Just off the top of my head, this makes me think of $c-$p nested loops to pick slots for the surplus cards like:
my $p = 7; # Number of pigeon holes.
my $c = 12; # Number of cards
for my $s1 ( 1..$p ) {
for my $s2 ( $s1..$p ) {
for my $s3 ( $s2..$p ) {
... Need $c-$p loops
Which translates to simple code if I don't worry about being more efficient, but it also handles your 7,12 "instantly" for me.
#!/usr/bin/perl -w
use strict;
use Algorithm::Loops 'NestedLoops';
my( $p, $c ) = @ARGV;
my $iter = NestedLoops(
[ [ 1..$p ], ( sub { [ $_..$p ] } ) x ($c-$p-1) ],
);
my @s;
while( @s = $iter->() ) {
my @p = (1) x $p;
$p[$_-1]++ for @s;
print "@p\n";
}