Asymptotic complexity of sorting using k-comparisons Sorting using 2-elements comparisons has an asymptotic worst-case complexity of $n \log_2(n)$ (reached by mergesort, heapsort, binary insertion, ford-johnson at least), which is optimal.

If we sort using comparisons that sort k elements as building blocks, the information theoretic lower bound is $n \log_{k!}(n)$.
We can easily reach $n \log_k(n)$ with k-ary insertion.

Question: Where is the optimal complexity between $n \log_k(n)$ and $n \log_{k!}(n)$ ?

Appropriate refs would be appreciated too.

Edit : because it wasn't clear :

I'm interested about the complexity in $n$, with $k=O(1)$. It has the asymptotic behavior of $\alpha n\log_2(n)$ with $\alpha \in [\frac{1}{\log_2(k)},\frac{1}{\log_2(k!)}]$, and i'd like more precision about $\alpha$.
As I was told by Abhishek Methuku, this has been studied already, in fact several times, see e.g. http://www.researchgate.net/publication/3042594_Sorting_n_objects_with_a_k-sorter

For $k$ large, the answer is close to $n \log_{k!}(n)$ but e.g. $k=3$ seems to be open.