] => 100101 => 2

{{1,2,3},{4},{5,6}} => [3,1,2] => 100110 => 2

{{1,2,3},{4},{5},{6}} => [3,1,1,1] => 100111 => 1

{{1,2,4,5,6},{3}} => [5,1] => 100001 => 1

{{1,2,4,5},{3,6}} => [4,2] => 100010 => 2

{{1,2,4,5},{3},{6}} => [4,1,1] => 100011 => 1

{{1,2,4,6},{3,5}} => [4,2] => 100010 => 2

{{1,2,4},{3,5,6}} => [3,3] => 100100 => 2

{{1,2,4},{3,5},{6}} => [3,2,1] => 100101 => 2

{{1,2,4,6},{3},{5}} => [4,1,1] => 100011 => 1

{{1,2,4},{3,6},{5}} => [3,2,1] => 100101 => 2

{{1,2,4},{3},{5,6}} => [3,1,2] => 100110 => 2

{{1,2,4},{3},{5},{6}} => [3,1,1,1] => 100111 => 1

{{1,2,5,6},{3,4}} => [4,2] => 100010 => 2

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 4\ q$

$F_{4} = 1 + 11\ q + 3\ q^{2}$

$F_{5} = 1 + 26\ q + 25\ q^{2}$

$F_{6} = 1 + 57\ q + 130\ q^{2} + 15\ q^{3}$

$F_{7} = 1 + 120\ q + 546\ q^{2} + 210\ q^{3}$

Description

The number of descents of 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

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

$0$ .