Find a balanced Neighborhood

Question

Your task will be to take a balanced-string and a integer representing a Levenshtein distance (the number of characters that have to be inserted, deleted or changed to make one string into another) and you must find the number of balanced strings with that distance from the original string (i.e. the neighborhood of that string).

Stipulations

• Balanced strings will consist only of the characters ()<>[]{}

• You will only be asked to find neighborhoods for positive even distances

• Input and output is flexible. As long as you take in all the proper data and output the correct answer without violating any loopholes I am happy with your answer.

• You may choose to divide all of your integer inputs by 2 if you choose.

• This is code-golf so the objective is to minimize the number of bytes in your answer

This was inspired by this CMC and this answer

Testcases

   Case   | Distance | Size of Neighborhood
--------------------------------------------
    ()    |    2     |         18
   ({})   |    2     |         33
   (())   |    2     |         32
    <>    |    4     |        186
   [][]   |    4     |        688
  <(){}>  |    4     |        1379
    {}    |    6     |        2270
  []{}[]  |    6     |        41097

Here are a few small examples with the actual neighborhoods included:

(), 2 :
{'', '<>', '()[]', '()()', '(())', '([])', '()<>', '{}', '{()}', '<>()', '(){}', '{}()', '<()>', '(<>)', '[()]', '[]()', '({})', '[]'}

---

({}), 2 :
{'([]{})', '()', '{}', '<({})>', '({<>})', '<{}>', '({()})', '(<>{})', '({}<>)', '({[]})', '(({}))', '({{}})', '({}[])', '{({})}', '({})()', '{}({})', '(())', '()({})', '([])', '<>({})', '({}{})', '({}){}', '({})<>', '(<{}>)', '({})[]', '((){})', '[{}]', '{{}}', '[]({})', '(<>)', '({}())', '([{}])', '[({})]'}

---

(()), 2 :
{'(())[]', '<>(())', '()', '{}(())', '{()}', '({()})', '{(())}', '(([]))', '(({}))', '(()[])', '(())<>', '((()))', '([])', '((<>))', '()(())', '(<()>)', '([()])', '[(())]', '(()){}', '(())()', '(()())', '(<>())', '(()<>)', '((){})', '<(())>', '<()>', '([]())', '(<>)', '({}())', '[()]', '({})', '[](())'}

---

<>, 4 :
{'<><<>>', '(<>)<>', '[<>][]', '<<><>>', '(){<>}', '(<>)()', '[<()>]', '<({})>', '<>()<>', '<[<>]>', '[][]<>', '<>[]<>', '<><><>', '[]<{}>', '[]<<>>', '[]<><>', '{<><>}', '[{<>}]', '<(<>)>', '(())<>', '{}<>{}', '()(<>)', '{()<>}', '(())', '{<>{}}', '(<><>)', '([])<>', '[]<[]>', '<{}<>>', '<><()>', '{()}<>', '{{}}<>', '{<>()}', '<<>>()', '{<<>>}', '<()>()', '<[]>()', '<>[<>]', '(<>())', '{}<>()', '(()<>)', '[{}]', '{{}}', '[]()', '[(<>)]', '<{}[]>', '<<>>[]', '{}<()>', '<>', '[()]<>', '<()><>', '[[]]<>', '[{}]<>', '[]<>[]', '()[<>]', '[]<>()', '{<>}{}', '{<[]>}', '<>(<>)', '(<>)[]', '<{}>()', '{}<><>', '{<>}()', '{[]}', '{[]}<>', '<<<>>>', '[]<()>', '<<[]>>', '<<{}>>', '[[]]', '()()<>', '[]{<>}', '<><[]>', '[[]<>]', '<{}()>', '<{<>}>', '<[]{}>', '{}<{}>', '<{}>[]', '()<<>>', '(<()>)', '[]{}', '{{}<>}', '{}()', '()<>[]', '<{}><>', '{[<>]}', '<><{}>', '<(())>', '<><>{}', '[()]', '<<>>{}', '{}{}<>', '[<<>>]', '<[][]>', '(<<>>)', '<[]><>', '[<>]<>', '[<>[]]', '[{}<>]', '{()}', '{<>[]}', '[]{}<>', '{(<>)}', '(<[]>)', '()[]<>', '<>{<>}', '{[]<>}', '(<>{})', '({}<>)', '[<><>]', '<><>()', '{}[<>]', '<{[]}>', '<<()>>', '<<>{}>', '([<>])', '<[]()>', '()()', '([])', '[[<>]]', '((<>))', '[](<>)', '(){}<>', '[()<>]', '<([])>', '<()()>', '[][]', '<<>[]>', '[<[]>]', '({})<>', '<{{}}>', '<[{}]>', '<{}{}>', '{}(<>)', '<<>><>', '[<>()]', '[][<>]', '({})', '{}[]<>', '{}<[]>', '<[()]>', '()[]', '<()>[]', '{{<>}}', '(<>){}', '{}{}', '({<>})', '{<()>}', '{}{<>}', '[]()<>', '<[]>[]', '(<>[])', '<[]>{}', '{}()<>', '()<[]>', '()<{}>', '{}<<>>', '<{}>{}', '{}[]', '()<>{}', '<()<>>', '[<>{}]', '{<>}[]', '<<>()>', '<><>[]', '{<{}>}', '<()[]>', '()<><>', '[<>]()', '()<>()', '{}<>[]', '<{()}>', '(<{}>)', '(){}', '()<()>', '<(){}>', '{<>}<>', '<[[]]>', '[]<>{}', '([]<>)', '<[]<>>', '[<>]{}', '<()>{}', '<>{}<>', '[<{}>]'}

Answer 1

Mathematica, 187 173 bytes

Length@Union@Select[""<>#&/@(Tuples[Characters@" ()[]<>{}",StringLength@#+#2]/." "->""),sFixedPoint[StringReplace["()"|"[]"|"{}"|"<>":>""],s]==""&&EditDistance[s,#]==#2]&

Brute force pure function. # represents the first argument (starting string) and #2 represents the second argument (distance).

Characters@" ()[]<>{}" is the list of possible characters (including " ")

Tuples[Characters@" ()[]<>{}",StringLength@#+#2] is the list of all tuples of those characters with length at most the original string length plus the distance.

Tuples[Characters@" ()[]<>{}",StringLength@#+#2]/." "->"" replaces all of the " " characters with the empty string.

""<>#&/@(...) joins all of those character lists into strings.

Next we Select all such strings which are balanced and which have the appropriate EditDistance with the following function:

s                                                                                                 String s
                                                                                                 maps to
  FixedPoint[StringReplace["()"|"[]"|"{}"|"<>":>""],s]                                              the fixed point of cancelling out pairs of brackets
                                                      ==                                             equals
                                                        ""                                          the empty string
                                                          &&                                        and
                                                            EditDistance[s,#]==#2                   the distance from s to # is #2

Next we use Union to delete the duplicates and take the Length.