Rank functions for A_n quivers

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In January 2021, I applied for a postdoc position with Joseph Chuang (and Andrey Lazarev). The main topic of the funded project was rank functions on triangulated categories. I did not get the job but I found it quite interesting and tried to understand rank functions on my own. I was first interested in rank functions on the singularity categories of simple singularities. But then I decided to try a "simpler" case: derived category of an A_n quiver. I tried to describe the space of all rank functions on this category in a simple way. I drew a lot of parallellograms before I realize that I am not getting all rank functions. But then the question is: what is special about the ones I am getting?