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Let $C_n$ denote subset of integer compositions of $n$ and $c=(c_1,c_2,\dots c_n)$

In a divergent sum, the sequence $$ a_n=\sum_{c\in C_n} \prod_{c_i\in c} c_i! $$ frequently shows up and one expects it to be asymptotic to the contributions containing its maximal part. I just want to know if this kind of integer composition has been studied and if I can find some explicit estimates/bounds for it that already exist in the literature.

Edit: OEIS has the standard case linked as A051296. I am trying to look for the proper "google term" to get the generalization of this sum.

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    $\begingroup$ I think you might find "integer partition" a more useful search term than "integer composition". (I am guessing that $c_i \leq c_j$ iff $i \leq j$.) You might find "multinomial" in combination with some other word also useful. Gerhard "Don't Know About The Sequence" Paseman, 2015.10.05 $\endgroup$
    – Gerhard Paseman
    Commented Oct 5, 2015 at 15:53
  • $\begingroup$ No I mean integer composition. These are not sorted. $\endgroup$
    – Daniel Parry
    Commented Oct 5, 2015 at 15:58
  • $\begingroup$ If the c_i's are not ordered, then you might find multinomial more useful than partition in searching. I would still look at partition as some sources might talk about the partition case being easier/different than the composition case (and give you something on compositions). Also, I read your post and (despite your worthy attempt) I still think partition. For the more dense among us, you might make it amazingly clear that Euler's partition function is not part of your picture. Gerhard "Not Quite 'MathOverflow For Dummies'" Paseman, 2015.10.05 $\endgroup$
    – Gerhard Paseman
    Commented Oct 5, 2015 at 16:22

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