A permutation $\sigma$ of a set $\{1,2,...,k\}$ is a bijective function mapping this set onto it self. Denote the set of all permutations of this set by $S_{k}$.
An elementary permutation $e_{i}$ of $S_{k}$ is a permutation that satisfies $e_{i}(j)=j$ if $j\notin\{i,i+1\}$, $e_{i}(i)=i+1$ and $e_{i}(i+1)=i$. That is, $e_{i}$ change $i$ with $i+1$ and preserves another numbers.
We know that every permutation $\sigma$ can be written by a composition of elementary permutations. But I'm thinking in something that I didn't saw at any book: Every permutation $\sigma$ can be written by a composition of elementary permutations with no repetition? By 'no repetition' I mean that a elementary permutation $e_{i}$ doesn't appear more than one time in the composition for $\sigma$
As discussed in the comments, the permutation $(13)\in S_3$ cannot be represented as a product of neighbour exchanges without either of the two available exchanges being used twice.
More generally, there are $n-1$ neighbour exchanges, and we can form at most $\sum_{k=0}^{n-1}k!$ different products from them. For $n\gt2$, this is less than $n!$ (which you can prove by induction), so we can't generate all $n!$ permutations with these products.