In Blom, Holst, Sandell, "Problems and snapshots from the world of probability", section 9.4, a model of records is discussed:
Elements are ordered in a sequence of increasing length according to some rule that leads to exchangeability. At each step, the element is inserted at its proper place among those already ordered. If an element comes first, it is called a record. The position of $r$-th record is denoted $N_r$. Naturally $N_1 = 1$.
The following probability is then derived: $ \mathbb{P}(N_r = n) = \frac{1}{n!} \left[ n-1 \atop r-1 \right] I\left( n \geqslant r\right) $ where $\left[ n \atop r \right]$ denotes the unsigned Stirling number of the first kind.
Question: How does one analytically prove that this probability mass function is normalized for all integer $r \geqslant 2$, i.e. $ \sum_{n=r}^\infty \frac{1}{n!} \left[ n-1 \atop r-1 \right] = 1 $
Using $\left[ n-1 \atop 1 \right] = (n-2)!$, and $\left[ n-1 \atop 2 \right] = H_{n-2} \cdot (n-2)!$, the normalization follows for special cases of $r=2$ and $r=3$.