Quick version: is there a definitive definition of weakly parsimonious counting reduction for #P?
Longer version: I am doing a gadget based reduction for NPC and would like to use it for #P. The reduction definitely isn't strongly parsimonious and to keep it simpler I'd like to avoid making it so.
Given the number of solutions to the reduced class, I can get a function which gives me the number of solutions to the original, IF I am allowed to put the original instance into the counting translation function. The number of solutions to the original is going to be something like the number of solutions to the translated problem divided by 2^n, where n is the number of variables in the original SAT problem.
Some definitions I've found on the web allow this and some don't, so I want to know if there is an important difference.