module Bell where

import Data.MemoTrie
import Stirling

--  A000110
-- number of partitions of a set
-- also the number of n-types in the structure (N, =)
bell :: Int -> Integer
bell = memo bell0 where
  bell0 n = if n <= 12
            then table !! n
            else sum [ secondStirling n k | k <- [0 .. n] ]
  table = [1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597]

-- number of n-types in the structure (N, <=)
orderedBell :: Int -> Integer
orderedBell = memo ob where
  ob n = if n <= 10
         then table !! n
         else sum [ fac k * secondStirling n k | k <- [0 .. n] ]
  table = [1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563]
  fac k = product [1 .. fromIntegral k]