- FindStat

Identifier

Mp00223: Permutations —runsort⟶ Permutations
St000317: Permutations ⟶ ℤ

Values

[1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 873 entries. <<<

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 6$

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

$F_{5} = 60 + 42\ q + 18\ q^{2}$

$F_{6} = 204 + 258\ q + 204\ q^{2} + 54\ q^{3}$

Description

The cycle descent number of a permutation.
Let

$(i_1,\ldots,i_k)$ be a cycle of a permutation

$\pi$ such that

$i_1$ is its smallest element. A **cycle descent** of

$(i_1,\ldots,i_k)$ is an

$i_a$ for

$1 \leq a < k$ such that

$i_a > i_{a+1}$ . The **cycle descent set** of

$\pi$ is then the set of descents in all the cycles of

$\pi$ , and the **cycle descent number** is its cardinality.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.