IMO that is just not good enough of your partner. You are not his slave!!!
My DH doesn't do heaps but he definitely does help out. He usually does the rubbish and bins. He usually baths Flynn while I wash the dishes. He usually puts the dishes away in the morning when they are dry. He mows the lawns and looks after our cars. He occasionally does the laundry (but only his own clothes cos I'm a bit pedantic about the laundry!!) and will bring in washing if I ask him to. He doesn't cook but that is not one of his strong points! I don't expect too much of him housework wise because he works outside of the house so that is generally my domain. I feel that we have a happy medium going on.
Perhaps you could ask him to do a few things (at leat the bins for a start). I don't think guys tend to see things which need doing and do them. They need to be alerted to the fact that something needs doing and you would appreciate them doing it!
HTH