Learn Swedish: Han har inte dammat av samlingen på länge. - He hasn't dusted off the collection for a long time.

When expressing how long an action lasts in a positive sentence, you normally use i (for example, i två veckor). However, when you state that something has not happened for a certain period, Swedish requires the preposition på. Since this sentence uses inte, we use på länge to mean "for a long time".

Is dammat av a particle verb, and how does that affect the word order?

Yes, damma av (to dust off) is a particle verb. In Swedish, the particle (av) almost always stays right next to the verb, coming before the object. While in English you can split them and say "dusted the collection off", in Swedish it must be dammat av samlingen. When speaking, remember to put the stress on av rather than the verb.

Could I say på en lång tid instead of på länge?

While på en lång tid makes literal sense and would be understood, på länge is the standard, fixed expression that native Swedish speakers use for "for a long time". It sounds much more natural.

How does the verb samla relate to the word samlingen?