The finitude of the fibers of the complementary Bell numbers

Subbarao and Verma introduced, in 1999, a number of open problems concerning the sequence $(f(n){)}_{n\ge 0}$ of complementary Bell numbers, which may be defined via Bell polynomials $B_n(x)={\sum }_{k=0}^n\left\{\begin{array}{c}n\\ k\end{array}\right\}{x}^k$ so that $f(n)=B_n(-1)$. Yang [Electron. J. Combin., 2001] subsequently solved the first two problems from Subbarao and Verma, but the third such problem has remained open, to the best of our knowledge. The first part of this third problem asks whether or not $f(n)$ takes any given value only a finite number of times. We solve this problem in the affirmative, through a combined application of finite difference-based methods, partial Motzkin paths, the completeness of the Tate algebra with respect to the Gauss norm, and Strassmann’s theorem.