Parentheses sequences in lexicographical order

Question

^{Challenge Taken from here and also here}

An n parentheses sequence consists of n (s and n )s.

A valid parentheses sequence is defined as the following:

You can find a way to repeat erasing adjacent pair of parentheses "()" until it becomes empty.

For example, (()) is a valid parentheses, you can erase the pair on the 2nd and 3rd position and it becomes (), then you can make it empty. )()( is not a valid parentheses, after you erase the pair on the 2nd and 3rd position, it becomes )( and you cannot erase any more

Task

Given a number n you need to generate all correct parenthesis sequence in lexicographical order

Output can be an array, list or string (in this case a sequence per line)

You can use a different pair of parenthesis such as {}, [], () or any open-close sign

Example

• n = 3

  ((()))    
  (()())    
  (())()    
  ()(())    
  ()()()
  

• n = 2

  (())
  ()()
  

Answer 1

Perl 6, 36 bytes

{grep {try !.EVAL},[X~] <[ ]>xx$_*2}

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Finds all lexographically sorted combinations of \$2n\$ []s and filters the ones that EVAL correctly. Note that all valid combinations (even stuff like [][]) evaluate to [] (which is falsey, but we not it (!) to distinguish from try returning Nil)

Explanation:

{                                  }  # Anonymous code block
                        <[ ]>         # Create a list of ("[", "]")
                             xx$_*2   # Duplicate this 2n times
                   [X~]               # Find all possible combinations
 grep {          },                   # Filter from these
            .EVAL                     # The EVAL'd strings
       try !                          # That do not throw an error

Answer 2

R, 112 107 99 bytes

Non-recursive approach. We use "<" and ">" because it avoids escape characters in the regex. To allow us to use a shorter specification for an ASCII range, we generate 3^2n 2n-character strings of "<", "=" and ">" using expand.grid (via their ASCII codes 60, 61 and 62) and then grep to see which combinations give balanced open and close brackets. The "=" possibilities will get ignored, of course.

Via http://rachbelaid.com/recursive-regular-experession/

function(n)sort(grep("^(<(?1)*>)(?1)*$",apply(expand.grid(rep(list(60:62),2*n)),1,intToUtf8),,T,T))

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Explanation

"^(<(?1)*>)(?1)*$" = regex for balanced <> with no other characters
^ # match a start of the string
  ( # start of expression 1
    < # open <>
       (?1)* # optional repeat of any number of expression 1 (recursive)
  # allows match for parentheses like (()()())(()) where ?1 is (\\((?1)*\\))
    > # close <>
  ) # end of expression 1
  (?1)* # optional repeat of any number of expression 1
$ # end of string

function(n)
  sort(
    grep("^(<(?1)*>)(?1)*$", # search for regular expression matching open and close brackets
      apply(
        expand.grid(rep(list(60:62),2*n)) # generate 3^(2n) 60,61,62 combinations
      ,1,intToUtf8) # turn into all 3^(2n) combinations of "<","=",">"
    ,,T,T) # return the values that match the regex, so "=" gets ignored
  ) # sort them

R, 107 bytes

Usual recursive approach.

-1 thanks @Giuseppe

f=function(n,x=0:1)`if`(n,sort(unique(unlist(Map(f,n-1,lapply(seq(x),append,x=x,v=0:1))))),intToUtf8(x+40))

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Answer 3

Python 2, 91 88 84 81 bytes

f=lambda n:n and sorted({a[:i]+'()'+a[i:]for a in f(n-1)for i in range(n)})or['']

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_{-3 bytes thanks to pizzapants184}

Answer 4

C (gcc), 114 bytes

f(n,x,s,a,i){char b[99]={};n*=2;for(x=1<<n;x--;s|a<0||puts(b))for(s=a=i=0;i<n;)a|=s+=2*(b[n-i]=41-(x>>i++&1))-81;}

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Should work for n <= 15.

