- FindStat

Identifier

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤ (values match St000246The number of non-inversions of a permutation.)

Values

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

search for individual values

searching the database for the individual values of this statistic

Description

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

Map

Ringel

Description

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

Map

to partition

Description

The cut-out partition of a Dyck path.
The partition

$\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition

$(n-1,\ldots,2,1)$ when cutting out

$D$ considered as a path from

$(0,0)$ to

$(n,n)$ .
In other words,

$\lambda_{i}$ is the number of down-steps before the

$(n+1-i)$ -th up-step of

$D$ .
This map is a bijection between Dyck paths of size

$n$ and partitions inside the staircase partition

$(n-1,\ldots,2,1)$ .

Map

to Dyck path

Description

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