- FindStat

Identifier

Mp00286: Ordered set partitions —to permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions

Images

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

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

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

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

[{1},{2},{3}] => [1,2,3] => 11 => [1,1,1]

[{1},{3},{2}] => [1,3,2] => 10 => [1,2]

[{2},{1},{3}] => [2,1,3] => 01 => [2,1]

[{3},{1},{2}] => [3,1,2] => 00 => [3]

[{2},{3},{1}] => [2,3,1] => 00 => [3]

[{3},{2},{1}] => [3,2,1] => 00 => [3]

[{1},{2,3}] => [1,2,3] => 11 => [1,1,1]

[{2},{1,3}] => [2,1,3] => 01 => [2,1]

[{3},{1,2}] => [3,1,2] => 00 => [3]

[{1,2},{3}] => [1,2,3] => 11 => [1,1,1]

[{1,3},{2}] => [1,3,2] => 10 => [1,2]

[{2,3},{1}] => [2,3,1] => 00 => [3]

[{1,2,3}] => [1,2,3] => 11 => [1,1,1]

[{1},{2},{3},{4}] => [1,2,3,4] => 111 => [1,1,1,1]

[{1},{2},{4},{3}] => [1,2,4,3] => 110 => [1,1,2]

[{1},{3},{2},{4}] => [1,3,2,4] => 101 => [1,2,1]

[{1},{4},{2},{3}] => [1,4,2,3] => 100 => [1,3]

[{1},{3},{4},{2}] => [1,3,4,2] => 100 => [1,3]

[{1},{4},{3},{2}] => [1,4,3,2] => 100 => [1,3]

[{2},{1},{3},{4}] => [2,1,3,4] => 011 => [2,1,1]

[{2},{1},{4},{3}] => [2,1,4,3] => 010 => [2,2]

[{3},{1},{2},{4}] => [3,1,2,4] => 001 => [3,1]

[{4},{1},{2},{3}] => [4,1,2,3] => 000 => [4]

[{3},{1},{4},{2}] => [3,1,4,2] => 000 => [4]

[{4},{1},{3},{2}] => [4,1,3,2] => 000 => [4]

[{2},{3},{1},{4}] => [2,3,1,4] => 001 => [3,1]

[{2},{4},{1},{3}] => [2,4,1,3] => 000 => [4]

[{3},{2},{1},{4}] => [3,2,1,4] => 001 => [3,1]

[{4},{2},{1},{3}] => [4,2,1,3] => 000 => [4]

[{3},{4},{1},{2}] => [3,4,1,2] => 000 => [4]

[{4},{3},{1},{2}] => [4,3,1,2] => 000 => [4]

[{2},{3},{4},{1}] => [2,3,4,1] => 000 => [4]

[{2},{4},{3},{1}] => [2,4,3,1] => 000 => [4]

[{3},{2},{4},{1}] => [3,2,4,1] => 000 => [4]

[{4},{2},{3},{1}] => [4,2,3,1] => 000 => [4]

[{3},{4},{2},{1}] => [3,4,2,1] => 000 => [4]

[{4},{3},{2},{1}] => [4,3,2,1] => 000 => [4]

[{1},{2},{3,4}] => [1,2,3,4] => 111 => [1,1,1,1]

[{1},{3},{2,4}] => [1,3,2,4] => 101 => [1,2,1]

[{1},{4},{2,3}] => [1,4,2,3] => 100 => [1,3]

[{2},{1},{3,4}] => [2,1,3,4] => 011 => [2,1,1]

[{3},{1},{2,4}] => [3,1,2,4] => 001 => [3,1]

[{4},{1},{2,3}] => [4,1,2,3] => 000 => [4]

[{2},{3},{1,4}] => [2,3,1,4] => 001 => [3,1]

[{2},{4},{1,3}] => [2,4,1,3] => 000 => [4]

[{3},{2},{1,4}] => [3,2,1,4] => 001 => [3,1]

[{4},{2},{1,3}] => [4,2,1,3] => 000 => [4]

[{3},{4},{1,2}] => [3,4,1,2] => 000 => [4]

[{4},{3},{1,2}] => [4,3,1,2] => 000 => [4]

[{1},{2,3},{4}] => [1,2,3,4] => 111 => [1,1,1,1]

[{1},{2,4},{3}] => [1,2,4,3] => 110 => [1,1,2]

[{1},{3,4},{2}] => [1,3,4,2] => 100 => [1,3]

[{2},{1,3},{4}] => [2,1,3,4] => 011 => [2,1,1]

[{2},{1,4},{3}] => [2,1,4,3] => 010 => [2,2]

