I would like to compute the Tutte polynomial of the complete graph $K\_n$ for n as large as possible. Using a program by Björklund, Husfeldt, Kaski, Koivisto ([here](https://github.com/thorehusfeldt/tutte_bhkk)), I managed to compute up to n=18 on my home computer (in serial) in less than a day. Overall, I've been very impressed by this program.


I'm likely to include the results of these computations in an upcoming paper, so, for comparison, I would also like to mention what people have done previously in the area. (Also, I'd like to check that the computations are consistent with one another.)


**Question**: How far have others computed the Tutte polynomial of the complete graph?



How many Tutte polynomials of complete graphs are known?
Our program "tutte" (<http://homepages.ecs.vuw.ac.nz/~djp/tutte/#download>) can compute the TP of $K\_{18}$ in 160s (on a recent machine with an i7).


However you wouldn't want to do it with this sort of program that works on general graphs, because the complete graph is special and by tackling it symbolically you can go much further. 


I can do $K\_{40}$ in under a minute with Maple.


Just FYI, some quick checks on consistency can be done by evaluating $T(1,1)$ and making sure it is equal to the number of spanning trees, and then $T(2,2)$ and making sure it is equal to $2^{n(n-1)/2}$.


