use List::Util 'min';
use POSIX 'ceil';

sub num_ways {
    my ($N, $S, $T) = @_;

    return 1 if $S == 0 and $T == 0;
    return 0 if $N < 1 or $S < 1 or $T < 1;

    my $min = (2*$T-$S+$S**2)/(2*$S); ## more correctly, ceil() of this
    my $max = min( $T-(($S-1)*$S/2), $N );

    my $sum = 0;
    for my $K ($min .. $max) {
        $sum += num_ways( $K-1, $S-1, $T-$K );
    }
    return $sum;
}

use Memoize;
memoize 'num_ways';
print num_ways(100, 10, 667), $/;

##</code><code>##

sub num_ways {
    my ($N, $S, $T, $callback, @so_far) = @_;

    return $callback->(@so_far) if $S == 0 and $T == 0;
    return if $N < 1 or $S < 1 or $T < 1;

    my $min = (2*$T-$S+$S**2)/(2*$S);
    my $max = min( $T-(($S-1)*$S/2), $N );

    for my $K ($min .. $max) {
        num_ways( $K-1, $S-1, $T-$K, $callback, (@so_far, $K) );
    }
}

num_ways( 100, 10, 667, sub { print "@_\n" } );