- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001550: Permutations ⟶ ℤ

Values

[1] => [1,0,1,0] => [3,1,2] => [1,3,2] => 0
[2] => [1,1,0,0,1,0] => [2,4,1,3] => [2,4,1,3] => 0
[1,1] => [1,0,1,1,0,0] => [3,1,4,2] => [3,1,4,2] => 0
[3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [2,5,1,3,4] => 0
[2,1] => [1,0,1,0,1,0] => [4,1,2,3] => [1,2,4,3] => 0
[1,1,1] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,1,4,5,2] => 0
[4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [2,6,1,3,4,5] => 0
[3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,3,5,4,2] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [2,1,4,3,5] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [1,5,2,4,3] => 0
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,1,4,5,6,2] => 0
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [3,4,6,1,5,2] => 0
[3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,3,5,1,4] => 0
[3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [1,3,2,5,4] => 0
[2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,2,5,3] => 0
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [4,6,1,2,5,3] => 0
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,4,6,5,1,3] => 1
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [1,4,3,6,5,2] => 0
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [2,1,5,3,4,6] => 0
[3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,2,3,5,4] => 0
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [1,6,3,2,5,4] => 0
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [2,1,4,5,3,6] => 0
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [5,1,6,2,4,3] => 0
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [2,3,6,1,4,5] => 0
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [1,2,4,6,5,3] => 0
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,4,1,2,6,5] => 0
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [5,1,3,6,4,2] => 0
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,6,3,5,1,4] => 0
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [1,2,6,3,5,4] => 0
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,2,5,6,3] => 0
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [7,5,4,1,2,3,6] => [1,2,5,7,6,4,3] => 1
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [1,3,4,6,5,2] => 0
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [2,4,1,6,3,5] => 0
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [1,2,6,4,5,3] => 0
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,1,5,6,3,4] => 0
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,1,5,2,6,4] => 0
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [1,6,2,3,5,4] => 0
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [1,2,7,6,3,5,4] => 0
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [7,3,5,1,2,4,6] => [1,3,5,7,2,6,4] => 0
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [7,4,1,5,2,3,6] => [1,4,5,2,7,6,3] => 0
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,3,4,6,1,5] => 0
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [1,3,4,2,6,5] => 0
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [1,4,2,3,6,5] => 0
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,2,3,6,4] => 0
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [7,3,1,5,2,4,6] => [1,3,5,2,7,6,4] => 0
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 0
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => [1,3,7,4,2,6,5] => 0
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [1,4,2,7,3,6,5] => 0
[5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0] => [4,3,1,7,2,5,6] => [1,4,5,3,7,6,2] => 0
[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [5,7,1,2,3,4,6] => [1,2,3,5,7,4,6] => 0
[5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => [1,3,7,5,2,6,4] => 0
[4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [7,3,1,2,6,4,5] => [1,3,2,4,7,6,5] => 0
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [1,2,3,6,4,7,5] => 0
[5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => [1,2,4,5,7,6,3] => 0
[5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [1,2,3,7,5,6,4] => 0
[4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [1,2,7,3,4,6,5] => 0
[5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => [1,3,4,5,7,6,2] => 0
[5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [1,2,4,7,5,6,3] => 0
[5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => [1,3,2,5,4,7,6] => 0
[5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [1,2,7,3,5,6,4] => 0
[5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [1,3,4,5,2,7,6] => 0
[5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => [1,2,4,3,5,7,6] => 0
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 0
[] => [] => [1] => [1] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3$

$F_{4} = 5$

Description

The number of inversions between exceedances where the greater exceedance is linked.
This is for a permutation

$\sigma$ of length

$n$ given by

$\operatorname{ile}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(j) < \sigma(i) \wedge \sigma^{-1}(j) < j \}.$

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.

Map

cactus evacuation

Description

The cactus evacuation of a permutation.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.