[{3},{1,2},{4}] => [3,1,2,4] => 001 => [3,1]

[{4},{1,2},{3}] => [4,1,2,3] => 000 => [4]

[{3},{1,4},{2}] => [3,1,4,2] => 000 => [4]

[{4},{1,3},{2}] => [4,1,3,2] => 000 => [4]

[{2},{3,4},{1}] => [2,3,4,1] => 000 => [4]

[{3},{2,4},{1}] => [3,2,4,1] => 000 => [4]

[{4},{2,3},{1}] => [4,2,3,1] => 000 => [4]

[{1},{2,3,4}] => [1,2,3,4] => 111 => [1,1,1,1]

[{2},{1,3,4}] => [2,1,3,4] => 011 => [2,1,1]

[{3},{1,2,4}] => [3,1,2,4] => 001 => [3,1]

[{4},{1,2,3}] => [4,1,2,3] => 000 => [4]

[{1,2},{3},{4}] => [1,2,3,4] => 111 => [1,1,1,1]

[{1,2},{4},{3}] => [1,2,4,3] => 110 => [1,1,2]

[{1,3},{2},{4}] => [1,3,2,4] => 101 => [1,2,1]

[{1,4},{2},{3}] => [1,4,2,3] => 100 => [1,3]

[{1,3},{4},{2}] => [1,3,4,2] => 100 => [1,3]

[{1,4},{3},{2}] => [1,4,3,2] => 100 => [1,3]

[{2,3},{1},{4}] => [2,3,1,4] => 001 => [3,1]

[{2,4},{1},{3}] => [2,4,1,3] => 000 => [4]

[{3,4},{1},{2}] => [3,4,1,2] => 000 => [4]

[{2,3},{4},{1}] => [2,3,4,1] => 000 => [4]

[{2,4},{3},{1}] => [2,4,3,1] => 000 => [4]

[{3,4},{2},{1}] => [3,4,2,1] => 000 => [4]

[{1,2},{3,4}] => [1,2,3,4] => 111 => [1,1,1,1]

[{1,3},{2,4}] => [1,3,2,4] => 101 => [1,2,1]

[{1,4},{2,3}] => [1,4,2,3] => 100 => [1,3]

[{2,3},{1,4}] => [2,3,1,4] => 001 => [3,1]

[{2,4},{1,3}] => [2,4,1,3] => 000 => [4]

[{3,4},{1,2}] => [3,4,1,2] => 000 => [4]

[{1,2,3},{4}] => [1,2,3,4] => 111 => [1,1,1,1]

[{1,2,4},{3}] => [1,2,4,3] => 110 => [1,1,2]

[{1,3,4},{2}] => [1,3,4,2] => 100 => [1,3]

[{2,3,4},{1}] => [2,3,4,1] => 000 => [4]

[{1,2,3,4}] => [1,2,3,4] => 111 => [1,1,1,1]

[{1},{2},{3},{4},{5}] => [1,2,3,4,5] => 1111 => [1,1,1,1,1]

[{1},{2},{3},{5},{4}] => [1,2,3,5,4] => 1110 => [1,1,1,2]

[{1},{2},{4},{3},{5}] => [1,2,4,3,5] => 1101 => [1,1,2,1]

[{1},{2},{5},{3},{4}] => [1,2,5,3,4] => 1100 => [1,1,3]

[{1},{2},{4},{5},{3}] => [1,2,4,5,3] => 1100 => [1,1,3]

[{1},{2},{5},{4},{3}] => [1,2,5,4,3] => 1100 => [1,1,3]

[{1},{3},{2},{4},{5}] => [1,3,2,4,5] => 1011 => [1,2,1,1]

[{1},{3},{2},{5},{4}] => [1,3,2,5,4] => 1010 => [1,2,2]

[{1},{4},{2},{3},{5}] => [1,4,2,3,5] => 1001 => [1,3,1]

>>> Load all 5316 entries. <<<

Map

to permutation

Description

Return the permutation obtained by concatenating the blocks sorted into increasing order.

Map

connectivity set

Description

The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is

$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$
For

$n > 1$ it can also be described as the set of occurrences of the mesh pattern

$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$
or equivalently

$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$
see [3].
The permutation is connected, when the connectivity set is empty.

Map

to composition

Description

The composition corresponding to a binary word.
Prepending

$1$ to a binary word

$w$ , the

$i$ -th part of the composition equals

$1$ plus the number of zeros after the

$i$ -th

$1$ in

$w$ .
This map is not surjective, since the empty composition does not have a preimage.