- FindStat

Identifier

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
St000684: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6

[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5

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

[[],[],[],[[],[]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => 5

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

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

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

[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => 5

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

[[],[],[[],[],[]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => 5

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

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

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

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

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

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

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

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

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

[[],[[],[]],[],[]] => [1,0,1,1,0,1,0,0,1,0,1,0] => 5

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

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

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

[[],[[],[],[]],[]] => [1,0,1,1,0,1,0,1,0,0,1,0] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The global dimension of the LNakayama algebra associated to a Dyck path.
An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with $n$ points for $n \geq 2$. Number those points from the left to the right by $0,1,\ldots,n-1$.
The algebra is then uniquely determined by the dimension $c_i$ of the projective indecomposable modules at point $i$. Such algebras are then uniquely determined by lists of the form $[c_0,c_1,...,c_{n-1}]$ with the conditions: $c_{n-1}=1$ and $c_i -1 \leq c_{i+1}$ for all $i$. The number of such algebras is then the $n-1$-st Catalan number $C_{n-1}$.
One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0].
Conjecture: that there is an explicit bijection between $n$-LNakayama algebras with global dimension bounded by $m$ and Dyck paths with height at most $m$.
Examples:

For $m=2$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192.
For $m=3$, the number of Dyck paths with global dimension at most $m$ starts for $n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.

Map

to Dyck path

Description

Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.