- FindStat

Identifier

Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5}} => {{1,2,3,4,5}} => [2,3,4,5,1] => 0000

{{1,2,3,4},{5}} => {{1},{2,3,4,5}} => [1,3,4,5,2] => 1000

{{1,2,3,5},{4}} => {{1,3,4,5},{2}} => [3,2,4,5,1] => 0000

{{1,2,3},{4,5}} => {{1,2},{3,4,5}} => [2,1,4,5,3] => 0100

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

{{1,2,4,5},{3}} => {{1,3},{2,4,5}} => [3,4,1,5,2] => 0000

{{1,2,4},{3,5}} => {{1,2,4,5},{3}} => [2,4,3,5,1] => 0000

{{1,2,4},{3},{5}} => {{1},{2,4,5},{3}} => [1,4,3,5,2] => 1000

{{1,2,5},{3,4}} => {{1,4,5},{2,3}} => [4,3,2,5,1] => 0000

{{1,2},{3,4,5}} => {{1,2,3},{4,5}} => [2,3,1,5,4] => 0010

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

{{1,2,5},{3},{4}} => {{1,4,5},{2},{3}} => [4,2,3,5,1] => 0000

{{1,2},{3,5},{4}} => {{1,3},{2},{4,5}} => [3,2,1,5,4] => 0010

{{1,2},{3},{4,5}} => {{1,2},{3},{4,5}} => [2,1,3,5,4] => 0110

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

{{1,3,4,5},{2}} => {{1,3,5},{2,4}} => [3,4,5,2,1] => 0000

{{1,3,4},{2,5}} => {{1,2,4},{3,5}} => [2,4,5,1,3] => 0000

{{1,3,4},{2},{5}} => {{1},{2,4},{3,5}} => [1,4,5,2,3] => 1000

{{1,3,5},{2,4}} => {{1,4},{2,3,5}} => [4,3,5,1,2] => 0000

{{1,3},{2,4,5}} => {{1,2,3,5},{4}} => [2,3,5,4,1] => 0000

{{1,3},{2,4},{5}} => {{1},{2,3,5},{4}} => [1,3,5,4,2] => 1000

{{1,3,5},{2},{4}} => {{1,4},{2},{3,5}} => [4,2,5,1,3] => 0000

{{1,3},{2,5},{4}} => {{1,3,5},{2},{4}} => [3,2,5,4,1] => 0000

{{1,3},{2},{4,5}} => {{1,2},{3,5},{4}} => [2,1,5,4,3] => 0100

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

{{1,4,5},{2,3}} => {{1,3,4},{2,5}} => [3,5,4,1,2] => 0000

{{1,4},{2,3,5}} => {{1,2,5},{3,4}} => [2,5,4,3,1] => 0000

{{1,4},{2,3},{5}} => {{1},{2,5},{3,4}} => [1,5,4,3,2] => 1000

{{1,5},{2,3,4}} => {{1,5},{2,3,4}} => [5,3,4,2,1] => 0000

{{1},{2,3,4,5}} => {{1,2,3,4},{5}} => [2,3,4,1,5] => 0001

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

{{1,5},{2,3},{4}} => {{1,5},{2},{3,4}} => [5,2,4,3,1] => 0000

{{1},{2,3,5},{4}} => {{1,3,4},{2},{5}} => [3,2,4,1,5] => 0001

{{1},{2,3},{4,5}} => {{1,2},{3,4},{5}} => [2,1,4,3,5] => 0101

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

{{1,4,5},{2},{3}} => {{1,4},{2,5},{3}} => [4,5,3,1,2] => 0000

{{1,4},{2,5},{3}} => {{1,3},{2,5},{4}} => [3,5,1,4,2] => 0000

{{1,4},{2},{3,5}} => {{1,2,5},{3},{4}} => [2,5,3,4,1] => 0000

{{1,4},{2},{3},{5}} => {{1},{2,5},{3},{4}} => [1,5,3,4,2] => 1000

{{1,5},{2,4},{3}} => {{1,5},{2,4},{3}} => [5,4,3,2,1] => 0000

{{1},{2,4,5},{3}} => {{1,3},{2,4},{5}} => [3,4,1,2,5] => 0001

{{1},{2,4},{3,5}} => {{1,2,4},{3},{5}} => [2,4,3,1,5] => 0001

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

{{1,5},{2},{3,4}} => {{1,5},{2,3},{4}} => [5,3,2,4,1] => 0000

{{1},{2,5},{3,4}} => {{1,4},{2,3},{5}} => [4,3,2,1,5] => 0001

{{1},{2},{3,4,5}} => {{1,2,3},{4},{5}} => [2,3,1,4,5] => 0011

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

{{1,5},{2},{3},{4}} => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => 0000

