It's not clear what exactly you're after, but my take is that you want to know the offsets within $string where $den occurs. The following code does just that (I've made $string shorter for posting purposes).

my @chars = ('A','D');
my $string = join '', map $chars[ rand @chars ], 1 .. 20;
my $den = 'DA';
my @positions;
while ( $string =~ /$den/g ) {
    push @positions, pos( $string ) - length $den;
}
if ( @positions ) {
    print "'$den' was found at offsets @positions\n";
}
else {
    print "DA is not present\n\n\n";
}

print +(0 .. 9) x 5, "\n";
print "$string\n";
[download]

Sample run:

'DA' was found at offsets 2 5 7 10
01234567890123456789012345678901234567890123456789
ADDAADADADDAAAAAAAAA
[download]

Notes

• pos returns the offset of the end of the match
• while is used to loop over the string
• I used a /g modifier to the m// operator to search the string. See perlre and perlreftut