- FindStat

Identifier

Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations

Images

[1] => [1] => [1] => [1]

[1,2] => [1,2] => [1,2] => [1,2]

[2,1] => [2,1] => [2,1] => [1,2]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 7307 entries. <<<

Map

Alexandersson Kebede

Description

Sends a permutation to a permutation and it preserves the set of right-to-left minima.
Take a permutation

$\pi$ of length

$n$ . The mapping looks for a smallest odd integer

$i\in[n-1]$ such that swapping the entries

$\pi(i)$ and

$\pi(i+1)$ preserves the set of right-to-left minima. Otherwise,

$\pi$ will be a fixed element of the mapping. Note that the map changes the sign of all non-fixed elements.
There are exactly

$\binom{\lfloor n/2 \rfloor}{k-\lceil n/2 \rceil}$ elements in

$S_n$ fixed under this map, with exactly

$k$ right-to-left minima, see Lemma 35 in [1].

Map

inverse

Description

Sends a permutation to its inverse.

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.