[Haskell-cafe] Course-of-value recursion by defining a sequence as a self-referential infinite list

[Vanessa McHale]

I came up with a one-liner for computing coefficients of the generating function for integer partitions:

part :: Int → [Integer ]
part n = take n $ product [cycle (1 : replicate n 0) | n ← [0 . . (n − 2)]]

Karczmarczuk’s solution via the Haskell prelude:

part = 1 : b 1
  where b n = (1 : b (n + 1)) + (replicate n 0 ++ b n)

cycle and replicate are really underrated!

> On Jan 20, 2025, at 8:54 AM, Douglas McIlroy <douglas.mcilroy at dartmouth.edu> wrote:
> 
> > catalanNumbers :: Num a => [a]
> > catalanNumbers =
> >   let xs = 1 : PowerSeries.mul xs xs
> >   in  xs
> 
> This example of a generating function come to life as a program deserves to be better known. Bill Burge presented it 50 years ago in "Recursive Programming Techniques", Addison-Wesley, 1975. I revisited it in "Power series, power serious", JFP 9 (1999) 323-335, where, with overloadied arithmetic, it became
>       ts = 1 : ts^2
> The technique is laid bare in ten one-liners at https://www.cs.dartmouth.edu/~doug/powser.html.
> 
> Doug
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