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