- FindStat

Identifier

Mp00223: Permutations —runsort⟶ Permutations
St000367: 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] => 0

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

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

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

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

[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] => 0

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

[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] => 0

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

[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] => 0

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

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

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

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

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

[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] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 0

>>> Load all 3609 entries. <<<

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 6$

$F_{4} = 24$

$F_{5} = 120$

$F_{6} = 720$

Description

The number of simsun double descents of a permutation.
The restriction of a permutation

$\pi$ to

$[k] = \{1,\ldots,k\}$ is given in one-line notation by the subword of

$\pi$ of letters in

$[k]$ .
A simsun double descent of a permutation

$\pi$ is a double descent of any restriction of

$\pi$ to

$[1,\ldots,k]$ for some

$k$ . (Note here that the same double descent can appear in multiple restrictions!)

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.