If the parts are of size at most $5$, why are considering numbers greater than $5$? For example, why can $z_5$ take on $10, 15, \dots$?
2026-04-07 22:59:21.1775602761
Question about: How many partitions of $12$ have parts of size at most $5$?
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The problem with using five $(1+x+x^2+x^3+x^4+x^5)^5$ is that you get duplicated partitions in that way. For instance $4+3+5=12$ and $3+5+4=12$ gives in fact the same partition.
The intuition behind the solution is that, $z_1$ represent the number of $1$s in the partition, then $z_2$ represent the number of $2$s in the partition and so on, in this way the partition is always sorted so that smaller numbers always come first and duplication is eliminated.