Wash the potatoes, but do not peel them (they may be sliced into large-as-possible chunks if necessary in order to fit the pot).
Boil in salted water to cover until barely soft -- about 15-20 minutes. Drain, cool and peel.
Slice potatoes (about 1/2 inch thick) into a buttered casserole, sprinkling in the diced apples as you go (about a 6-8 cup casserole or baking pan is fine).
In a small saucepan, bring the brown sugar, cider, butter and raisins to a low boil, then add the rum. (If desired, you can then bring the mixture to a high boil for a minute or two to boil off the alcohol, but it's optional.)
Pour the rum mixture over the potatoes.
Sprinkle the chopped pecans over the top and poke them between the potatoes.
Bake uncovered at 350 for 30 minutes, basting two or three times with the syrup in the pan.
Serve hot from the oven, or cover with tin foil and put in a box with newspapers crumpled around to maintain the heat, transport it to your next potluck dinner and serve warm.
Leftovers are great.