Preheat oven to 350 degrees. Line a standard muffin pan with papers or grease the pan.
In a dish with a sieve, drain the pineapple and shredded apple, pressing as much juice as possible into the dish. Set the fruit aside and pour the juice into a small saucepan. Bring the juice to a medium boil, and cook it down to ¼ cup. Allow the juice to cool.
Melt the stick of butter and allow it to cool as well.
Meanwhile, in a bowl combine the flour, sugar, baking soda, baking powder, salt, and cinnamon. Set aside.
In a mixing bowl, whisk together the melted butter, juice, eggs, and vanilla until combined.
Gradually mix in the dry ingredients, in two batches, until just mixed.
Stir in the chopped pecans, carrots, pineapple, and apple.
Scoop the batter into the muffin tin. It is okay to fill the cups, as these do not rise a great deal.
Bake 22-26 minutes, or until a toothpick inserted in the center comes out clean.