- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
St001202: Dyck paths ⟶ ℤ

Values

[1] => [1,0] => 1

[1,1] => [1,1,0,0] => 1

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

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

[1,1,1] => [1,1,1,0,0,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1] => [1,1,1,1,0,0,0,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra.
Associate to this special CNakayama algebra a Dyck path as follows:
In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra to which we can associate a Dyck path as the top boundary of the Auslander-Reiten quiver of the LNakayama algebra.
The statistic gives half the dominant dimension of hte first indecomposable projective module in the special CNakayama algebra.

Map

to Dyck path

Description

Sends a parking function to its support Dyck word.