How do I understand the proof of Laplace's Theorem in wikipedia?

Question

See http://en.wikipedia.org/wiki/Laplace_expansion

What does $\tau\,=(n,n-1,\ldots,i)\sigma'(j,j+1,\ldots,n)$ stand for as well as the statements follow? "Since the two cycles can be written respectively as $n - i$ and $n - j$ transpositions..."

Answer 1

Follow the internal link to “cycles” in that proof. You will learn that $(n,n-1,\ldots,i)$ is the permutation that maps $n$ to $n-1$, …, and $i$ to $n$, and similarly for the parenthesis on the right. The whole expression is the product of three permutations, with $\sigma'$ in the middle.

A cycle of length $k$ is the product of $k-1$ transpositions (cycles of length 2).

Addendum: The comment thread makes it seem that a formal definition of a cycle like $(a_1,\ldots,a_k)$ is called for. Assuming $a_1$, …, $a_k$ are distinct elements of $\{1,2,\ldots,n\}$, you can define the cycle as the following permutation on $\{1,2,\ldots,n\}$: $$(a_1,\ldots,a_k)(q) =\begin{cases} a_{j+1}&\text{if $q=a_j$ with $1\le j<k$},\\ a_1&\text{if $q=a_k$,}\\ q&\text{otherwise.} \end{cases}$$