Counting QF-1 algebras inside quotient algebras of upper triangular matrices via Dyck paths

Question

Sorry, I modified the question. I first used the weaker version of QF-1. Now the problem is about weakly QF-1 and QF-1. QF-1 rings are generalisations of Quasi-Frobenius rings. See for example https://en.wikipedia.org/wiki/Quasi-Frobenius_ring#Thrall.27s_QF-1.2C2.2C3_generalizations. The problem is as follows: Count the number of (weakly) QF-1 rings inside the class of (interesting) quotient algebras of the algebra of upper triangular matrices (we just look at admissible quotients here and those algebras are also called Nakayama algebras with a line as a quiver).

Algebraic background:

Let A always be an algebra of dominant dimension at least one (which our algebras always have). Look at the following two properties:

1.A has dominant dimension at least two (which is equivalent to the condition that A has a double centralizer property with a minimal faithful projective-injective module)

2.Every module has dominant dimension or codominant dimension at least one.

Using a theorem of Morita for algebras with dominant dimension at least one (see the article "Frobenius algebras" by Yamagata corollary 3.5.1.), an algebra is QF-1 iff it has property 1. and 2. Call it weakly QF-1 if it has only property 2.

First some definitions: For a Dyck path look also at its interior: Namely the place between the Dyck path and the x-axis. Look at all the points, where a subDyck path of the original Dyck path passes trough. The Dyck path corresponding to UUDUUDDUDD has as its points: (0,0),(1,1),(2,0),(2,2),(3,1),(4,0),(4,2),(5,1),(5,3),(6,0),(6,2),(7,1),(8,0),(8,2),(9,1),(10,0). Call a point inside a Dyck path good in case the diagonal to the right connecting the point and the boundary of the Dyck path ends to the right with a peak of the Dyck path OR the diagonal to the left connecting the point and the boundary of the Dyck path ends to the left with a peak of the Dyck path. Call a Dyck path nice in case all its points are good. The above example is not nice, since the point (5,1) is not good. An example of a nice Dyck path is UDUDUDUD. Now the definition of dominant dimension, which is a little confusing (sorry). Now define a projective point as a point on the top boundary connecting a point of height 0 to the left to the boundary (thus there are as many projective points as there are points of height 0 (which correspond to the simple modules of the algebra)). Define the dimension of a point as the height+1 of the point. Define $(a,b)=\Omega^{-1}(x,y)$ of a projective point (x,y) as the unique third point having dimension dim((c,d))-dim((x,y)) on the triangle with points (x,y),(c,d),(a,b), where (c,d) is the unique first peak right of the projective point. Say the algebra has dominant dimension at least two iff for every projective point (x,y) , the point \Omega^{-1}(x,y) is on the top boundary and thus lying on a diagonal to the right to a peak. Example: UUDUDUDD should have dominant dimension at least two, while UUDD has not.

If I made no mistake, the counting of weakly QF-1 algebras should reduce to the following: Count the number of nice $2n-2$ -Dyck paths. For n=2,3,... the numbers start as follows: 1, 1, 2, 4, 8, 17, 37, 81, ... .

If I made no mistake, the counting of QF-1 algebras should reduce to the following: Count the number of nice $2n-2$ -Dyck paths with dominant dimension at least two. The numbers start as 1,1,2,4,8,16,33,69,145,... (this looks like the sequence Martin Rubey obtained in the comments for the irreducible nice Dyck paths). The examples with n=5 are UDUDUDUD and UUDUDUDD. So questions: Is there a nice description of nice Dyck paths and what is the formula to count them?

edit: Extended questions suggested by the work of Martin Rubey: Find the generating function $F(q)=\sum\limits_{k=0}^{r}{a_k q^k}$ , when $a_k$ denotes the number of bad (=non-good points) in a Dyck path. Homological interpretation is: Bad points are exactly the indecomposable modules in the algebra with dominant and codominant dimension equal to zero. (See the book by Auslander-Reiten-Smalo or https://www.math.uni-bielefeld.de/~ringel/lectures/tachi/tachikawa/dominant.htm for motivation/definition of dominant dimension).

Answer 1

I think the following should work, but I'm off to bed now.

Let $G_k$ be the set of Dyck paths not containing a sub-path $U(UD)^\ell D$ where the first $U$ step begins at height $h\geq\ell-k$. $G_1$ should be the set of good paths. A path in $G_{k+1}$ is good even when prepending $k$ up and appending $k$ down steps, except for the paths $(UD)^\ell$ for $1\leq\ell\leq k$.

To get a recurrence, let $P_k$ be the set of paths in $G_k$ which do not return to the axis except at the end.

Then $P_k=\{U\, p\, D: p \in G_{k+1}\setminus \{UD,\dots,(UD)^k\}\}$ and $G_k$ is a (possibly empty) list of paths in $P_k$.

It remains to find out whether it is possible to extract $P_1$ from this.