@Jan Grabowski, thank you very much!

@Sam Hopkins, I added the method to produce $P_i$. Thanks a lot.

@Sam Hopkins, thank you very much.

@Per Alexandersson, thanks. Yes, exactly. The rule I know to get T_i from P_i is not complete. So I am trying to find a some more straightforward rule.

@Per Alexandersson, thank you very much. For $P_1$, we can first reorder each row from small to large. Then we reorder each column from small to large. And we move some numbers down when a column is not strictly increasing. But this does not work for $P_3$. We need to interchange $7$ and $8$ in the end. Do you have some uniform rule which work for every $P_i$?

thank you very much!

@JeremyRickard, thank you very much for your answer.

@JeremyRickard, thank you very much for your help. I have another question. I think that this property is true for the Auslander-Reiten sequence of the algebra $B_{k,n}$ in the post. Does your proof also work for the algebra $B_{k,n}$?