- FindStat

Identifier

Mp00069: Permutations —complement⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000628: Binary words ⟶ ℤ

Values

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

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

[1,2,3] => [3,2,1] => [1,2,3] => 00 => 0

[1,3,2] => [3,1,2] => [1,2,3] => 00 => 0

[2,1,3] => [2,3,1] => [1,2,3] => 00 => 0

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

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

[3,2,1] => [1,2,3] => [1,2,3] => 00 => 0

[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 000 => 0

[1,2,4,3] => [4,3,1,2] => [1,2,3,4] => 000 => 0

[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => 000 => 0

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

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

[1,4,3,2] => [4,1,2,3] => [1,2,3,4] => 000 => 0

[2,1,3,4] => [3,4,2,1] => [1,2,3,4] => 000 => 0

[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => 000 => 0

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

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

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

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

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

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

[3,2,1,4] => [2,3,4,1] => [1,2,3,4] => 000 => 0

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

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

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

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

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

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

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

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

[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 000 => 0

[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0000 => 0

[1,2,3,5,4] => [5,4,3,1,2] => [1,2,3,4,5] => 0000 => 0

[1,2,4,3,5] => [5,4,2,3,1] => [1,2,3,4,5] => 0000 => 0

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

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

[1,2,5,4,3] => [5,4,1,2,3] => [1,2,3,4,5] => 0000 => 0

[1,3,2,4,5] => [5,3,4,2,1] => [1,2,3,4,5] => 0000 => 0

[1,3,2,5,4] => [5,3,4,1,2] => [1,2,3,4,5] => 0000 => 0

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

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

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

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

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

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

[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => 0000 => 0

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

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

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

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

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

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

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

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

[1,5,4,3,2] => [5,1,2,3,4] => [1,2,3,4,5] => 0000 => 0

[2,1,3,4,5] => [4,5,3,2,1] => [1,2,3,4,5] => 0000 => 0

[2,1,3,5,4] => [4,5,3,1,2] => [1,2,3,4,5] => 0000 => 0

[2,1,4,3,5] => [4,5,2,3,1] => [1,2,3,4,5] => 0000 => 0

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

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

[2,1,5,4,3] => [4,5,1,2,3] => [1,2,3,4,5] => 0000 => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,2,1,4,5] => [3,4,5,2,1] => [1,2,3,4,5] => 0000 => 0

[3,2,1,5,4] => [3,4,5,1,2] => [1,2,3,4,5] => 0000 => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The balance of a binary word.
The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1].
A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.

Map

descent word

Description

The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.

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

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.