How can we map a partition of $n$ to some permutation of [1,2, ... , n]?

Question

Here is the question I was reading:

Does every partition of n correspond to some permutation of [1,2, ... n]?

And here is a statement in the answer given there that I want to use:

If the partition is $j_1 + \dots + j_k = n$ where $j_1 \geq \dots \geq j_k$, then we can map it uniquely to the permutation $(1 \dots j_1) (j_1 + 1 \dots j_1 + j_2) \dots (n-j_k+1 \dots n)$

My question is:

I want a reference (including specifically which edition)that contains this statement and its proof, could anyone help me in this please?

EDIT:

1-My question is why is this function unique? could not we define a different function? or unique here means well defined?

2- Also, how to prove that this function is well defined?