I sure my question is simple. We have, for example $$\sum\limits_{a+b+c+\cdots=n}\frac{(a+b+c+\cdots)!}{a!b!c!\cdots}$$ then for $n=6$ exists 5 pairs: (5,1), (4,2), (3,3), (2,4), (1,5)
or 10 triples: (4,1,1), (3,2,1), (3,1,2), (2,3,1), (2,2,2), (2,1,3), (1,4,1), (1,3,2), (1,2,3), (1,1,4) and so on.
Is there a specific notation, where we use for sum only 3 pairs (5,1), (4,2), (3,3) or only 3 triples (4,1,1), (3,2,1), (2,2,2)?
Is also there a special symbol or something, which denote, that if some elements of sum are equal, ex. $a=d=h=2$, $b=e=k=r=3$, we need to divide that term by $q_{1}!q_{2}!\cdots$ (where $q_{k}$ - quantity of elements (a,d,h) or (b,e,k,r) and so on)?
For your first question, when you go from order mattering to order not mattering, you are going from "compositions" to "partitions".
For your second question, if this is supposed to be used with partitions, then there's just a scaling involved. There are $18!$ permutations of distinct $(a,b,\ldots,r)$, and there are $18!/(3!4!)$ permutations where $a=d=h$ and $b=e=k=r$ and no other things are equal. Therefore, if you just do the sum over the compositions and divide by $18!$, you'll get the appropriately adjusted sum over the partitions.