Learn English – Help understanding “to have” in sentences like “I would have liked to have invited few people.”

This construction with the perfect infinitive is regular and idiomatic.

The past-form modal would expresses some sort of ‘unreality’, but exactly what sort and in what degree can only be inferred from the context. With an infinitive complement would love usually signifies a great wish or hope (greater than mere would like) for some future event; the particular ‘unreality’ in play might be a matter of

• improbability: I would love to win the lottery = I wish I would win the lottery
• impossibility (counterfactuality): I would love to be six feet tall = I wish I were six feet tall
• contingency: I would love to go, if I can find the time = I hope I can go...
• politeness (non-insistence): I would love to have your help on this = I hope I will have your help ...

But with a perfect infinitive like to have seen the sense is somewhat different. The perfect, HAVE seen, names an event prior to the time you are speaking of, not a future event. That event is the origin of some state at the time you are speaking of. With like or love the prior event will ordinarily be the cause of your present state of happiness:

I love having seen her = I am very happy because I saw her

This, however, employs a perfect gerund (-ing form), not a perfect infinitive. In theory you could substitute an infinitive for the gerund (I love having seen her → I love to have seen her), but the perfect infinitive is very rare with the ‘real’ I love. (I find only 27 instances on Google, and none in either Google Books or COCA.) By contrast, the perfect infinitive is standard, and the perfect gerund very rare, with ‘unreal’ I would love. The ‘unreality’ expressed here is almost always counterfactuality:

I would love to have seen her = I would have been very happy if I had seen her, but alas! I did not

Here the two senses, ‘source of happiness’ and ‘wish’ flow together:

I wish I had seen her: it would have caused me great happiness.

The time reference in this construction can be a little quirky. Ordinarily the event which did not occur is one which you hoped would take place in the past:

I would love to have seen her when she was in town last week.

But it can also refer to a future event which in the past appeared to be possible but is now known to be impossible:

I would love to have seen her this evening, but alas! I have to go to dinner with a client.

This site has a pretty good explanation:

We use would have as the past tense form of will have:

• I phoned at six o’clock. I knew he would have got home by then.
• It was half past five. Dad would have finished work.

We use would have in past conditionals to talk about something that did not happen:

• If it had been a little warmer we would have gone for a swim.
• He would have been very angry if he had seen you.