Here a combinatorics problem. I offer 30 euro for a proof and 100 bounty points for a counterexample:
Let $n \geq 2$.


An $n$-*Kupisch series* is a list of $n$ numbers $c:=[c\_1,c\_2,...,c\_n]$ with $c\_n=1$, $c\_i \ge 2$ for $i \neq n$ and $c\_i-1 \leq c\_{i+1}$ for all $i=1,...,n-1$ and setting $c\_0:=c\_n$.
The number of such $n$-Kupisch series is equal to $C\_{n-1}$ (Catalan numbers).
The *CoKupisch series* $d$ of $c$ is defined as $d=[d\_1,d\_2,...,d\_n]$ with $d\_i:= \min \{k | k \geq c\_{i-k} \} $ and $d\_1=1$. One can show that the $d\_i$ are a permutation of the $c\_i$. A number $a \in \{1,...,n \}$ is a *descent* if $a=1$ or $c\_a >c\_{a-1}$. Define a corresponding set, indexed by descents: $X\_1 := \{1,2,...,c\_1-1 \}$, and $X\_a := \{ c\_{a-1}, c\_{a-1}+1 ,..., c\_a -1 \}$ for descents $a > 1$.


A $n$-Kupisch series is called $2-$Gorenstein if it satisfies the following condition:


for each descent $a$, and each $b \in X\_a$:
 either $c\_{a+b} \geq c\_{a+b-1}$ or $d\_{a+b-1} = d\_{a+b + c\_{a+b}-1} - c\_{a+b}$ is satisfied. 


Conjecture: The number of $n$-Kupisch series that are $2$-Gorenstein for $n \geq 2$ equals 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, ... which are the Motzin numbers. <https://oeis.org/A001006>


My computer says it is true for $n=2,3,...,14$. (also for n=1 if [1] is the unique 1-Kupisch series, by a representation-theoretic interpretation)


Here an example which visualises the c and d sequences in a picture of a Dyck path:
<https://drive.google.com/file/d/0B9hKtvQe-4-bQlpjcURfYnNzUGs/view>


Background:
This has a representation theoretic background (see <http://www.sciencedirect.com/science/article/pii/0022404994900442> )and would give a certain classification result together with a new categorification of the Motzkin numbers. There is a very natural bijection of n-Kupisch series to Dyck paths from (0,0) to (2n-2,0) and probably the 2-Gorenstein algebras among them might give a new combinatorial interpretation of Motzkin paths as subpaths of Dyck paths.


I found in fact many such things and translated them into elementary problems, but I have no talent in complicated combinatorics :(


example for n=5:


All 5-Kupisch series (number is 14):
[ [ 2, 2, 2, 2, 1 ], [ 3, 2, 2, 2, 1 ], [ 2, 3, 2, 2, 1 ], [ 3, 3, 2, 2, 1 ], 
 [ 4, 3, 2, 2, 1 ], [ 2, 2, 3, 2, 1 ], [ 3, 2, 3, 2, 1 ], [ 2, 3, 3, 2, 1 ], 
 [ 3, 3, 3, 2, 1 ], [ 4, 3, 3, 2, 1 ], [ 2, 4, 3, 2, 1 ], [ 3, 4, 3, 2, 1 ], 
[ 4, 4, 3, 2, 1 ], [ 5, 4, 3, 2, 1 ] ]


All 2-Gorenstein 5-Kupisch series (number is 9):
[ [ 2, 2, 2, 2, 1 ], [ 3, 2, 2, 2, 1 ], [ 2, 3, 2, 2, 1 ], [ 4, 3, 2, 2, 1 ], 
 [ 2, 2, 3, 2, 1 ], [ 3, 2, 3, 2, 1 ], [ 3, 3, 3, 2, 1 ], [ 2, 4, 3, 2, 1 ], 
 [ 5, 4, 3, 2, 1 ] ]


Representation theoretic conjecture/background:


Conjecture:


The number of 2-Gorenstein algebras that are Nakayama algebras with n simple modules with a linear oriented line as a quiver is equal to the sequence of Motzkin numbers.


Background/explanations:


Here the $c\_i$ are the dimension of the indecomposable projective modules which determine the algebra uniquely (assuming that the algebras are connected quiver algebras over a field). The $d\_i$ are the dimension of the indecomposable injective modules at point $i$. 2-Gorenstein means that the dual of the regular module $D(A)$ has a projective presentation $P\_1 \rightarrow P\_0 \rightarrow D(A) \rightarrow 0$ with $P\_1$ having injective dimension bound by 1. ($P\_0$ is always projective-injective for Nakayama algebras)


Here a link for the program I used to test things (copy all programs, and the finaltest program does the job then. 1 means the Kupisch series is 2-Gorenstein and 0 means it is not):
<https://docs.google.com/document/d/1U9mriuvCEE9FeXY1TfY_yJ1mi05TCdHRfppwCSwi1S8/pub>


edit: Since we have 2 people with proofs now, I award 30 Euro each to them in case their proof is correct (I still have to check) and then no more money for new proofs.


I award additionally 200 bounty points to the nicest proof (which might be one of the two first posted proofs or another not yet posted proof), which can be decided by the community in terms of upvotes near the end when the bounty expires (16.08.)


edit 2: Now at 16.08. I give the bounty points to Anton, since it is the most complete answer and the details of findstats answer are not posted yet. I will give 30 Euro to each after checking findstats proof (when it is posted and correct).


edit 3: Since findstat updated the answer and now has a full proof of an extension of the conjecture just a little time after I awarded the first bounty, I also will give a bounty to them and accept their answer (second bounty had to be 400 or higher and can be awarded after 24 hours).



