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The probability distribution of rolling $n$ dice and keeping $k$ highest

In many roleplaying games one rolls a handful of dice and calculates their sum. In some games there are bonus or penalty dice, so that we roll, for example, 4 dice with six sides and take the sum of the three highest, ignoring the lowest.
So let us fix some notation. We are rolling $n \ge 0$ dice with $s \ge 1$ sides. The dice are iid distributions selecting uniformly random number from $\{1, 2, \ldots, n\}$. We want to keep $k \le n$ highest of the results and calculate their sum. **We want to know the probability distribution**, or at least as much as we can of the distribution; what is the average, for example?
An analytical formula would be the best, of course, but probably out of reach. If $k = n$, that is, we are not discarding any dice, the way I would calculate the probability distribution is to represent the single die as a probability generating function and then use multiplication of polynomials for the addition of probability distributions. I don't thing anything similar is possible here, but maybe I am wrong.
Of course, just going through every possible permutation of die results is technically possible, but it provides little general insight.

In many roleplaying games one rolls a handful of dice and calculates their sum. In some games there are bonus or penalty dice, so that we roll, for example, 4 dice with six sides and take the sum of the three highest, ignoring the lowest.
So let us fix some notation. We are rolling $n \ge 0$ dice with $s \ge 1$ sides. The dice are iid distributions selecting uniformly random number from $\{1, 2, \ldots, s\}$. We want to keep $k \le n$ highest of the results and calculate their sum. **We want to know the probability distribution**, or at least as much as we can of the distribution; what is the average, for example?
An analytical formula would be the best, of course, but probably out of reach. If $k = n$, that is, we are not discarding any dice, the way I would calculate the probability distribution is to represent the single die as a probability generating function and then use multiplication of polynomials for the addition of probability distributions. I don't thing anything similar is possible here, but maybe I am wrong.
Of course, just going through every possible permutation of die results is technically possible, but it provides little general insight.

probability dice