grammatical number - Singular or plural usage for ellipsis in direct object

Suppose I have the following sentences:

1. There should be an X and a Y chromosome.


2. There should be an X and a Y chromosomes.

Is the second grammatically correct? If the last word had to be plural for the same meaning of the sentence and an ellipsis, would the following be correct?

3. There should be X and Y chromosomes.

Answer

The first and the last sentence are correct. The middle sentence is not correct.

The reason is this sentence, which is the original one:

• There should be an X chromosome and a Y chromosome.

Notice that this is not

• *There should be an X chromosomes and a Y chromosomes.

Neither chromosome should be plural.

That's what is meant.
Now the rule of Conjunction Reduction deletes the first chromosome, leaving sentence 1

• There should be an X [...] and a Y chromosome.

Since chromosome wasn't plural before Conjunction Reduction, it isn't plural afterwards.
So the second sentence above is ungrammatical.
Conjunction Reduction only deletes; it doesn't do arithmetic.

However, the speaker can do the arithmetic.
There is, after all, one X chromosome and one Y chromosome involved,
and that makes two chromosomes, should one need to speak of them.
(There is, of course, no article, since a/an is only singular.)

But X and Y is a perfectly reasonable conjoined NP that can modify
a plural chromosomes, which leads to the third sentence:

• There should be X and Y chromosomes.