Definition. Let $A$ be a non-empty set. A collection $P$ of subsets $A_1,A_2,A_3,\ldots$ of $A$ is called a partition of $A$ if the following three conditions hold:
- $\emptyset \notin P$,
- $A_1 \cup A_2 \cup A_3 \cup \cdots = A$, and
- For all $i$ and $j$, if $i \ne j$, then $A_i \cap A_j = \emptyset$.
Definition. Let $n$ be a positive integer. A partition of $n$ is a finite sequence $\lambda_1,\lambda_2,\lambda_3,\ldots\lambda_k$ of positive integers such that $\lambda_1 \ge \lambda_2 \ge \lambda_3 \ge \cdots \ge \lambda_k$ and $$\lambda_1+\lambda_2+\lambda_3+\cdots+\lambda_k = n.$$
I want to know the relationship/link between the partition of set with a partition of a positive integer, since the word 'partition' used is the same. But, I still didn't get it. What I tried was as follows: Let $n$ be a positive integer. Define a set $A$ of finite positive integers $\lambda_1,\lambda_2,\lambda_3,\ldots,\lambda_k$ for which their sum is equal to $n$ and form a non-increasing sequence, that is, $\lambda_1 \ge \lambda_2 \ge \lambda_3 \ge \ldots \ge \lambda_k$.
Next, form clopen subintervals $[\lambda_k, \lambda_{k-1}), \ldots, [\lambda_2, \lambda_1), [\lambda_1,\lambda_1]$. Then, the set which contain these subintervals is a partition of $A$, and so, such $\lambda$ is called a partition of $n$.
Is this correct? Any help please? Thanks in advanced.
Suppose you have a set $A$ with $n$ elements and a partition of $A$ in subsets $A_1, \dots, A_k$. If you call $\lambda_i$ the cardinality of $A_i$, after ordering the cardinalities of the subsets, then you get a partition of $n$, because the $A_i$ are disjoint, so $\lambda_1 + \dots + \lambda_k = n$.