Thanks, very nice to see that open-source pari can do that.

I found maple somewhere and got Z12 := 6.1338480888612326740223759609725595265762100140504896275875‌​38049554517487250070‌​39482363166526364236‌​7 - 0. I

Nice, thanks a lot. Too bad that I cannot reproduce the strategy with my open-source tools, not able to perform the inner symbolic summation. Would you be so kind to provide me with the value of $Z(1,2)$ too ?

Thanks. Björner article is here : doi.org/10.1016/S0012-365X(81)80027-1

Well, one can certainly wonder about the asymptotic growth of the numbers of configurations on the n times n grid. I would guess the standard method would be to use the transfer matrix method and the powerful techniques of the australian school of combinatorial enumeration to guess asymptotics from the initial terms.