Categorification of monotone maps via tilting modules? It is well known that for the algebra of $n \times n$-upper triangular matrices over a field the number of tilting modules is equal to the Catalan number $C_n$. This is just the (hereditary) Nakayama algebra with Kupisch series $[n,n-1,...,2,1]$ and can be viewed as the "mother" of all Nakayama algebras with a linear quiver because it has each such algebra as a quotient.
The cyclic analogue of this algebra is the Nakayama algebra with Kupisch series $[n,2n-1,2n-2,....,n+2,n+1]$ for $n \geq 2$, which can be viewed as the "mother" of all Nakayama algebras with a cyclic quiver of finite global dimension because it has each such algebra as a quotient. Now let $A$ be this Nakayama algebra with Kupisch series $[n,2n-1,2n-2,....,n+2,n+1]$. This is an algebra with global dimension 2 (while the algebra of upper triangular matrices has global dimension 1).
I wondered what the tilting modules over this algebra are. The problem seems to contain the problem of classifying the tilting modules over the algebra of upper triangular matrices as a special case because the indecomposable modules with projective dimension one in $A$ just behave like modules over the algebra of upper triangular matrices and thus the number of 1-tilting modules of $A$ should be also equal to the Catalan numbers.
The number of tilting modules of $A$ starts with 1,3,10,35,126 and this suggests that the number of tilting modules equals $\binom{2n-1}{n}$ which are the monotone maps $\{1,...,n \} \rightarrow \{1,...,n \}$, see https://oeis.org/A001700.
This leads to the following guess:

There is a natural bijection from the set of monotone maps $\{1,...,n \} \rightarrow \{1,...,n \}$ to the set of tilting modules of $A$.

Note that the monotone maps $f$ with $f(i) \leq i$ are counted by the Catalan numbers (see for example exercise 78. in the book "Catalan numbers" by Richard Stanley) and thus the above bijection (if it exists) should restrict to a bijection between monotone maps $f$ with $f(i) \leq i$ and the 1-tilting modules of $A$ (at least if it is a nice bijection).
I wanted to ask whether there is a quick proof of this guess in case it is true using some advanced tools. I am able to translate the problem into a purely combinatorial problem but it looks very complicated at the moment and maybe there is an easy trick to obtain such a bijection or maybe this is even known.
The combinatorial translation gives the problem where $n$ points (corresponding to the indecomposable summands of the basic tilting module) are drawn into two triangles (whose points correspond to the 2-rigid indecomposable modules in the Auslander-Reiten quiver of the algebra). There is one bigger triangle with $\frac{n(n+1)}{2}$ points and one smaller triangle with $\frac{n(n-1)}{2}$ points so that both triangles have together $n^2$ points.
Here the tilting modules for $n=3$ (maybe someone can see how they correspond to monotone sequences?):
https://www.docdroid.net/YwBhi0k/monotonetilting.pdf
Here the configurations where the red market points only occur in the smaller triangle or on the leftmost boundary of the bigger triangle count the 1-tilting modules so that for n=3 we get 5 1-tilting modules.
 A: I'm missing some details, but I think the answer is lurking in Buan-Krause (2004) and Adachi (2016). 
Neither of these papers asks exactly the same question as Mare. Let Q be the oriented $n$-cycle, let $k[Q]$ be the path algebra and let $k[[Q]]$ be the completion of the path algebra. The Buan-Krause paper studies tilting modules for $k[[Q]]$; the Adachi paper studies Nakayama algebra $k[Q]/I$ where all the indecomposable projectives have length $\geq n$ (such as the OP's) but studies "basic $\tau$-tilting" and "basic proper support $\tau$-tilting" rather than "tilting". I'm not completely clear on the relation between these concepts, but I hope the OP is.
In any case, both these papers biject the modules they study to $n$-tuples $(a_1, \ldots, a_n)$ of nonnegative integers with $\sum a_i = n$. These correspond easily to monotone functions $f: [n] \to [n]$; put $a_i = \# f^{-1}(i)$.
These papers give bijections to the variants of tilting modules they study in Theorem D of Baun-Krause, and Theorems 2.16 and 2.19 in Adachi.
