Anyone managing communities is always looking for ways to optimize their users engagement.

I just came across this white paper Optimizing Facebook Engagement written by Track Social and I believe you should read it. It has some interesting pieces of advice like

It turns out that although posting more often does have consequences, it is possible to post far more frequently on Facebook than many brands believe.

or

Can you blatantly ask for Likes and Comments without alienating your Fans? Answer: Definitely.

You can skim it in under 5 minutes while still gaining some valuable insights.