How could I use a convolution to find a generating function for the number of ways that I can fill a closet with n suits, which are chosen from either from casual suits (choose from 10, as many of each as I like) and business suits (choose from 20, but at most 1 of each).
2026-04-08 09:02:51.1775638971
Find Generating Function using convolution
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Convolution of ordinary generating functions corresponds to concatenation of objects, i.e. appending a string of objects of type 2 to a string of objects of type 1. Concatenation of exponential generating functions adds relabeling to this procedure, i.e. after concatenation, the resulting string is shuffled so as to preserve the order of objects of each type. Looks like that's the situation in your case: the order of suits of each type does not matter (i.e. they can be considered to be arranged in some canonical order), but the positions of suits of each type do matter (it's not necessarily, say, all casual suits followed by all business suits).