A $n$-LNakayama algebra (for $n \geq 2$) is a list $[c_0,c_1,...,c_{n-1}]$ with $c_{n-1}=1$, $c_i \geq 2$ for $i \neq n-1$ and $c_i -1 \leq c_{i+1}$ for all $i$. They are in bijection with Dyck paths, where one associates to such a list simply the Dyck path with area sequence $[c_0,c_1,...,c_{n-1}]$ , see page 6 of https://arxiv.org/pdf/1811.05846.pdf. Thus $n$-LNakayama algebras are enumerated by the Catalan numbers $C_{n-1}$. Call an $n$-LNakayama algebra strong in case for all $i \neq n-1$ we have that $c_{i+1}=c_i -1$ or ($c_{i+1}>c_i -1$ and $c_{i+c_i}=c_{i+1}-c_i +1$) (Those are exactly the $n$-LNakayama algebras with global dimension at most 2). Strong $n$-LNakayama algebras are enumerated by $2^{n-2}$ and thus in bijection with integer compositions. (although Im not sure what the best bijection might be)

The coarea sequence $[d_0,d_1,...,d_{n-1}]$ is the area sequence of the "opposite" Dyck path. Formally : $d_i = min \{ k | k \geq c_{i-k} \}$.

Now define a statistic $f(A)$ on the strong $n$-LNakayama algebras $A$: $f(A):= \{ i \in \{0,...,n-1 \} | c_i < d_{i+c_i -1}$ and $d_{i+c_i-1}-c_i=d_{i-1} \}$ (those are the number of indecomposable projective modules with injective dimension one. One can show that this is also the number of indecomposable injective modules with projective dimension one, since Nakayama algebras are QF-3 algebras).

I can prove that the strong $n$-LNakayama algebras with $f(A)=0$ are enumerated by the Fibonacci numbers (note that Fibonacci numbers are enumerated by integer compositions without parts equal to one, see https://oeis.org/A000045). Experimenting with the computer and findstat suggests that much more is true.

Guess 1: Via the Billey-Jockusch-Stanley bijection $g$ from Dyck paths to 321-avoiding permutations we have that $f(A)= p(g(A))$, where for a permutation $\pi$ we have that $p(\pi)$ is the number of fixed points of $\pi$.

See http://www.findstat.org/StatisticsDatabase/St001008 .

Guess 2: There is a bijection $h$ from strong $n$-LNakayama algebras to integer compositions such that $f(A)= t(h(A))$, where $t(U)$ for an integer composition $U$ counts the number of parts equal to one.

I arrived at guess 2 since the strong $n$-LNakayama algebras with $f(A)=1$ seem to be counted by https://oeis.org/A105423 (integer compositions with exactly one part equal to 1) and those with $f(A)=2$ are counted by https://oeis.org/A105423 (integer compositions with exactly two parts equal to one) (and of course the $f(A)=0$ case counted by the Fibonacci numbers as mentioned before).

For guess 2 there is a cyclic analogue for CNakayama algebras (Nakayama algebras with a cyclic quiver). But for simplicity I do not state this for now (since they do not have such a nice combinatorial model as Dyck paths). For the cyclic case the algebras with $f(A)=0$ where counted by the cyclic analogue of the Fibonacci numbers : https://oeis.org/A032190. And those with $f(A)=1$ were counted surprisingly again by the usual Fibonacci numbers.

Guess 1 is better tested I guess since it comes from findstat, while guess 2 is based just on small values of $f(A)$ (namely $f(A) \leq 2$) and oeis. Here is the statistic f for the 4-LNakayama algebras:

[ [ [ 2, 2, 2, 1 ], 0 ]

[ [ 3, 2, 2, 1 ], 1 ]

[ [ 2, 3, 2, 1 ], 1 ]

[ [ 3, 3, 2, 1 ], 0 ]

[ [ 4, 3, 2, 1 ], 3 ] ]

Here is the statistic f for the strong 5-LNakayama algebras:

[ [ 2, 3, 2, 2, 1 ], 0 ]

[ [ 4, 3, 2, 2, 1 ], 2 ]

[ [ 3, 2, 3, 2, 1 ], 2 ]

[ [ 4, 3, 3, 2, 1 ], 1 ]

[ [ 2, 4, 3, 2, 1 ], 2 ]

[ [ 3, 4, 3, 2, 1 ], 1 ]

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

[ [ 5, 4, 3, 2, 1 ], 4 ]

edit: Here is the statistic on strong $n$-LNakayama algebras for $n \leq 6$ where they are displayed as permutations (via the Billey-Jockusch-Stanley bijection). Has anyone seens those $2^{n-2}$ or can recognize them?

n=2: [1] => 1

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

[1, 2] => 2

n=4: [1, 3, 2] => 1

[2, 1, 3] => 1

[3, 1, 2] => 0

[1, 2, 3] => 3

n=5: [2, 1, 4, 3] => 0

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

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

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

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

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

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

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

n=6: [2, 1, 4, 3, 5] => 1

[2, 1, 3, 5, 4] => 1

[2, 1, 5, 3, 4] => 0

[2, 1, 3, 4, 5] => 3

[1, 3, 2, 5, 4] => 1

[1, 3, 2, 4, 5] => 3

[3, 1, 2, 5, 4] => 0

[3, 1, 2, 4, 5] => 2

[1, 2, 4, 3, 5] => 3

[1, 4, 2, 3, 5] => 2

[4, 1, 2, 3, 5] => 1

[1, 2, 3, 5, 4] => 3

[1, 2, 5, 3, 4] => 2

[1, 5, 2, 3, 4] => 1

[5, 1, 2, 3, 4] => 0

[1, 2, 3, 4, 5] => 5