- FindStat

Identifier

Mp00223: Permutations —runsort⟶ Permutations
St000124: Permutations ⟶ ℤ

Values

[1] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,5,2,3,4] => [1,5,2,3,4] => 6

[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] => 6

[1,5,4,2,3] => [1,5,2,3,4] => 6

[1,5,4,3,2] => [1,5,2,3,4] => 6

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

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

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

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

[2,1,5,3,4] => [1,5,2,3,4] => 6

[2,1,5,4,3] => [1,5,2,3,4] => 6

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

[2,3,1,5,4] => [1,5,2,3,4] => 6

[2,3,4,1,5] => [1,5,2,3,4] => 6

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,1,5,4,2] => [1,5,2,3,4] => 6

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

[3,2,1,5,4] => [1,5,2,3,4] => 6

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

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

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

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

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

[3,4,1,5,2] => [1,5,2,3,4] => 6

[3,4,2,1,5] => [1,5,2,3,4] => 6

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

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

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

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

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

>>> Load all 2034 entries. <<<

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 6\ q$

$F_{4} = 18\ q + 6\ q^{2}$

$F_{5} = 6 + 54\ q + 42\ q^{2} + 18\ q^{6}$

$F_{6} = 114 + 162\ q + 204\ q^{2} + 18\ q^{4} + 168\ q^{6} + 54\ q^{24}$

Description

The cardinality of the preimage of the Simion-Schmidt map.
The Simion-Schmidt bijection transforms a [3,1,2]-avoiding permutation into a [3,2,1]-avoiding permutation. More generally, it can be thought of as a map

$S$ that turns any permutation into a [3,2,1]-avoiding permutation. This statistic is the size of

$S^{-1}(\pi)$ for each permutation

$\pi$ .
The map

$S$ can also be realized using the quotient of the

$0$ -Hecke Monoid of the symmetric group by the relation

$\pi_i \pi_{i+1} \pi_i = \pi_{i+1} \pi_i$ , sending each element of the fiber of the quotient to the unique [3,2,1]-avoiding element in that fiber.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.