Step by Step solution about How to resolve the algorithm Bell numbers step by step in the Haskell programming language

Explanation:

Bell's Triangle

Bell's Triangle is a triangular array of numbers with the following properties:

• The left and right edges contain 1s.
• Each number in the triangle is the sum of the two numbers above it.
• The nth row of the triangle contains the nth Bell number, which counts the number of ways to partition a set of n elements into non-empty subsets.

Haskell Implementation

The provided Haskell code defines a function bellTri that generates Bell's Triangle and a function bell that extracts the Bell numbers from the triangle.

Detailed Explanation:

bellTri :: [[Integer]]

This is the main function that generates Bell's Triangle. It returns a list of lists of integers, representing the rows of the triangle.

bellTri =
 let f xs = (last xs, xs)
  in map snd (iterate (f . uncurry (scanl (+))) (1, [1]))

• The code uses a nested let expression to define a helper function f and the main computation.
• f xs is a function that takes a list xs and returns a tuple containing the last element of xs and the original list itself.
• uncurry (scanl (+)) combines two lists element-wise, using the + function as the binary operator. This function accumulates the sums of the elements in each row of the triangle.
• iterate (f . uncurry (scanl (+))) iteratively applies f to the result of uncurry (scanl (+)), starting with the initial value (1, [1]).

bell :: [Integer]
bell = map head bellTri

This function extracts the Bell numbers from the triangle. It returns a list of integers, where each element is the first (leftmost) number in a row of Bell's Triangle.

main :: IO ()
main = do
 putStrLn "First 10 rows of Bell's Triangle:"
 mapM_ print (take 10 bellTri)
 putStrLn "\nFirst 15 Bell numbers:"
 mapM_ print (take 15 bell)
 putStrLn "\n50th Bell number:"
 print (bell !! 49)

This function is the program entry point:

• It prints the first 10 rows of Bell's Triangle.
• It prints the first 15 Bell numbers.
• It prints the 50th Bell number.

Alternative Implementations

The code also includes alternative implementations of bellTri using different Haskell idioms:

• The first alternative uses the Control.Arrow module to combine functions.
• The second alternative uses the Control.Applicative module to apply functions to values within a data structure (here, a list of lists).
• The third alternative uses the operator .<*> (infix application) to combine functions.

bellTri :: [[Integer]]
bellTri =
  let f xs = (last xs, xs)
   in map snd (iterate (f . uncurry (scanl (+))) (1, [1]))

bell :: [Integer]
bell = map head bellTri

main :: IO ()
main = do
  putStrLn "First 10 rows of Bell's Triangle:"
  mapM_ print (take 10 bellTri)
  putStrLn "\nFirst 15 Bell numbers:"
  mapM_ print (take 15 bell)
  putStrLn "\n50th Bell number:"
  print (bell !! 49)


import Control.Arrow

bellTri :: [[Integer]]
bellTri = map snd (iterate ((last &&& id) . uncurry (scanl (+))) (1,[1]))


import Control.Applicative

bellTri :: [[Integer]]
bellTri = map snd (iterate ((liftA2 (,) last id) . uncurry (scanl (+))) (1,[1]))


bellTri :: [[Integer]]
bellTri = map snd (iterate (((,) . last <*> id) . uncurry (scanl (+))) (1, [1]))