`[[[[]]]], [[[][]]], [[[]][]], [[][[]]], [[[]]][], [[][][]], [][[[]]],
 [[][]][], [[]][[]], [][[][]], [[]][][], [][[]][], [][][[]], [][][][]`

nest[n_] :=
DeleteCases[Quiet[ToExpression /@ (StringReplace[#, "}{" -> "},{"] & /@
StringJoin /@ Permutations[Flatten@Array[{"{", "}"} &, n], {2 n}])], $Failed]

DyckWordQ[ s_String ] := With[
    {
        f = StringReplace[
              {
                  Shortest[ "[" ~~ Whitespace ~~ "]" ] -> "",
                  "[]" -> ""
              }
        ]
    },
    If[
        FixedPoint[ f, s ] === "",
        True,
        False,
        False
    ]
]

DyckWord[ n_Integer ] /; EvenQ[n] := With[
    {
        p = Permutations[ ConstantArray[ 0, n/2 ] ~ Join ~ ConstantArray[ 1,      n/2]
    },
    p // RightComposition[
        ReplaceAll[ { 0 -> "[", 1 -> "]" } ],
        Map[ StringJoin ],
        Select[#, DyckWordQ] &
    ]
]

DyckWord[8]

d = 16
If[EvenQ[d], StringJoin @@@ (Pick[#, Sign@*Min@*Accumulate /@ #, 0] &[
 Permutations[Join[#, -#] &[ConstantArray[1, d/2]]]] /. {-1 -> "]", 1 -> "["})]

ClearAll[f]

f[n_] := f[n, 1, 0, "["]

f[n_, n_, c_, r_] := {r <> ConstantArray["]", n - c]}

f[n_, o_, c_, r_] /; c < o :=
  f[n, o + 1, c, r <> "["] ~Join~ f[n, o, c + 1, r <> "]"]

f[n_, o_, c_, r_] := f[n, o + 1, c, r <> "["]

f[4]

{"[[[[]]]]", "[[[][]]]", "[[[]][]]", "[[[]]][]",
 "[[][[]]]", "[[][][]]", "[[][]][]", "[[]][[]]",
 "[[]][][]", "[][[[]]]", "[][[][]]", "[][[]][]",
 "[][][[]]", "[][][][]"}

(* D24 *)

f[12] // Length // RepeatedTiming

{1.15, 208012}

{6.87, 208012}

f[u : {a_, x___}, v : {b_, y___}, c___] := f[{x}, v, c, a] ⋃ f[u, {y}, c, b]

f[{x___}, {y___}, c___] := {{c, x, y}}

f2[a_, b_] := Union @@ (f[#, b] & /@ a)

d = 8;

StringJoin /@ Fold[f2, {{}}, Table[{"[", "]"}, {d/2}]]

{"[[[[]]]]", "[[[][]]]", "[[[]][]]", "[[[]]][]",
 "[[][[]]]", "[[][][]]", "[[][]][]", "[[]][[]]",
 "[[]][][]", "[][[[]]]", "[][[][]]", "[][[]][]",
 "[][][[]]", "[][][][]"}