My Haskell implementation represents numbers as the ratio of products of ordered integer streams. For example, I represent 3!/(4*5) as (R numerator=[1,2,3] denominator=[4,5]). In this representation, multiplication becomes merging the numerator and denominator streams and then canceling the first stream by the second. In this way I can remove all cancelable original terms in the
P_{cutoff} formula before finally multiplying the terms that remain.
*FishersExactTest> fac 6
R {numer = [2,3,4,5,6], denom = []}
*FishersExactTest> fac 3
R {numer = [2,3], denom = []}
*FishersExactTest> fac 6 `rdivide` fac 3
R {numer = [4,5,6], denom = []}
Here's the example from the MathWorld page:
*FishersExactTest> rpCutoff [ [5,0],
[1,4] ]
R {numer = [2,3,4,5], denom = [7,8,9,10]}
*FishersExactTest> fromRational . toRatio $ it
2.3809523809523808e2
The code:
module FishersExactTest (pCutoff) where
import Data.Ratio
import Data.List (transpose)
pCutoff = toRatio . rpCutoff
rpCutoff rows =
facproduct (rs ++ cs) `rdivide` facproduct (n:xs)
where
rs = map sum rows
cs = map sum (transpose rows)
n = sum rs
xs = concat rows  cells
facproduct = rproduct . map fac
fac n  n < 2 = runit
 otherwise = R [2..n] []
 I represent numbers as ratios of products of integer streams
 R [1,2,3] [4,5] === (1 * 2 * 3) / (4 * 5)
data Rops = R { numer :: [Int], denom :: [Int] } deriving Show
runit = R [] []  the number 1
toRatio (R ns ds) = bigProduct ns % bigProduct ds
bigProduct = product . map toInteger
 multiplication is merging numerator and denominator streams
 and then canceling the first by the second
rtimes (R xns xds) (R yns yds) =
uncurry R $ (merge xns yns) `cancel` (merge xds yds)
rproduct = foldr rtimes runit
 division is multiplication by the inverse
rdivide x (R yns yds) = rtimes x (R yds yns)
 helpers
merge (x:xs) (y:ys)
 x < y = x : merge xs (y:ys)
 otherwise = y : merge (x:xs) ys
merge [] ys = ys
merge xs [] = xs
cancel (x:xs) (y:ys)
 x == y = cancel xs ys
 x < y = let (xs', ys') = cancel xs (y:ys) in (x:xs', ys')
 otherwise = let (xs', ys') = cancel (x:xs) ys in (xs', y:ys')
cancel xs ys = (xs, ys)

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