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Identifier

Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001583: Permutations ⟶ ℤ (values match St000246The number of non-inversions of a permutation.)

Values

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

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$F_{1} = q$

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

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

Description

The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.

Map

to non-crossing permutation

Description

Sends a Dyck path

$D$ with valley at positions

$\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation

$\pi$ having descents

$\{i_1,\ldots,i_k\}$ and whose inverse has descents

$\{j_1,\ldots,j_k\}$ .
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to

$n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

Map

to Dyck path

Description

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