# Perl 6, 117 bytes

    say $_ <2??1!!permutations(+[(my@n=(^$_).join.comb)]).elems÷[*] ([*] 2..$_ for @n.classify(&unique).values)for lines

and in a more readable fasion

    for (lines) -> $number {
        say 1 and next if $number < 2;
        my @digits = (^$number).join.comb;
        my @duplicates = @digits.classify(&unique).values;
        my $unique_permutations = permutations(+@digits).elems ÷ [*] ([*] 2..$_ for @duplicates);
        say $unique_permutations;
    }