Parentheses... Again – Rich 4Clojure Problem 195 – See: https://github.com/PEZ/rich4clojure

rich4clojure_medium_problem_195.clj

(ns rich4clojure.medium.problem-195
  (:require [hyperfiddle.rcf :refer [tests]]))

;; = Parentheses... Again =
;; By 4Clojure user: djtrack16
;; Difficulty: Medium
;; Tags: [math combinatorics]
;; 
;; In a family of languages like Lisp, having balanced
;; parentheses is a defining feature of the language.
;; Luckily, Lisp has almost no syntax, except for these
;; "delimiters" -- and that hardly qualifies as "syntax",
;; at least in any useful computer programming sense.
;; 
;; 
;; It is not a difficult exercise to find all the
;; combinations of well-formed parentheses if we only have
;; N pairs to work with. For instance, if we only have 2
;; pairs, we only have two possible combinations: "()()"
;; and "(())". Any other combination of length 4 is
;; ill-formed. Can you see why?
;; 
;; 
;; Generate all possible combinations of well-formed
;; parentheses of length 2n (n pairs of parentheses). For
;; this problem, we only consider '(' and ')', but the
;; answer is similar if you work with only {} or only [].
;; 
;; 
;; There is an interesting pattern in the numbers!

(def __ :tests-will-fail)

(comment
  
  )

(tests
  [#{""} #{"()"} #{"()()" "(())"}] := (map (fn [n] (__ n)) [0 1 2])
  #{"((()))" "()()()" "()(())" "(())()" "(()())"} := (__ 3)
  16796 := (count (__ 10))
  (nth (sort (filter #(.contains ^String % "(()()()())") (__ 9))) 6) := "(((()()()())(())))"
  (nth (sort (__ 12)) 5000) := "(((((()()()()()))))(()))")

;; To participate, fork:
;; https://github.com/PEZ/rich4clojure

;; Post your solution below, please!