Learn Danish: Han må have fundet mønten i løbet af udgravningen. - He must have found the coin during the excavation.

Even though the action of finding happened in the past, the speaker's deduction is happening right now. Just like in English, Danish uses the present tense modal må followed by the perfect infinitive have fundet to express a strong current assumption about a past event.

Does må in this structure ever mean 'is allowed to'?

While må can mean 'is allowed to' in simple sentences, when it is combined with have and a past participle like fundet, it almost exclusively expresses logical deduction, meaning 'must have'. If you wanted to say 'might have found' to express a weaker possibility, you would use kan have fundet.

Could I use under instead of i løbet af to say 'during'?

Yes, under udgravningen would also be perfectly natural here. The phrase i løbet af literally means 'in the course of' and emphasizes the passage of time, suggesting the discovery happened at some unspecified point within that timeframe. Under simply places the event concurrently with the excavation, but in this context, they mean the same thing.

Can we drop have in casual Danish and just say Han må fundet?

No, Danish grammar strictly requires the infinitive have after the modal verb in this construction. Unlike informal English where 'must have' blends into 'musta', you cannot omit have in written Danish. However, in fast spoken Danish, have is often reduced to a short vowel sound, which can make it easy to miss.