@FLYERS1240,
My inclination is to simply pay half, except for expenses related to the children. She should be okay with in view of the child support you mentioned on the other thread.
If you wanted to put a really sharp pencil to the situation, exclude child suport from the equation, and allocate expenses on the basis of income. That is, if you have 60% of the partnership's income, pay 60% of the expenses. Don't quibble over the children's share of the water bill, or whatever. Remember that we are excluding child support payments from her income.
Alternatively, you may essentially charge the children their fair share of rent, utilities, etc. If you go this route, it is no longer fair to exclude their support payments from income. > Adding the income adds the related expense.
Again, I would just go the 50 - 50 answer and not worry about it. If it's a big deal, you might consider another home.