There are rock musicians like John Petrucci from Dream Theater who is said to play around 20.000 notes per concert, all from memory without looking at notes or tabs. I guess (correct me if I'm wrong) this relates to the amount of notes a classical solo pianists plays at a concert. They however will have the sheets with them. What is the share of notes played from memory vs. read from the paper for classic solo artist, e.g. a piano player? Or do pianists typically know the piece perfectly but have the sheets mainly as a backup just in case?
A lot of concert players - pianists and others - may well not have the dots with them to refer to. At that stage, they won't need them to read, usually.
Like a lot of singers, though, some do like to have the music there. It won't usually be referred to, but it's somewhat like a placebo to them. A lot of players aren't necessarily brilliant sight-readers. They don't need to be: their job is to learn pieces inside out, so they can just play the notes, and put their own interpretation to them - working solo or with a conducted orchestra. So having the music for some is just a back-up, them having played a pieces literally hundreds if not thousands of times, in private practice.
They however will have the sheets with them.
Where do you get that from?
There's some discussion going on on whether it makes sense or not for a concert pianist (or any other concert instrumentist) to perform without sheets, but the norm is nowadays to play without them. I've never seen a concert pianist perform with sheets, although some exceptions may be made in some cases (very old pianist, or some more informal concert).
The only scenario where you see a pianist playing with the sheets is in chamber music or as an accompanying pianist (which is, again, chamber music).
Or do pianists typically know the piece perfectly but have the sheets mainly as a backup just in case?
It depends on the pianist. The pianist certainly isn't reading the sheet live. It is used more for cues and as backup.
As per the number of notes played in a single performance: I think it's not representative of the difficulty to memorize music. You can have a section full of chords and ornamentation which adds up to a large number of notes but might be easy to memorize while having some passage which is full of modulations and little variations, with much fewer notes, but which will a lot tougher to memorize.
That being said, I don't really know how many notes is the average of a classical concert. For reasons beyond the ones I exposed earlier, too, it would have no meaning to me.
Hope this helps!