How to construct an example for defined Partition Suppose $ m \geq n \geq 1$ are two integers. An ordered $n$-tuple of integers $ \pi = (m_{1}, \dots , m_{n})$, $ m _{i} \geq 1$ is called an $n$-partition of $m$  if $ m_{1}+ \dots m_{n} = m$. The set of all such partitions is denoted by $\Pi (m,n)$. 
Question 1: How can we show that a partition $ \pi \in \Pi(m,n)$ determines  a one-to-one correspondence between the sets $ \{ 1 , \dots , m \}$ and $\{ (i,j) | i=1, \dots , n , j=1, \dots ,m_{i} \}$ ?
Question 2: For two partitions $ \tau = (p_{1} , \dots, p_{m} ) \in \Pi (p,m)$ , $\pi \in (m_{1} , \dots , m_{n}) \in \Pi (m,n)$, then $\tau \pi$ is defined $ \tau \pi \in \Pi(p,n)$ in a natural way: 
$$ \tau \pi = ( p_{1}+ \dots + p_{m_{1}}, p_{m_{1}+1}+ \dots + p_{m_{1}+ m_{2}}, \dots , p_{m-m_{n+1}} + \dots + p_{m})$$.
I am confused of the product definition, may you give me an example for that?
 A: First, a partition is more than just an ordered tuple because we want to say that $1 + 2 = 3$ and $2 + 1 = 3$ are the same partition of $3$. One way to do this is to require that your tuple be decreasing. So we would write this tuple as $(2,1)$ and not $(1,2)$. It is possible that this is what you meant by "ordered" tuple but the term "ordered tuple" means that, for example, $(1,2) \ne (2,1)$ (i.e. the order matters, but for partitions we want the order not to matter). With that clarification out of the way, let's discuss your questions.
Q1: Consider the partition $7 + 5 + 5 + 2 + 1 + 1 = 21$. Then we can associate to this the following diagram:
*
*
* *
* * * * *
* * * * *
* * * * * * *

Where the bottom row has 7 asterisks, the next 5, then 5, then 2, then 1, then 1. If we give these coordinates, then the bottom row is $(1,1),(1,2),\dots,(1,7)$ and the next $(2,1),(2,2),\dots,(2,5)$. I think the picture here is clear enough, but let me know if you are still confused.
Q2: What's going on here is you are partitioning one partition with a second partition. For instance, with the partition $\tau = (7,5,5,2,1,1)$ above and a partition $\pi = (3,2,1)$ we are looking at
$$ (7 + 5 + 5) + (2 + 1) + (1) = 21 $$
where we group according to $\pi$. Then, combining terms in parentheses, we obtain the partition
$$ \tau\pi : 17 + 3 + 1 = 21. $$
