- FindStat

Identifier

Mp00310: Permutations —toric promotion⟶ 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]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 9623 entries. <<<

Map

toric promotion

Description

Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.

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.