- FindStat

Identifier

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000684: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,5] => [1,1] => [2] => [1,1,0,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5,1] => [1,1] => [2] => [1,1,0,0] => 1

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

[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 7

[1,1,1,1,1,2] => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,5] => [2,1] => [1,2] => [1,0,1,1,0,0] => 2

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

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

[1,2,1,2,1] => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0] => 1

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

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

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

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

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

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

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

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

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

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

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

[1,5,1] => [1,1,1] => [3] => [1,1,1,0,0,0] => 1

[1,6] => [1,1] => [2] => [1,1,0,0] => 1

[2,1,1,1,1,1] => [1,5] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 5

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

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

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

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

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

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$F_{1} = q$

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

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

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

$F_{5} = 7\ q + 6\ q^{2} + 2\ q^{3} + q^{5}$

$F_{6} = 14\ q + 10\ q^{2} + 5\ q^{3} + 2\ q^{4} + q^{6}$

$F_{7} = 23\ q + 23\ q^{2} + 11\ q^{3} + 4\ q^{4} + 2\ q^{5} + q^{7}$

$F_{8} = 39\ q + 50\ q^{2} + 22\ q^{3} + 10\ q^{4} + 4\ q^{5} + 2\ q^{6} + q^{8}$

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

bounce path

Description

The bounce path determined by an integer composition.

Map

delta morphism

Description

Apply the delta morphism to an integer composition.
The delta morphism of a composition

$C$ is the compositions composed of the lengths of consecutive runs of the same integer in

$C$ .

Map

complement

Description

The complement of a composition.
The complement of a composition

$I$ is defined as follows:
If

$I$ is the empty composition, then the complement is also the empty composition. Otherwise, let

$S$ be the descent set corresponding to

$I=(i_1,\dots,i_k)$ , that is, the subset

$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$
of

$\{ 1, 2, \ldots, |I|-1 \}$ . Then, the complement of

$I$ is the composition of the same size as

$I$ , whose descent set is

$\{ 1, 2, \ldots, |I|-1 \} \setminus S$ .
The complement of a composition

$I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to

$I$ .