Note that for integers $i > 2$ and. $h > 0$, and except for the case $(i,h) = (3,1)$, one has
$i^h(h!) < 2^{\lfloor ih/2 \rfloor}(\lfloor ih/2 \rfloor !)$. We can account for the exception and bound
from above the denominator of the left hand side of the posted inequality by
$(3)2^{(n - c_1)/2}(((n - c_1)/2)!)(c_1!) \leq (3)2^{(n - c_1)/2}(((n + c_1)/2)!)$. This latter term is increasing
in $c_1$, and for large $n$ one can have $c_1 \leq n - 10$ and still be less than $(n - 4)!$. One can use
this to show the inequality of the post is satisfied for $n > 15$ and $c_1 < 4$; likely the inequality holds
for more $n$ and more $c_1$.

**BEGIN EDIT 2012.01.05**
I decided to add some detail to the post.

Letting $f(i)=i^{c_i}(c_i)!$, we can rearrange the poster's inequality to $$(n - 4)! \gt \prod_{1 \le i \le n}f(i)$$, and ask for which values of $n, i,$ and $c_i$ the inequality holds. Given $c_i$, let $g(i) = 2^{\lfloor ic_i/2 \rfloor} (\lfloor ic_i /2 \rfloor)! $ for $i \gt 1$ and $g(1) = f(1)$. Now, when $i = 3$ and $c_i = 1$ we have $f(3) \lt 2g(3)$, and for other pairs $(i,c_i)$ with $i \gt 2$ and $c_i \gt 0$ one has $f(i) \le g(i)$, so one can have the original inequality follow from $$(n-4)! \gt 2\prod_{1 \le i \le n} g(i) $$. However, the product of the $g(i)$ is itself bounded from above by $h=2(c_1)! 2^{(n - c_1)/2} {\lceil(n - c_1)/2\rceil}!$. When $c_1 \le 4$ and $n \ge 15$, $2h$ is less than $(n-4)!$.

It is clear that the original inequality implies $c_1 \lt n-4$, and that there is no solution for $n \lt 8$. For $n=8$, the product of the $f(i)$ has to be less than $24$, so $c_1 \lt 4$. For $n= 8,9,10,11$ it is routine to find restrictions on $c_1$ that will permit solutions of the inequality. From another direction, if $c_1 = n-5$ then case-by-case examination gives that $f(i) = 1$ for $ i \gt 5$ and $f(2)f(3)f(4)f(5)$ is at most $8$, so that $n > 12$ for the original inequality to hold. When $c_1 = n - 6$, a similar analysis requires $n > 11$, and for larger values of $n - c_1$ the inequality holds for all the remaining meaningful cases.
**END EDIT 2012.01.05**

Gerhard "Ask Me About System Design" Paseman, 2011.12.26