Learn Danish: Er det tilladt at ryge i området? - Is it permitted to smoke in the area?

This sentence actually avoids the s-passive by using the adjective tilladt (permitted) and the active infinitive at ryge (to smoke). If you wanted to use an s-passive in the infinitive to ask the same thing, you would rephrase the question as Må der ryges i området? (May smoking be allowed in the area?).

Is tilladt a passive verb here?

It functions like an adjective in this sentence. It is the past participle of the verb at tillade (to permit). Danish often uses this form with the verb at være (to be) to express rules, just like saying 'is permitted' in English.

Why is there an at before ryge?

at is the infinitive marker, just like 'to' in English. When you use the base form of a verb like ryge (smoke) to state an action, it needs at. If we rephrased the sentence using a modal verb like må (may), we would drop the at, as in Må man ryge? (May one smoke?).

Why is it området instead of just område?

område means 'area', and it is a neuter noun (a t-word). To say 'the area' instead of just 'an area', you attach the definite ending -t directly to the end of the word, giving you området.