Ved ankomst til stationen fandt vi heldigvis en ledig vinduesplads.

Ved ankomst is a prepositional phrase built from:

• ved = at / upon / on
• ankomst = arrival

So it literally means upon arrival or on arriving.

This structure is a bit more compact and slightly more formal/written than da vi ankom (when we arrived). Both are correct, but they feel different:

• Ved ankomst til stationen ... = more concise, a little formal
• Da vi ankom til stationen ... = very natural, everyday style

Because ved ankomst is a common fixed-style expression in Danish. With abstract nouns like ankomst (arrival), Danish often leaves out the article in this kind of phrase.

So:

• ved ankomst = upon arrival
• efter ankomst = after arrival

If you want to make it more specific, you can add a determiner:

• ved vores ankomst = upon our arrival
• ved hans ankomst = upon his arrival

So the version without an article is normal here.

Because the noun ankomst normally takes til when you say where someone arrives.

So:

• ankomst til stationen = arrival at the station
• ankomst til København = arrival in Copenhagen

English uses at here, but Danish typically uses til with ankomst.

Compare:

• Vi ankom til stationen = We arrived at the station
• Vi var på stationen = We were at the station

So til is about the destination reached, while på / ved can describe location.

Because stationen is the definite form: the station.

In Danish, the definite article is often added to the end of the noun:

• en station = a station
• stationen = the station

Here it is definite because it refers to a specific station, either already known from context or understood as the relevant one in the situation.

This is because Danish main clauses follow the V2 rule: the finite verb usually comes in the second position.

Here, the sentence starts with the adverbial phrase:

• Ved ankomst til stationen = first position

So the finite verb must come next:

• fandt = second position

Then the subject follows:

• vi

That gives:

• Ved ankomst til stationen fandt vi ...

Compare the normal order without the fronted phrase:

• Vi fandt heldigvis en ledig vinduesplads.

So the inversion is completely normal.

Fandt is the past tense of finde (to find).

• finde = infinitive
• finder = present
• fandt = past
• fundet = past participle

Examples:

• Jeg finder en plads. = I find a seat.
• Jeg fandt en plads. = I found a seat.
• Jeg har fundet en plads. = I have found a seat.

It is an irregular verb, so learners usually just have to memorize finde – fandt – fundet.

Because heldigvis is a sentence adverb, and in a main clause it commonly comes after the subject.

So in a basic clause:

• Vi fandt heldigvis en ledig vinduesplads.

When another element is moved to the front, the verb still stays second, and the subject comes right after it:

• Ved ankomst til stationen fandt vi heldigvis en ledig vinduesplads.

So the order here is:

• fronted phrase
• finite verb
• subject
• sentence adverb
• object

That is very normal Danish word order.

Because plads is a common gender noun, so it takes en, not et.

• en plads = a seat / a place
• pladsen = the seat / the place

Since vinduesplads is a compound whose final part is plads, the whole compound also has common gender:

• en vinduesplads

So:

• en ledig vinduesplads = correct

not

• et vinduesplads = incorrect

Because it agrees with an indefinite singular common-gender noun:

• en ledig vinduesplads

That is the basic adjective form.

Compare:

• en ledig plads = common gender singular indefinite
• et ledigt sæde = neuter singular indefinite
• den ledige plads = definite
• ledige pladser = plural

So ledig is the correct form because vinduesplads is common gender and indefinite.

Ledig means free, available, vacant.

For a seat, it usually means the seat is not taken and can be used.

Tom means empty. That is similar, but not always identical. A seat can be empty without necessarily being available.

So:

• en ledig vinduesplads = an available/vacant window seat
• en tom vinduesplads = an empty window seat

In this sentence, ledig is the better choice because the point is that they were able to take the seat.

Because Danish normally writes compound nouns as a single word.

So:

• vindue = window
• plads = seat / place
• vinduesplads = window seat

This is very typical in Danish. English often writes such things as two words, but Danish usually combines them into one.

Also, the last part of the compound is the main noun. That means plads determines things like:

• gender
• plural
• definite form

So:

• en vinduesplads
• vinduespladsen
• vinduespladser

That -s- is a common linking element in Danish compounds. It does not necessarily mean possession.

So vinduesplads is built like this:

• vindue
• linking -s-
• plads

Many Danish compounds use such linking sounds or letters. Unfortunately, they are not always fully predictable, so learners often have to get used to them word by word.

The important point is that vinduesplads is the normal standard form.

Yes, but with a small nuance: ved ankomst sounds a bit more formal or written than everyday speech.

The whole sentence is perfectly good Danish, but in casual conversation many speakers would probably say something like:

• Da vi kom til stationen, fandt vi heldigvis en ledig vinduesplads.
• Da vi ankom til stationen, fandt vi heldigvis en ledig vinduesplads.

So the sentence is natural, just slightly more polished or written in style because of ved ankomst.