- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000441: Permutations ⟶ ℤ

Values

[1] => [1] => [1] => [1] => 0

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

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

[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0

[1,3,2] => [1,2,3] => [1,2,3] => [3,2,1] => 0

[2,1,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0

[2,3,1] => [1,2,3] => [1,2,3] => [3,2,1] => 0

[3,1,2] => [1,3,2] => [3,1,2] => [1,3,2] => 0

[3,2,1] => [1,3,2] => [3,1,2] => [1,3,2] => 0

[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0

[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0

[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0

[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0

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

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

[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0

[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0

[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0

[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0

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

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

[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => [2,4,3,1] => 0

[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => [2,4,1,3] => 0

[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => [2,4,3,1] => 0

[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => [2,4,1,3] => 0

[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => [2,4,3,1] => 0

[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => [2,4,3,1] => 0

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

[4,1,3,2] => [1,4,2,3] => [4,1,2,3] => [1,4,3,2] => 0

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

[4,2,3,1] => [1,4,2,3] => [4,1,2,3] => [1,4,3,2] => 0

[4,3,1,2] => [1,4,2,3] => [4,1,2,3] => [1,4,3,2] => 0

[4,3,2,1] => [1,4,2,3] => [4,1,2,3] => [1,4,3,2] => 0

[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[1,2,5,3,4] => [1,2,3,5,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0

[1,2,5,4,3] => [1,2,3,5,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0

[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[1,3,5,2,4] => [1,2,3,5,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0

[1,3,5,4,2] => [1,2,3,5,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0

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

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

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

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

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

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

[1,5,2,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0

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

[1,5,3,2,4] => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0

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

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

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

[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[2,1,5,3,4] => [1,2,3,5,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0

[2,1,5,4,3] => [1,2,3,5,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0

[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0

[2,3,5,1,4] => [1,2,3,5,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0

[2,3,5,4,1] => [1,2,3,5,4] => [1,2,5,3,4] => [5,4,1,3,2] => 0

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

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

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

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

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

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

[2,5,1,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0

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

[2,5,3,1,4] => [1,2,5,4,3] => [1,2,4,5,3] => [5,4,2,1,3] => 0

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

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

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

[3,1,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 0

[3,1,2,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 0

[3,1,4,2,5] => [1,3,4,2,5] => [3,1,4,2,5] => [3,5,2,4,1] => 0

[3,1,4,5,2] => [1,3,4,5,2] => [3,1,4,5,2] => [3,5,2,1,4] => 0

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

[3,1,5,4,2] => [1,3,5,2,4] => [3,1,5,2,4] => [3,5,1,4,2] => 0

[3,2,1,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 0

[3,2,1,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 0

[3,2,4,1,5] => [1,3,4,2,5] => [3,1,4,2,5] => [3,5,2,4,1] => 0

[3,2,4,5,1] => [1,3,4,5,2] => [3,1,4,5,2] => [3,5,2,1,4] => 0

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

[3,2,5,4,1] => [1,3,5,2,4] => [3,1,5,2,4] => [3,5,1,4,2] => 0

[3,4,1,2,5] => [1,3,2,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 0

[3,4,1,5,2] => [1,3,2,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 0

[3,4,2,1,5] => [1,3,2,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 0

[3,4,2,5,1] => [1,3,2,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 0

[3,4,5,1,2] => [1,3,5,2,4] => [3,1,5,2,4] => [3,5,1,4,2] => 0

[3,4,5,2,1] => [1,3,5,2,4] => [3,1,5,2,4] => [3,5,1,4,2] => 0

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

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

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Description

The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as small ascents or 1-rises.

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$

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

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.