Combinatorics problem related to Motzkin numbers with prize money I
UPDATE
------


The promised writeup, proving both conjectures, is now finished. It will be available on the arxiv on Friday, in the meantime it is available on the [arXiv](https://arxiv.org/abs/1811.05846).


FindStat's original answer
--------------------------


I have the following refined conjecture, which hopefully can be transformed into a proof without too much trouble:


let's call a pair $(a, b)$ where $a$ is a descent and $b\in X\_a$ which does not satisfy the condition



> 
> $c\_{a+b} \geq c\_{a+b-1}$ or $d\_{a+b-1} = d\_{a+b + c\_{a+b}-1} - c\_{a+b}$
> 
> 
> 


a $2$-Gorenstein failure.


Then it appears that the number of $2$-Gorenstein failures is the composition of <http://findstat.org/Mp00129> (The Billey-Jockusch-Stanley bijection to 321-avoiding permutations) and <http://findstat.org/St000732> (The number of double deficiencies of a permutation).


In the above mentioned statistic there is a reference to a paper by Sergi Elizalde, "Continued fractions for permutation statistics", which contains a bijection between $321$-avoiding permutations without double excedances and Motzkin paths.


Thus it remains to show that $2$-Gorenstein failures indeed correspond to double deficiencies.


Outline of proof
----------------


Throughout the proof, a Dyck path starts at the origin, has north and east steps and never goes below the diagonal $y=x$. We think of the Dyck path as running in an array of columns labelled $1$ to $n$ from left to right, and rows labelled $1$ to $n$ from bottom to top.


Let us first recall the bijection due to Billey, Jockusch and Stanley, sending a Dyck path $D$ to a $321$-avoiding permutation $\pi$. To obtain the (horizontally flipped) permutation matrix of $\pi$ corresponding to $D$, put a cross into each valley (an east step followed by a north step) and fill the remaining slots with an increasing sequence of crosses.


Our claim is an immediate consequence of the following three statements.


**Lemma 1.** $c\_{x+1} = c\_x - 1$ if and only if $x$ is a weak deficiency of $\pi$, that is $\pi(x)\leq x$. Equivalently, there is a double east step spanning columns $x$ and $x+1$.


**Lemma 2.** $c\_{x+1} = c\_x - 1$ and $c\_{x+1} + d\_x = d\_{x+c\_{x+1}}$ if and only if $x$ is a fixed point of $\pi$.


**Lemma 3.** Let $x$ be a strict deficiency of $\pi$. Then $\pi^{-1}(x)$ is also a strict deficiency if and only if $x+1=a+b$ where $a$ is a descent and $b\in X\_a$.


We intend to provide a detailed and illustrated version of this argument within the next few days.


Sage code used to discover the conjecture
-----------------------------------------



```
def kupisch(D):
    """
    sage: [kupisch(D) for D in DyckWords(3)]
    [[2, 2, 2, 1], [2, 3, 2, 1], [3, 2, 2, 1], [3, 3, 2, 1], [4, 3, 2, 1]]
    """
    H = D.heights()
    return [1+H[i] for i, s in enumerate(D) if s == 0]+[1]

def cokupisch(D):
    """
    sage: [cokupisch(D) for D in DyckWords(3)]
    [[1, 2, 2, 2], [1, 2, 2, 3], [1, 2, 3, 2], [1, 2, 3, 3], [1, 2, 3, 4]]
    """
    H = D.heights()
    return [1]+[1+H[i+1] for i, s in enumerate(D) if s == 1]

def descents(D):
    """
    sage: [(D, kupisch(D), descents(D)) for D in DyckWords(3)]
    [([1, 0, 1, 0, 1, 0], [2, 2, 2, 1], [1]),
     ([1, 0, 1, 1, 0, 0], [2, 3, 2, 1], [1, 2]),
     ([1, 1, 0, 0, 1, 0], [3, 2, 2, 1], [1]),
     ([1, 1, 0, 1, 0, 0], [3, 3, 2, 1], [1]),
     ([1, 1, 1, 0, 0, 0], [4, 3, 2, 1], [1])]
    """
    c = kupisch(D)
    return [1] + [a+1 for a in range(1, len(c)) if c[a] > c[a-1]]

def gorenstein_failures(D):
    D = DyckWord(D)
    c = kupisch(D)
    d = cokupisch(D)
    D = descents(D)
    f = 0
    for a in D:
        if a == 1:
            X = range(1, c[0])
        else:
            X = range(c[a-1-1], c[a-1])
        for b in X:
            if c[a+b-1] >= c[a+b-1-1] or d[a+b-1-1] == d[a+b+c[a+b-1]-1-1] - c[a+b-1]:
                continue
            else:
                f += 1
    return f

findstat("Dyck Paths", gorenstein_failures)
0: (St000732: The number of double deficiencies of a permutation., [Mp00129: to 321-avoiding permutation (Billey-Jokusch-Stanley)], 200)
1: (St000366: The number of double descents of a permutation., [Mp00129: to 321-avoiding permutation (Billey-Jokusch-Stanley), Mp00087: inverse first fundamental transformation], 200)
2: (St000731: The number of double exceedences of a permutation., [Mp00129: to 321-avoiding permutation (Billey-Jokusch-Stanley), Mp00066: inverse], 200)

```

