I do not know too much about the topic, but Valiant also defined the VNP classes: http://delivery.acm.org/10.1145/810000/804419/p249-valiant.pdf?key1=804419&key2=2034149521&coll=GUIDE&dl=GUIDE&CFID=65423911&CFTOKEN=83322167
There is also a notion of reducibility there and PERMANENT (with many other counting problems is complete) while the number of colorings is not there, so probably there is no simple reduction but unfortunately I am really not familiar with these notions.