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Identifier

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St001778: Permutations ⟶ ℤ

Values

[1] => [1,0] => [(1,2)] => [2,1] => 1
[2] => [1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => 1
[1,1] => [1,1,0,0] => [(1,4),(2,3)] => [4,3,2,1] => 1
[3] => [1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => 1
[2,1] => [1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => 3
[1,1,1] => [1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => 1
[2,2] => [1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => 1

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 2\ q + q^{3}$

Description

The largest greatest common divisor of an element and its image in a permutation.

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

Map

to tunnel matching

Description

Sends a Dyck path of semilength n to the noncrossing perfect matching given by matching an up-step with the corresponding down-step.
This is, for a Dyck path

$D$ of semilength

$n$ , the perfect matching of

$\{1,\dots,2n\}$ with

$i < j$ being matched if

$D_i$ is an up-step and

$D_j$ is the down-step connected to

$D_i$ by a tunnel.

Map

to permutation

Description

Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.