I'm quite sure that I don't understand your reference to "randomly select a string with four elements from a data frame with 1553 strings..." -- so, as in Kenosis reply above, I've worked with a string comprised of "A" .. "Z" and taken a couple additional steps to obtain the one-thousandth permutation ^*1 (shuffle) thereof:

#!/usr/bin/perl
use 5.016;
use warnings;
use List::Util qw/shuffle/;

my $runs  = 10^3;
my $range = 1553;
my @letterRange = ( 'A' .. 'Z' );
my $i = 0;
my (@array,  @arr2, $key, $valref, $permutes);
my %permutes = ();

for ($i=1;$i<=$range;$i++) {       # C-style, start w/"1"; "0" would g
+ive 1554 elements
    @array = shuffle @letterRange for 1 ..$runs;   # @arr will hold th
+e 1000th shuffle
    $key = "@array";
    $permutes {$key } = $i;
}

for my $key ( keys %permutes ) {
        my $value = $permutes{$key};
        push @arr2, "$value => $key";
}

print "size of hash:  " . keys( %permutes ) . ".\n";

no warnings 'numeric';
my @sorted = sort {$a <=> $b} @arr2;
use warnings;

for $_ (@sorted) {
    say $_;
}
[download]

The output looks like this:

size of hash:  1553.
1 => V Y N H D R W Q C B E F T L S I J M O U Z A G X P K
2 => W E C B M V P I Y U D K G A T Z R O N F S J L X Q H
3 => L A T Y N R C Z J O V U G Q H F B S K W I E X P M D
...
[download]

... no dupes, no missing iteration numbers found but a selection of the output is in the readmore which follows:

4 => G O R J Z B I V Q D H W E N U L T F K X A M C P S Y
5 => M W E F R T K H N Y D Q P S A C O Z B V U J I G X L
6 => D Q B M C P A O I U H J W G S Z T L Y V N K R F E X
7 => I C F P M U K T W E D Z V H A X S Q O J G B Y N R L
8 => B P Y R Q C U T X L Z J F K I D V H M O W A E N G S
9 => C D V X A P T F S Y E W H G K U Q N B Z I O M R L J

... (Including all the output led to truncation of this node)

1550 => G M V L S P H F A J D Y I E X W K R B T O Z U Q N C
1551 => Y A J X K Q S O F U Z C W E H V B G I R N M L T D P
1552 => F W H T R S K E M U X D O I B N A J L Z G V P Y C Q
1553 => X V E F J N W A Z C G H R K O Q U P L I Y M B D T S

^*1: Reiterating, I doubt this is the solution to your problem -- as noted above, I simply don't understand a part of your problem statement (and the code doesn't illuminate the meaning for /me) but I do hope this may offer an approach to your specific case.