Given n distinct numbers, k numbers are picked out, 1 by 1 w/ replacement to get a seq of numbers. What is the prob that this seq is increasing?
I know that since non-decreasing seq are allowed, there are in addition n constant seq on top of the strictly increasing probability of C(n,k)/n^k. But I am stuck from here...
Let $F(k,n)$ be number of $k$-tuples such that $X_1 \le \ldots \le X_k$. Conditioning on $X_k$, we get $$F(k,n) = \sum_{m=1}^n F(k-1, m)$$ Thus $F(k,n) - F(k,n-1) = F(k-1,n)$. Boundary conditions are $F(k,1) = 1$ and $F(1,n) = n$.
It is easy to show that $$ F(k,n) = {n+k-1 \choose k-1} $$
Then the probability is $F(k,n)/n^k$.