Write a transposition as a product of adjacent transpositions I just read that any transposition can be written as a product of adjacent transpositions. I thought that I knew the right proof of this, but then I read that $\tau_{i,j} = \tau_{i,i-1} \circ...\circ \tau_{j+1,j},$ where $i > j$ and $\tau_{a,b}$ permutes the a-th and b-th element. 
Somehow I don't think that this is true or is it?

The way I go about solving something like this is by drawing a bunch of $x$'s to denote generic numbers and $o$'s to denote the numbers I want to transpose in a line.  Then the game is to only be allowed to do adjacent transpositions if to achieve your desired affect.  Your first goal is to get your first $o$ to where the second $o$ is, and you can do that by what you have written.  After playing around like this you can convince yourself that the formula should be:
$$ (i\, j) = (j+1\,\,\,j)\dots(i-1\,\,\,i-2)(i\,\,\,i-1)(i-1\,\,\,i-2)\dots(j+1\,\,\,j).$$
Now you just have to prove this formula is correct.