{{1},{2,5},{3},{4}} => {{1,4},{2},{3},{5}} => [4,2,3,1,5] => 0001

{{1},{2},{3,5},{4}} => {{1,3},{2},{4},{5}} => [3,2,1,4,5] => 0011

{{1},{2},{3},{4,5}} => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => 0111

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

{{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 00000

{{1,2,3,4,5},{6}} => {{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => 10000

{{1,2,3,4,6},{5}} => {{1,3,4,5,6},{2}} => [3,2,4,5,6,1] => 00000

{{1,2,3,4},{5,6}} => {{1,2},{3,4,5,6}} => [2,1,4,5,6,3] => 01000

{{1,2,3,4},{5},{6}} => {{1},{2},{3,4,5,6}} => [1,2,4,5,6,3] => 11000

{{1,2,3,5,6},{4}} => {{1,3},{2,4,5,6}} => [3,4,1,5,6,2] => 00000

{{1,2,3,5},{4,6}} => {{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => 00000

{{1,2,3,5},{4},{6}} => {{1},{2,4,5,6},{3}} => [1,4,3,5,6,2] => 10000

{{1,2,3,6},{4,5}} => {{1,4,5,6},{2,3}} => [4,3,2,5,6,1] => 00000

{{1,2,3},{4,5,6}} => {{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => 00100

{{1,2,3},{4,5},{6}} => {{1},{2,3},{4,5,6}} => [1,3,2,5,6,4] => 10100

{{1,2,3,6},{4},{5}} => {{1,4,5,6},{2},{3}} => [4,2,3,5,6,1] => 00000

{{1,2,3},{4,6},{5}} => {{1,3},{2},{4,5,6}} => [3,2,1,5,6,4] => 00100

{{1,2,3},{4},{5,6}} => {{1,2},{3},{4,5,6}} => [2,1,3,5,6,4] => 01100

{{1,2,3},{4},{5},{6}} => {{1},{2},{3},{4,5,6}} => [1,2,3,5,6,4] => 11100

{{1,2,4,5,6},{3}} => {{1,3,5,6},{2,4}} => [3,4,5,2,6,1] => 00000

{{1,2,4,5},{3,6}} => {{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => 00000

{{1,2,4,5},{3},{6}} => {{1},{2,4},{3,5,6}} => [1,4,5,2,6,3] => 10000

{{1,2,4,6},{3,5}} => {{1,4},{2,3,5,6}} => [4,3,5,1,6,2] => 00000

{{1,2,4},{3,5,6}} => {{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => 00000

{{1,2,4},{3,5},{6}} => {{1},{2,3,5,6},{4}} => [1,3,5,4,6,2] => 10000

{{1,2,4,6},{3},{5}} => {{1,4},{2},{3,5,6}} => [4,2,5,1,6,3] => 00000

{{1,2,4},{3,6},{5}} => {{1,3,5,6},{2},{4}} => [3,2,5,4,6,1] => 00000

{{1,2,4},{3},{5,6}} => {{1,2},{3,5,6},{4}} => [2,1,5,4,6,3] => 01000

{{1,2,4},{3},{5},{6}} => {{1},{2},{3,5,6},{4}} => [1,2,5,4,6,3] => 11000

{{1,2,5,6},{3,4}} => {{1,3,4},{2,5,6}} => [3,5,4,1,6,2] => 00000

>>> Load all 2590 entries. <<<

Map

inverse Wachs-White

Description

The inverse of a transformation of set partitions due to Wachs and White.
Return the set partition of

$\{1,...,n\}$ corresponding to the set of arcs, interpreted as a rook placement, applying Wachs and White's bijection

$\gamma^{-1}$ .
Note that our index convention differs from the convention in [1]: regarding the rook board as a lower-right triangular grid, we refer with

$(i,j)$ to the cell in the

$i$ -th column from the right and the

$j$ -th row from the top.

Map

to permutation

Description

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.

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.