ArrayFormula - Number Partitioning row wise - Attempt to use a single column formula for all rows

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Problem description. From a CSV string ("10,13,25,17") each number is partitioned X times such that sum is equal to the original number. Only interested in the average number(s).

Eg: 10 partition 3 = { 3,3,4 }, 15 partition 2 = { 7,8 }, 10 partition 2 = { 5, 5 }

So the input to output (JOIN-ed) looks like ==>

Part("22,25,30",12) = "1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,2,2,2,2,3,3,3,3,3,3"

I have successfully created the ArrayFormula for above. But am stuck on how to apply it to a whole column (vertically). Any ideas? (recently someone in StackOverflow introduced me to QUERY header smush - but still stuck on this).

On a separate note - I wonder if "Named Formulas" or "Formula Aliases" are ever coming to Google Sheets. Reusing formula components will be a life saver with these huge ArrayFormulas. And yes - I have sent this query to Google many times over a few years.