Learn Finnish: Vanhaa perintöä ei säilytettäisi ilman uutta sukupolvea. - The old heritage would not be preserved without a new generation.

You use the negative verb ei alongside the passive conditional form without its final -in. The affirmative is säilytettäisiin (would be preserved), so you simply drop the -in to get säilytettäisi.

Why is the object vanhaa perintöä in the partitive case instead of the basic nominative form?

In affirmative passive sentences, the object is usually in the nominative case. However, this is a negative sentence (ei säilytettäisi), and the rule that negative sentences require a partitive object overrides the passive object rules.

Why are the words uutta sukupolvea in the partitive case?

They follow the word ilman (without). In Finnish, ilman requires the noun and any attached adjectives to be in the partitive case.

Could ilman be placed after the noun instead of before it?

Yes, ilman is one of the few words in Finnish that can act as either a preposition (before the word) or a postposition (after the word). So, uutta sukupolvea ilman is grammatically correct, but using it as a preposition is much more common.