There is no such jQuery setting called rm, is there? That is just a query parameter, so treat it is as such, as you tried in your second attempt. The reason why i believe someurl.pl?rm=my_runmode is not working for you is because your Ajax call uses POST, not GET. Failure tends to happen when you mix your HTTP methods. Try passing rm via POST.

L-LL-L--L-LL-L--L-LL-L--
-R--R-RR-R--R-RR-R--R-RR
B--B--B--B--B--B--B--B--
H---H---H---H---H---H---
(the triplet paradiddle with high-hat)