Take a multiset $S=\{s_1,\ldots,s_n\} \subset \mathbb{N}$ and some $k \in \mathbb{N}$.  Take $T$ to be the set of all sequences of the form $s_{\pi(1)},\ldots,s_{\pi(i)}$, such that $\sum_{j=1}^{i-1} s_{\pi(j)} < k \leq \sum_{j=1}^{i} s_{\pi(j)}$ where $\pi$ is a permutation of the indices of $S$ and $i\leq n$ . Is there an efficient algorithm to compute the average of the sum of members of $T$?  This is of interest to me because of its application to scheduling.
Scheduling problem of average sums with condition
complexity theory;combinatorics;scheduling