How do most people write permutations? I'd like to know how people prefer to write permutations, or elements of the symmetric group $S_n$ for $n\ge0$.
The most natural way to define a permutation in $S_n$ is as a bijection on the set $\{1,2,....,n\}$. Then the set of permutations (bijections) becomes a group under composition of maps. If $f,g\in S_n$ then there are two ways to define the composition $f\circ g$ depending on whether our functions act from the left or the right:
$$(x)(f\circ g) = ((x)f)g\quad\text{and}\quad (f\circ g)(x)=f(g(x)).$$
I think that the latter is by far the most common these days. Of course, in the first case I could just define $(f\circ g)(x)=g(f(x))$ but it really is a right action so it should be written this way.
In terms of multiplying permutations using cycle notation the two ways of writing composition correspond, respectively, to whether we read the cycles left-to-right or right-to-left. For example:
$$(1,2)(2,3) = \begin{cases}
    (3,2,1), &\text{using the $(x)f$ convention},\\
    (1,2,3), &\text{using the $f(x)$ convention}.
\end{cases}$$
To me it has always seemed more natural to read permutations from left-to-right, as in the first case, but this implicitly uses the less common convention for composition of maps.
So the question: do you prefer to read products of permutations, written as cycles, from left-to-right or right-to-left?
 A: The GAP convention is to multiply permutations from the left to the right, i.e.
$(1,2) \cdot (1,3) = (1,2,3)$, to write down each cycle with the smallest moved point first
and to sort cycles in ascending order w.r.t. their smallest moved points.
I think meanwhile this convention is a somewhat widely used standard.
A: For a topologist considering representations of the fundamental group of a space to a symmetric group, it is very helpful to multiply permutations from left to right, since it is natural to interpret path concatenation, which defines the group operation in the fundamental group, by first doing the left-hand loop and then the right hand loop.
A: I draw permutations in TikZ. Compose permutations by drawing the permutation pictures beside each other: . Invert by reversing the arrows. Of course there is more data in these pictures than just a permutation, but students understand the pictures immediately. 
A: I asked a well-known veteran of finite permutation group theory what specialists in that field did.  The answer was that "almost everyone" does it left to right.  That is, if $g,h$ are permutations, then $gh$ means "do $g$ then do $h$".  Actions are usually written using exponential notation: $x^g$ is the image of $x$ under $g$.  The left-to-right convention means that $(x^g)^h=x^{gh}$. As to why the convention is like it is, the widespread influence of Wielandt's book Finite Permutation Groups was proposed as one reason.
A: This is something that greatly annoys me, so I tend to use the following rule:

Does this make sense if permutations are replaced by permutation
  matrices?

Thus, I follow the convention of matrix multiplication, where I almost always act on vectors on the left side. 
