In the following I present a conjecture on Nakayama algebras that I have for nearly 2 years now. Since I was not able to solve it and it can be stated purely combinatorically, I thought it might be worth a try to post it on mathoverflow. A proof of conjecture 1 would have very nice consequences and applications that I present at the end. I will first present everything in elementary terms so that no knowledge of representation theory or homological algebra is needed.

**Combinatorial definitions**

An **$n$-CNakayama algebra** is a list of $n$ natural numbers $[c_0,...,c_{n-1}]$ (where we read the indices modulo $n$ so that $c_i$ is defined for all $i \in \mathbb{Z}$) satisfying $c_i \geq 2$, $c_{n-1}-1=c_0=\min \{ c_0,c_1,...,c_{n-1} \}$ and $c_i -1 \leq c_{i+1}$ for all $i$.

See also https://arxiv.org/pdf/1811.05846.pdf where CNakayama algebras are identified with periodic Dyck paths (two $n$-CNakayama algebras are identified in case their lists are cyclic shifts of eachother).

The **defect** $\operatorname{Def(A)}$ of an n-Cnakayama algebra A is by definition the cardinality of the set $\{ i \in \{0,...,n-1 \} | c_{i-1} > c_i \}$.
A **module** is a tuple $(i,k)$ with $i \in \mathbb{Z}$ and $k \in \{1,...,c_i \}$ and we identify $(i_1,k_1)$ with $(i_2,k_2)$ in case $i_1 \equiv i_2 \ mod \ n$ and $k_1=k_2$.
Formally, we also introduce modules $(i,k)$ with $k \leq 0$ that we call zero-modules and they are all identified. The index $i$ is called the **top** of a module.

A module $(i,k)$ is called **projective** in case $k=c_i$ and it is called **projective-injective** in case it is projective and additionally $c_{i-1} \leq c_i$.

Note that the defect simply counts how many projective modules there are that are not injective.

The **first syzygy** of a module $M=(i,k)$ is defined as $\Omega^1(M):=(i+k,c_i-k)$. The $i$-th syzygy module $\Omega^i(M)$ of $M$ is then defined inductively as $\Omega^i(M):=\Omega^{1}(\Omega^{i-1}(M))$. Set $\Omega^0(M):=M$.
The **i-th projective cover** of a module $M$ is defined as the module $(l,c_l)$ when $l$ is the top of the module $\Omega^l(M)$.
The **dominant dimension** $\operatorname{domdim(M)}$ of a module $M$ is defined as the smallest integer $i$ such that the $i$-th projective cover of $M$ is not projective-injective. By convention, the dominant dimension of a projective-injective modules and zero modules is infinite.

A module $(i,k)$ is called **special** in case $c_{i-1} < c_i$ and $k \in \{c_{i-1},c_{i-1} +1 , ..., c_i -1 \}$.

The dominant dimension $\operatorname{domdim(A)}$ of an $n$-CNakayama algeba A is defined as the minimum of all dominant dimensions of the special modules.

Conjecture 1: For any $n$-CNakayama algebra $A$, we have the inequality $$\operatorname{Def(A)} \cdot \operatorname{domdim(A)} \leq 2n-2.$$

Conjecture 2: In case an $n$-Cnakayama algebra $A$ has dominant dimension larger than or equal to $n$, we have $\operatorname{Def(A)}=1$.

Note that conjecture 1 implies conjecture 2 since the inequality gives $\operatorname{domdim(A)} \leq \frac{2n-2}{\operatorname{Def(A)}} \leq n-1$ in case $\operatorname{Def(A)} \geq 2$. Also note that an $n$-CNakayama algebra has $\operatorname{Def(A)}=1$ if and only if it is of the form $[a,a,....a,a+1,...,a+1]$.

$\mathbf{Examples}$

a) We look at the 6-CNakayama algebra [5,5,5,7,7,6]. It has defect equal to 2. The special modules are $(3,6)$ with dominant dimension 4 and $(3,5)$ with dominant dimension 4. Thus $A$ has dominant dimension 4 and the inequality $2 *4 \leq 10$ is true.

b) The $n$-CNakayama algebra $[n,n+1,n+1,....,n+1]$ has defect 1 and dominant dimension equal to $2n-2$, which shows that the inequality in conjecture 1 is optimal in case it is true.

$\mathbf{Remarks}$

A proof of conjecture 1 would have some nice applications. I name two here. First, it would substantially improve the main result on the dominant dimension of Nakayama algebras in https://www.sciencedirect.com/science/article/pii/S0021869317305999 .

A proof of conjecture 2 would be a needed key result to connect and complete two classification result that Dag Madsen and I work on.

The evidence for conjecture 1 (and thus also conjecture 2) is very good. Conjecture 1 is proven for $n$-CNakayama algebras with $n \leq 12$ and thus for over 2 million examples (here I mean 2 million different difference classes, see below).

