There's no requirement that an "array" be regular, but chaining down the 0'th element of each subarray, we get:

my @dims = ();
my $thing = \@array;
while (ref $thing eq "ARRAY") {
  push @dims, scalar @$thing;
  $thing = $thing->[0];
}
print "@dims\n";
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-- Randal L. Schwartz, Perl hacker