Suppose you want to say something like
There are two crates, having three and one balls respectively.
How to say this correctly?
Is it ‘…one balls’ or ‘three balls and one ball’, or something else?
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As with a comment under the question, I'm not sure about the overall construction of the sentence, but I will focus on the actual question first:
There are two crates, having three balls and one ball, respectively.
That is the full version version of the sentence.
The problem you're having is how your shortening it. Regardless of actual grammar, neither of these sounds right:
✘ three and one balls ✘ three and one ball
There are two ways of getting around it. One is to provide the longer phrase, as you suggested. The other is to reverse the position of the numbers:
✔ There are two crates, having three balls and one ball, respectively.
✔ There are two crates, having one and three balls, respectively.
The formation of the second variation matches some similar situations.
✘ Your pens and notebook are out of date.
✔ Your notebook and pens are out of date.
Although the combined subject is plural (and, in context, is grammatical), the plural verb only sounds natural if it comes immediately after a plural subject.
I am eating an apple and cherry.
That sentence is correct. But based on ellipsis, if we were to literally apply the written article to each of the items, we would get this:
✘ I am eating an apple and an cherry.
But we don't do that. Along with being able to assume the missing article in front of the second item, we also assume the correct article:
✔ I am eating an apple and a cherry.
Applying both of the above to the second variation of your sentence, the following is observed in terms of subject-verb agreement and ellipsis:
There are two crates, having one (ball) and three balls, respectively.
Finally, although that answers your actual question, I believe the sentence itself would read better if it had a different construction.
Using the second variation I mentioned to get around the problem you are having, I suggest restating it in one of these ways:
There are two crates, containing one and three balls, respectively.
There are two crates that have one and three balls, respectively.
There are two crates that contain one and three balls, respectively.
(The / these / those) two crates have one and three balls, respectively.
(The / these / those) two crates contain one and three balls, respectively.
It makes little sense (to me at least) to say "respectively" here. If you say "respectively" then you mean that a list of things corresponds to a previously cited list of things, in a 1:1 mapping.
Your second list of things is (1) three balls and (2) one ball. But what's your first list of things? If you don't provide an explicit list then it makes little sense to use "respectively". If your only description of the first list is "two crates" then that's a set, not a list -- there's no order specified, and neither crate is identified.
Yes, you'll see such sentences here and there, and they're understandable. "Respectively" is used in them just to make clear that it is not the case that each of the crates contains both three balls and one ball (e.g. different colors or whatever). But in my book "respectively" refers to order and identity, and there's no order specified in "two crates", neither of which is identified.
Identify the crates as "Crate 1" and "Crate 2" and it's a different ballgame. Then you can say something like "Crate 1 and Crate 2 contain three balls and one ball, respectively."
To solve your problem, here is a suggestion:
There are two crates, one with three balls and one with one ball.
(Other possibilities instead of "with": "holding", "containing".)
Adverb: respectively ri'spek-tiv-lee
In the order given --
"the brothers were called Felix and Max, respectively" -- WordWeb
in precisely the order given; sequentially.
(of two or more things, with reference to two or more things previously mentioned) referring or applying to in a parallel or sequential way:
"Joe and Bob escorted Betty and Alice, respectively." -- Dictionary.com
respectively adverb re·spec·tive·ly | \ ri-ˈspek-tiv-lē \
in particular : SEPARATELY
"could not recognize the solutions as salty or sour, respectively"
in the order given
"Mary and Anne were respectively 12 and 16 years old" -- Merriam-Webster
The first Merriam-Webster definition employs an unordered set: "solutions". But its most legitimate (clearest) use is following a statement that identifies "the solutions", in order. The first is salty, and the second is sour.
Otherwise it makes little sense, saying only that the first of two solutions is salty and the second is sour. In that case it's clearer to say that one is salty, the other sour - there's no special order.
You can say that - you can say that you met two men; the first was tall and the second short. I'd say that using "respectively" there is overkill, an attempt to give an impression of precision where there is none. OK in literature, if that's the desired effect. It kind of begs the question of which is the first one.
It's also OK to do that as a preliminary to following up with identification of the first and second. IOW, it's not a requirement that the use of an ordering ("first", "second", or "respectively") follow another ordering that it maps against; it can precede it. But there really should be two (or more) sequences, for clarity.
To be clear, it's about clarity. Consider this:
What does it add, in terms of meaning/clarity, to say this instead?
Nothing. It subtracts rather than adds. It reduces clarity (or at best just adds noise), because it causes us to look back, to see if in fact the "respectively" maps Jane to a particular person and John to another one. "Respectively" doesn't do that here, so we're left scratching our heads.
Use it when it makes sense. It's real aim is to point out a 1:1 mapping. Using it superfluously is "OK" -- it's certainly done. But it's generally not helpful. IMO it reflects a bad habit: people try to beef up the information content of what they say without actually adding any information. Academese - an occupational handicap.