import Data.Ord
import Data.List
import Data.Function

dice n                  = [1..n]

combinations xs ys      = concat $ map (\y -> map (\x -> x : y) xs) ys


diceCombinations n k    = foldr (combinations) (map return diceN) (replicate (k - 1) diceN)
    where
        diceN           = dice n

mapFst f xs             = map (\(x, y) -> (f x, y)) xs

isEq :: Ordering -> Bool
isEq EQ                 = True
isEq _                  = False

elementCounts :: (a -> a -> Ordering) -> [a] -> [(a, Int)]
elementCounts f xs      = map (\ys -> (head ys, length ys)) $ groupBy (\x y -> isEq $ f x y) $ sortBy f xs

diceModuloCounts k      = groupModCounts
    where
        diceCounts      = elementCounts compare $ map sum $ diceCombinations 6 k
        modCounts       = mapFst (`rem`k) diceCounts
        groupedMods     = groupBy ((==) `on` fst) $ sortBy (compare `on` fst) modCounts
        groupModCounts  = map (\l -> (fst $ head l, sum $ map snd l)) groupedMods