For $n\in \mathbb{N}$ let $S_n$ denote the set of permutations (bijections) $\pi: \{0,\ldots,n-1\}\to \{0,\ldots,n-1\}$. A transposition swaps exactly $2$ elements and is often denoted by $(i \; k)$ if $i\neq k\in\{0,\ldots,n-1\}$ are the elements being swapped.

For $n\in\mathbb{N}$ let $E_n$ be the number of elements of $S_n$ that can be obtained by a composition of all the transposition, such that every transposition is used exactly once.

What is the value of $\lim\sup\frac{E_n}{n!}$?