TITLE: How many Tutte polynomials of complete graphs are known?
QUESTION [9 upvotes]: 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), 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?

REPLY [13 votes]: There is a simple formula for the generating function of $T_{K_n}(x,y)$, which is more cleanly expressed in terms of the (equivalent) "coboundary polynomial" $X_M(q,t) = (t-1)^{r(M)} T_M(1+\frac{q}{t-1}, t)$:
$$
1+q\sum_{n \geq 1} X_{K_n}(q,t) \frac{x^n}{n!} = \left( \sum_{n \geq 0} t^{n \choose 2}\frac{x^n}{n!}\right)^q.
$$
This is essentially due to Tutte in the 50s; see this paper and the references in it.
For what it's worth, using similar methods one easily obtains formulas for other similar families such as  complete bipartite graphs (for graphs) and classical root systems (for matroids).