- FindStat

Identifier

Mp00223: Permutations —runsort⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000358: Permutations ⟶ ℤ

Values

[1] => [1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 6$

$F_{4} = 22 + 2\ q$

$F_{5} = 92 + 26\ q + 2\ q^{2}$

$F_{6} = 426 + 236\ q + 50\ q^{2} + 8\ q^{3}$

Description

The number of occurrences of the pattern 31-2.
See Permutations/#Pattern-avoiding_permutations for the definition of the pattern

$31\!\!-\!\!2$ .

Map

invert Laguerre heap

Description

The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let

$\pi$ be a permutation. Following Viennot [1], we associate to

$\pi$ a heap of pieces, by considering each decreasing run

$(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of

$\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.