Usage of "expect to" and "expectation to/of"

Solution 1:

One of the properties of individual words in a language - and often one of the hardest for foreigners to learn - is what linguists call their "subcategorization frame" : the particular kinds of word and phrase that they require, or allow, to follow them in a sentence.

Rhodri's answer is generally correct: "expect" requires a direct object, which may be of various types including a "that" clause ("I expect that he will come", and an infinitive clause ("I expect to win"). But "expectation" has different requirements: it may take a direct object only if that is a "that" clause ("His expectation that I would come"; any other object must be expressed in a PP (prepositional phrase) introduced by "of": "my expectation of winning").

The bit I disagree with in Rhodri's answer is the implication that because it is a noun, "expectation" cannot take a direct infinitive. This is not so: "expectation" does not take an infinitive, but "ability" can: ("His ability to climb came in useful").

Solution 2:

Your friend is right that "expectation of seeing" is preferred to "expectation to see" (that is, it sounds more natural).

Still, I think you could say it still another way:

Expecting to see you on Monday, I'm counting the days.

Solution 3:

In "I expect to see you," to is introducing the infinitive "to see". It happens to be the way that the verb expect works that it takes an infinitive.

When you rearrange the sentence to start "With the expectation...", expectation is a noun. It doesn't get to take an infinitive, so that rule doesn't apply. English then heads off into a maze of special cases about what words to use when, or indeed whether to use an intervening particle at all. In this case, we would say:

With the expectation of seeing you...

With the expectation that I will see you...

Edit: I originally said that "expectation" didn't take an infinitive because it wasn't a verb. As Colin points out, I was completely wrong there.