Finding all combinations of well-formed brackets

Question

This came up while talking to a friend and I thought I\'d ask here since it\'s an interesting problem and would like to see other people\'s solutions.

The task is to

Answer 1

F#:

UPDATE: this answer is wrong. My N=4 misses, for example "(())(())". (Do you see why?) I will post a correct (and more efficient) algorithm shortly.

(Shame on all you up-voters, for not catching me! :) )

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Inefficient, but short and simple. (Note that it only prints the 'nth' line; call in a loop from 1..n to get the output asked for by the question.)

#light
let rec brackets n =
    if n = 1 then
        ["()"]
    else
        [for s in brackets (n-1) do
            yield "()" ^ s
            yield "(" ^ s ^ ")"
            yield s ^ "()"]

Example:

Set.of_list (brackets 4) |> Set.iter (printfn "%s")
(*
(((())))
((()()))
((())())
((()))()
(()(()))
(()()())
(()())()
(())()()
()((()))
()(()())
()(())()
()()(())
()()()()
*)