julia> for i in 1:12
           n = 2^i
           print(lpad(n, 4)); @btime sortperm(x) setup=(x=rand($n));
       end
   2  228.013 ns (2 allocations: 96 bytes)  =>    2  503.767 ns (4 allocations: 128 bytes)
   4  237.002 ns (2 allocations: 112 bytes) =>    4  521.963 ns (4 allocations: 144 bytes)
   8  260.322 ns (2 allocations: 144 bytes) =>    8  550.038 ns (4 allocations: 176 bytes)
  16  326.942 ns (2 allocations: 208 bytes) =>   16  742.277 ns (5 allocations: 432 bytes)
  32  481.206 ns (2 allocations: 352 bytes) =>   32  1.200 μs (5 allocations: 720 bytes)
  64  850.224 ns (2 allocations: 592 bytes) =>   64  2.091 μs (5 allocations: 1.17 KiB)
 128  2.159 μs (2 allocations: 1.08 KiB)    =>  128  4.026 μs (5 allocations: 2.17 KiB)
 256  5.409 μs (2 allocations: 2.14 KiB)    =>  256  8.060 μs (5 allocations: 4.30 KiB)
 512  17.091 μs (2 allocations: 4.14 KiB)   =>  512  15.894 μs (5 allocations: 8.30 KiB)
1024  44.933 μs (2 allocations: 8.14 KiB)   => 1024  33.661 μs (5 allocations: 16.30 KiB)
2048  99.723 μs (2 allocations: 16.14 KiB)  => 2048  71.643 μs (5 allocations: 32.30 KiB)
4096  220.581 μs (3 allocations: 32.06 KiB) => 4096  154.905 μs (7 allocations: 64.14 KiB)

julia> VERSION
v"1.8.3" => v"1.10.0-DEV.91"