grammatical number - When indicating separate items linked by a common word?

Let's say my sentence is:

I am a part of Generations X, Y, and Z.

In a case like this, should "generations" be plural or singular? I feel it should be plural, since singular makes it sound like "X, Y, and Z" are one item, but:

I am a part of Generation X, Y, and Z.

I feel someone would argue that since there is context in the word "generation" (as in "everyone will know you mean three separate generations because no one generation is called "X, Y, and Z"), Y and Z do not need to be "counted".

Answer

Your initial impression is exactly correct. Consider:

I am a member of teams Alpha, Bravo, and Charlie.

I have sailed the seas Mediterranean, Black, Red, and Caspian, but have yet to sail the seas Arabian, Indian, or Adriatic.

While you could conceivably find a house style guide that requires proper-name groupings like "Generation X" to never be shortened, if you omit the term you should indicate that it is plural and not singular.