Random partitions with two-side bounds and $(r,s)$-Bell polynomials in a parametric probabilistic model

On the set of all partitions of an $n$-element set $X_n = \{1,2,\ldots,n\}$ into blocks with sizes exceeding the number $r \geq 0$ and not exceeding the number $s \leq n$ a probability measure is defined such that the probability of each partition with $k$ blocks is proportional to $\theta^k$, where $\theta > 0$ is the parameter of measure. The $(r,s)$-Bell polynomials are introduced and their asymptotic are investigated for $n,r,s \to \infty$. The asymptotic normality of the numbers of blocks in a random partition of $X_n$ in this model is proved, a statistical test for the uniformity hypothesis $H_0\colon \theta = 1$ against the alternatives $H_0\colon \theta \ne 1$ is constructed.