A sequence $[c_0,c_1,...,c_{n-1}]$ with $n \geq 2$ is called a Dyck path in case $c_{n-1}=1$, $c_i \geq 2$ for $i \neq n-1$ and $c_i-1 \leq c_{i+1}$ for each $i$. For example the Dyck paths for $n=4$ are the 5 sequences [ 2, 2, 2, 1 ], [ 3, 2, 2, 1 ], [ 2, 3, 2, 1 ], [ 3, 3, 2, 1 ], [ 4, 3, 2, 1 ]. In general they are enumerated by the Catalan sequence. Note that those sequences are just the area sequence in the classical definition of Dyck paths, see https://arxiv.org/pdf/1811.05846.pdf page 6. One also has the coarea sequence associated to a Dyck path $[d_0,d_1,...,d_{n-1}]$ which can be formally defined via $d_0=1, d_1=2$ and $d_i = \min \{k \geq 2 | k \geq c_{i-k} \}$ for $i \geq 1$.

A module of a Dyck path $[c_0,c_1,...,c_{n-1}]$ is a tuple $(i,m)$ with $0 \leq i \leq n-1$ and $1 \leq m \leq c_i$.

We call a Dyck path $[c_0,c_1,...,c_{n-1}]$ shod in case the following is satisfied: For each tuple $(i,m)$ with $1 \leq m \leq c_i -1$ and $m < c_{i-1}$ we have (($c_i-m=c_{i+m}$) or ($d_{i-1}=d_{i+m-1}-m$)).

This condition looks complicated but has a nice pictorial interpretation in a Dyck path.

Question 1: Is there a nice combinatorial description of shod Dyck paths?

Question 2: Is it true that the number of shod Dyck paths is equal to $\frac{(n-2)^3+2(n-2)}{3}+1$, which is also equal to the number of permutations of length n which avoid the patterns 321, 2143, 3124 , see https://oeis.org/A116731 . This is true for $n \leq 9$. So the sequence starts with 1,2,5,12,25,46,77,120....

Question 2' (equivalent to Question 2): Call a shod Dyck path special in case it additionally satisfies that there exists an $i \neq n-1$ such that $c_{i+1}>c_i-1$ and $c_{i+c_i}>c_{i+1}-c_i+1$. Then the number of special shod Dyck paths $[c_0,c_1,...,c_{n+2}]$ seems to be given by the Square pyramidal numbers $\sum\limits_{k=0}^{n}{k^2}$, see https://oeis.org/A000330. It counts for example the Number of permutations avoiding 13-2 that contain the pattern 32-1 exactly once.Is this true?

Background: A finite dimensional algebra is called shod, in case each indecomposable module has projective dimension at most one or injective dimension at most one. Recently in https://www.sciencedirect.com/science/article/pii/S0001870815300219 such algebras are characterised as endomorphism algebras of 2-term silting complexes in derived categories of Ext-finite hereditary abelian categories. Dyck paths correspond to Nakayama algebras with a linear quiver (which are also exactly the admissible quotient algebras of the algebra of upper triangular matrices) and the elementary question asks for a combinatorial classification of Nakayama algebras that are shod. Note also that Dyck paths are in bijection with permutations of length n which avoid the pattern 321 and maybe viewing Dyck paths as such pattern avoiding permutations might be better for this question. But there are various bijections from Dyck paths to 321 avoiding permutations and it is not clear what is the "best" one for such questions.

edit: Here is another approach to question 2 leading to a possibly much nicer enumeration result, namely the sum of all squares: Call a shod Dyck path special in case it additionally satisfies that there exists an $i \neq n-1$ such that $c_{i+1}>c_i-1$ and $c_{i+c_i}>c_{i+1}-c_i+1$. Then the number of special shod Dyck paths $[c_0,c_1,...,c_{n-1}]$ seems to be given by the Square pyramidal numbers $\sum\limits_{k=0}^{n-3}{k^2}$: https://oeis.org/A000330 , which is a much more well known sequence. They enumerate for example according to oeis "Number of rhombi in an n X n rhombus" and "Number of permutations avoiding 13-2 that contain the pattern 32-1 exactly once". I can classifly the shod non-special Dyck paths, so it would be enough to solve question 1 and 2 for special shod Dyck paths.

For $n=4$ the unique special shod Dyck path is [2,2,2,1], for $n=5$ we have [ [ 3, 2, 2, 2, 1 ], [ 3, 3, 2, 2, 1 ], [ 2, 2, 3, 2, 1 ], [ 2, 3, 3, 2, 1 ], [ 3, 3, 3, 2, 1 ] ] and for $n=6$ we got [ [ 4, 3, 2, 2, 2, 1 ], [ 4, 3, 3, 2, 2, 1 ], [ 4, 4, 3, 2, 2, 1 ], [ 3, 2, 2, 3, 2, 1 ], [ 3, 3, 2, 3, 2, 1 ], [ 3, 2, 3, 3, 2, 1 ], [ 4, 3, 3, 3, 2, 1 ], [ 4, 4, 3, 3, 2, 1 ], [ 2, 2, 4, 3, 2, 1 ], [ 2, 3, 4, 3, 2, 1 ], [ 3, 3, 4, 3, 2, 1 ], [ 2, 4, 4, 3, 2, 1 ], [ 3, 4, 4, 3, 2, 1 ], [ 4, 4, 4, 3, 2, 1 ] ].