Denote the Stirling number of the second kind as $S(n,k)$, and the (signed) Stirling number of the first kind as $s(n,k)$. It is known that \begin{align} \sum_{j = 0}^n S(n,j) s(j,k) = \delta_{nk}. \end{align}
The natural question is, how can we upper and lower bound the following sum in terms of $n$ and $k$: \begin{align} \left | \sum_{j = 0}^n S(n,j) s(j,k) p^j \right|, \quad p \in [0,1], n\neq k. \end{align}
In some sense, it challenges us to give a continuous interpolation for two extreme values of $p = 0$ and $p = 1$, which both lead to the value zero in the sum.