- FindStat

Identifier

Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000650: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1200 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{2} = 2$

$F_{3} = 6$

$F_{4} = 18 + 6\ q$

$F_{5} = 78 + 36\ q + 6\ q^{2}$

$F_{6} = 426 + 234\ q + 54\ q^{2} + 6\ q^{3}$

Description

The number of 3-rises of a permutation.
A 3-rise of a permutation

$\pi$ is an index

$i$ such that

$\pi(i)+3 = \pi(i+1)$ .
For 1-rises, or successions, see St000441The number of successions of a permutation., for 2-rises see St000534The number of 2-rises of a permutation..

Map

inverse first fundamental transformation

Description

Let

$\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps

$\sigma$ to the permutation

$[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.

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

inverse

Description

Sends a permutation to its inverse.