- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤ

Values

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

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$F_{1} = q^{2}$

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

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

$F_{4} = 4\ q^{2} + q^{3}$

Description

The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices

$\{0,1,2,\ldots,n\}$ and root

$0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also St000325The width of the tree associated to a permutation. for the width of this tree.

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

Map

Ringel

Description

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

Map

to Dyck path

Description

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