On multisets, interpolated multiple zeta values and limit laws

In this work we discuss a parameter $\sigma$ on weighted $k$-element multisets of $[n]= \{1,\dots ,n\}$. The sums of weighted $k$-multisets are related to $k$-subsets, $k$-multisets, as well as special instances of truncated interpolated multiple zeta values. We study properties of this parameter using symbolic combinatorics. We rederive and extend certain identities for $\zeta^{t}_n(\{m\}_k)$. Moreover, we introduce random variables on the $k$-element multisets and derive their distributions, as well as limit laws for $k$ or $n$ tending to infinity.