Probability of $k$ of a kind (or greater) on $n$ dice with $m$ sides, with duplicates allowed

How would I calculate the probability of at least one matching set of $k$ (or more) dice, from a roll of $n$ dice with $m$ sides?

I can do this for $k$ > $n/2$, I believe, by summing a formula for an exact match of $k$ dice (e.g., exactly 7 of a kind on 10 dice with 6 sides), from $k$ to $n$.

If I had a general formula, given $k \leq n/2$), for the largest match being exactly $k$ dice (with duplicates allowed): then I could continue the sum-of-probabilities approach for all $k$. However, the two existing questions I could find on this are either asking for no duplicate $k$-of-a-kind matches, or are asking about a specific case where $k = (n/2)-1$. I'm not sure how to generalize either of those to a case where duplicates (but not larger matches) are allowed.

A possibly useful sanity check that I've been using: when $n > (k-1)m$, the probability of getting a match of at least $k$ (the main probability I want to calculate) is $1$; and when $n > km$, the probability of the largest match being exactly $k$ (a useful but non-essential intermediate probability, for me) is $0$.