Explanation

f(n,x,s,a,i){
  char b[99]={};   // Output buffer initialized with zeros.
  n*=2;            // Double n.
  for(x=1<<n;x--;  // Loop from x=2**n-1 to 0, x is a bit field
                   // where 1 represents '(' and 0 represents ')'.
                   // This guarantees lexicographical order.
      s|a<0||puts(b))  // Print buffer if s==0 (as many opening as
                       // closing parens) and a>=0 (number of open
                       // parens never drops below zero).
    for(s=a=i=0;i<n;)  // Loop from i=0 to n-1, note that we're
                       // traversing the bit field right-to-left.
      a|=              // a is the or-sum of all intermediate sums,
                       // it becomes negative whenever an intermediate
                       // sum is negative.
        s+=            // s is the number of closing parens minus the
                       // number of opening parens.
                        x>>i++&1   // Isolate current bit and inc i.
                    41-(        )  // ASCII code of paren, a one bit
                                   // yields 40 '(', a zero bit 41 ')'.
             b[n-i]=               // Store ASCII code in buffer.
          2*(                    )-81;  // 1 for ')' and -1 for '(' since
                                        // we're going right-to-left.
}

Answer 5

05AB1E, 13 bytes

„()©s·ãʒ®õ:õQ

Try it online or verify some more test cases.

Explanation:

„()            # Push string "()"
   ©           # Store it in the register without popping
    s·         # Swap to get the (implicit) input, and double it
      ã        # Cartesian product that many times
       ʒ       # Filter it by:
        ®      #  Push the "()" from the register
         õ:    #  Infinite replacement with an empty string
           õQ  #  And only keep those which are empty after the infinite replacement

Answer 6

Japt, 15 13 bytes

ç>i<)á Ôke/<>

Try it

---

Explanation

ç                 :Input times repeat the following string
 >i<              :  ">" prepended with "<"
    )             :End repeat
     á            :Get unique permutations
       Ô          :Reverse
        k         :Remove any that return true (non-empty string)
         e/<>     :  Recursively replace Regex /<>/

Answer 7

Ruby, 70 bytes

f=->n{n<1?['']:(0...n).flat_map{|w|f[n-1].map{|x|x.insert w,'()'}}|[]}

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Answer 8

K (ngn/k), 36 35 bytes

{"()"(&/-1<+\1-2*)#(x=+/)#+!2|&2*x}

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+!2|&2*x all binary vectors of length 2*n

(x=+/)# only those that sum to n

(&/-1<+\1-2*)# only those whose partial sums, treating 0/1 as 1/-1, are nowhere negative

"()" use 0/1 as indices in this string

Answer 9

JavaScript (ES6), 112 102 bytes

This is heavily based on nwellnhof's C answer.

f=(n,i)=>i>>2*n?'':(g=(o,k)=>o[2*n]?s|m<0?'':o:g('()'[m|=s+=k&1||-1,k&1]+o,k/2))(`
`,i,m=s=0)+f(n,-~i)

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Answer 10

Perl 6, 42 bytes

{grep {!S:g/\(<~~>*\)//},[X~] <( )>xx$_*2}

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Uses a recursive regex. Alternative substitution: S/[\(<~~>\)]*//

38 bytes with 0 and 1 as open/close symbols:

{grep {!S:g/0<~~>*1//},[X~] ^2 xx$_*2}

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Explanation

{                                        }  # Anonymous block
                              <( )>         # List "(", ")"
                                   xx$_*2   # repeated 2n times
                         [X~]  # Cartesian product with string concat
                               # yields all strings of length 2n
                               # consisting of ( and )
 grep {                },  # Filter strings
        S:g/             # Globally replace regex match
            \(           #   literal (
              <~~>       #   whole regex matched recursively
                  *      #   zero or more times
                   \)    #   literal )
                     //  # with empty string
       !                 # Is empty string?

Answer 11

Red, 214, 184 136 bytes

func[n][g: func[b n][either n = length? b[if not error? try[load b][print b]return 0][g append copy b"["n g append copy b"]"n]]g""2 * n]

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Uses Jo King's approach. Finds all possilble arrangements of brackes using recursion (they are generated in lexicographic order) and print it if the arrangement evaluates as a valid block.

Answer 12

Perl 5 -n, 41 39 bytes

-2 bytes with angle brackets

s/<(?R)*>//gr||say for glob'{<,>}'x2x$_

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Port of my Perl 6 answer.

Answer 13

Retina 0.8.2, 50 bytes

.+
$*
1
11
+%1`1
<$'¶$`>
Gm`^(?<-1>(<)*>)*$(?(1).)

Try it online! Uses <>s. Explanation:

.+
$*

Convert to unary.

1
11

Double the result.

+%1`1
<$'¶$`>

Enumerate all of the 2²ⁿ 2n-bit binary numbers, mapping the digits to <>.

Gm`^(?<-1>(<)*>)*$(?(1).)

Keep only balanced sequences. This uses a balanced parentheses trick discovered by @MartinEnder.

Answer 14

Jelly, 19 bytes

Ø+xŒ!QÄAƑ>ṪƊ$Ƈ=1ịØ(

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Output clarified over TIO.