Yes, because it is called $#{$valid{$key}}. To pick apart, $# gets the index of the last element of an array. The following curlies imply we want to get it from an arrayref. And inside them, we have $valid{$key} which is standard fare.

However your approach is not good Perl. You're iterating over a list; you shouldn't do that with an index variable unless you really need the index number.

foreach my $key (keys %valid) {
    foreach my $element (@{$valid{$key}}) {
        print STDERR "$key: $element\n";
    }
}
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And in this case, since you're not sorting the keys or otherwise in need of a specific order, you could use the more efficient each %hash function:

while(my($key, $val) = each %valid) {
    foreach my $element (@$val) {
        print STDERR "$key: $element\n";
    }
}
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Makeshifts last the longest.

#!/usr/bin/perl -w

use strict;

my %valid=(
        first   => [1..8],
        second  => [1..3],
        third   => [1..6]
);

for(keys %valid){

        print "the length of \$valid{$_} is ", 
                 scalar @{$valid{$_}}, 
                 "\n";
}
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the length of $valid{first} is 8
the length of $valid{third} is 6
the length of $valid{second} is 3
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Beware though that this method will not return the number of elements in the array, only the last index position. If the array only has one element,  $#{$valid{$key}} will return 0.

-- vek --