Suppose that a person has a problem with his memory. Every night forgets a certain amount of people that he knew. For example if he knows k people the night before ,then the next morning he will remember 0,1,...,k with probability $1/(k+1)$ each.Also his doctor every morning teach him a new person.If we define as $X_{n}$ the number of people that he knows the n-night before ,we should determine transition probabilities p(i,k) of $X_{n}$ where $j,k\in\mathbb{N}$.
So I thought that if we claim that he remembers $k$ people then the transition matrix will be a lower triangular matrix , because in the first row we will have the probability $p(0,0)=1$.
Second row $p(1,0)=\frac{1}{2}$ and $p(1,1)=\frac{1}{2}$.
Third row $p(2,0)=\frac{1}{3}$,$p(2,1)=\frac{1}{3}$and $p(2,2)=\frac{1}{2}$. The same way for all the other rows.
My problem is that I'm not sure if I found the correct transition matrix because when I'm trying to calculate stationary distribution I get a vector full of zeros.
You're forgetting the fact that the doctor teaches the man a new person every morning, so you need to shift everything by $1$. So if the man remembers $k$ people one night, the next night he will remember $1,2,\ldots, k,k+1$ with equal probability. Hence the transition probabilities are:
$$p(i,j)=\begin{cases} \frac1{i+1}&\text{if }j\in\{1,\ldots,i+1\},\\ 0&\text{otherwise.}\end{cases}$$