Using monads for trivial tasks like list manipulation?

Question

Whenever I read about Monad example, they always present IO as a case study.

Are there any examples of monads doing list manipulation which somebody could present? I apr

Answer 1

Another example is constructing all the divisors of a number from its prime factorization (though out of order):

divs n = map product 
           . mapM (\(p,n)-> map (p^) [0..n]) 
           . primeFactorization $ n

-- mapM f = sequence . map f

-- primeFactorization 12  ==>  [(2,2),(3,1)]  -- 12 == 2^2 * 3^1

The function f in mapM f == sequence . map f produces a list of powers of a factor, for each entry in the input list; then sequence forms all paths through the lists picking one number from each at a time; then this list of all possible combinations is fed to map product which calculates the divisors:

12                                      -- primeFactorization:
[(2,2),(3,1)]                           -- map (\(p,n)-> ...):
[ [1,2,4], [1,3] ]                      -- sequence:
[[1,1],[1,3],[2,1],[2,3],[4,1],[4,3]]   -- map product:
[1,3,2,6,4,12]                          --   the divisors of 12

The great description given in Luis Casillas's answer applies here too:

In the context of the list monad, mapM :: (Monad m) => (a -> m b) -> [a] -> m [b] is a nondeterministic list map: a mapping operation that maps a nondeterministic function - such that produces a list of alternative results - over a list, creating all the alternative possible lists of results.