$\mathbf{More \ examples}$

Say that two $n$-CNakayama algebra $[c_0,c_1,...,c_{n-1}]$ and $[e_0,e_1,...,e_{n-1}]$ are in the same difference class in case $c_i \equiv e_i $ mod $n$ for all $i$. Note that dominant dimension and defect are the same for two such lists in the same difference class. Thus the inequality has to be checked only for finitely many $n$-CNakyama algebras for a given $n$. Here are all (difference classes of) $n$-CNakayama algebras together with their dominant dimension for $n=4$:

```
[ [ 2, 2, 2, 3 ], 4 ],
[ [ 2, 2, 3, 3 ], 1 ],
[ [ 2, 2, 4, 3 ], 1 ],
[ [ 2, 3, 2, 3 ], 2 ],
[ [ 2, 3, 3, 3 ], 2 ],
[ [ 2, 3, 4, 3 ], 1 ],
[ [ 2, 4, 3, 3 ], 1 ],
[ [ 2, 4, 4, 3 ], 1 ],
[ [ 2, 5, 4, 3 ], 1 ],
[ [ 3, 3, 3, 4 ], 5 ],
[ [ 3, 3, 4, 4 ], 3 ],
[ [ 3, 3, 5, 4 ], 2 ],
[ [ 3, 4, 3, 4 ], 1 ],
[ [ 3, 4, 4, 4 ], 1 ],
[ [ 3, 4, 5, 4 ], 1 ],
[ [ 3, 5, 4, 4 ], 1 ],
[ [ 3, 5, 5, 4 ], 1 ],
[ [ 3, 6, 5, 4 ], 1 ],
[ [ 4, 4, 4, 5 ], 2 ],
[ [ 4, 4, 5, 5 ], 4 ],
[ [ 4, 4, 6, 5 ], 1 ],
[ [ 4, 5, 4, 5 ], 2 ],
[ [ 4, 5, 5, 5 ], 6 ],
[ [ 4, 5, 6, 5 ], 1 ],
[ [ 4, 6, 5, 5 ], 1 ],
[ [ 4, 6, 6, 5 ], 2 ],
[ [ 4, 7, 6, 5 ], 1 ],
[ [ 5, 5, 5, 6 ], 1 ],
[ [ 5, 5, 6, 6 ], 2 ],
[ [ 5, 5, 7, 6 ], 1 ],
[ [ 5, 6, 5, 6 ], 1 ],
[ [ 5, 6, 6, 6 ], 3 ],
[ [ 5, 6, 7, 6 ], 1 ],
[ [ 5, 7, 6, 6 ], 1 ],
[ [ 5, 7, 7, 6 ], 1 ],
[ [ 5, 8, 7, 6 ], 1 ].
```

$\mathbf{Extension \ of \ conjecture \ 1}$

The following extension is suggested by a comment of Gjergji Zaimi.
Define the **super dominant dimenson** $\operatorname{sdomdim(A)}$ of an $n$-CNakayama algebra as the sum of all dominant dimension of the special modules.
A positive answer of the following question would be an extension of conjecture 1.

Question (by Gjergji Zaimi): Do we have $\operatorname{sdomdim(A)} \leq 2n-2$ for any $n$-CNakayama algebra?

The question has a positive answer for $n \leq 12$ and I try to look at it for higher $n$ with the computer.

Here are the values of the super dominant dimension for difference classes of $n$-CNakayama algebras $n=4$:

```
[ [ 2, 2, 2, 3 ], 4 ]
[ [ 2, 2, 3, 3 ], 1 ]
[ [ 2, 2, 4, 3 ], 4 ]
[ [ 2, 3, 2, 3 ], 4 ]
[ [ 2, 3, 3, 3 ], 2 ]
[ [ 2, 3, 4, 3 ], 4 ]
[ [ 2, 4, 3, 3 ], 4 ]
[ [ 2, 4, 4, 3 ], 2 ]
[ [ 2, 5, 4, 3 ], 4 ]
[ [ 3, 3, 3, 4 ], 5 ]
[ [ 3, 3, 4, 4 ], 3 ]
[ [ 3, 3, 5, 4 ], 4 ]
[ [ 3, 4, 3, 4 ], 2 ]
[ [ 3, 4, 4, 4 ], 1 ]
[ [ 3, 4, 5, 4 ], 3 ]
[ [ 3, 5, 4, 4 ], 3 ]
[ [ 3, 5, 5, 4 ], 5 ]
[ [ 3, 6, 5, 4 ], 4 ]
[ [ 4, 4, 4, 5 ], 2 ]
[ [ 4, 4, 5, 5 ], 4 ]
[ [ 4, 4, 6, 5 ], 3 ]
[ [ 4, 5, 4, 5 ], 4 ]
[ [ 4, 5, 5, 5 ], 6 ]
[ [ 4, 5, 6, 5 ], 5 ]
[ [ 4, 6, 5, 5 ], 5 ]
[ [ 4, 6, 6, 5 ], 4 ]
[ [ 4, 7, 6, 5 ], 4 ]
[ [ 5, 5, 5, 6 ], 1 ]
[ [ 5, 5, 6, 6 ], 2 ]
[ [ 5, 5, 7, 6 ], 2 ]
[ [ 5, 6, 5, 6 ], 2 ]
[ [ 5, 6, 6, 6 ], 3 ]
[ [ 5, 6, 7, 6 ], 3 ]
[ [ 5, 7, 6, 6 ], 3 ]
[ [ 5, 7, 7, 6 ], 3 ]
[ [ 5, 8, 7, 6 ], 3 ]
```