Sometimes the rule S(n,k) = 0 if k>n is given for completeness

VERY important if you dont realize n elements ... k non-empty subsets and you just start with

 
for $n (0..10) {
  for $k (0..10) { 
    print "$n $k '.fs($n,$k)."\n"; 
    }
  }
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use strict; use warnings; 

my $l='k '; 
$l.=sprintf "%3d ",0; 
for my $k (0..10){ 
    $l.=sprintf " %10s",$k; 
    } 
print $l."\n";
$l=~s/./-/g; 
print $l."\n";

for my $n (0..10){ 
  printf "n %3d ",$n; 
  for my $k (0..10){ 
    printf " %10s",fs($n,$k); 
    }
  print "\n"; 
  }

sub fs { 
  my $n=shift; 
  my $k=shift; 

  if ( $n ==0 && $k == 0 ) {return 1}; 
  if ( $k > $n ) {return 0};           # important 
  if ( $n > 0 && $k == 0 ) {return 0} 
  if ( $k == 1 )            {return 1} 
  if ($n == $k )           {return 1} 
  my $p1=fs($n-1,$k-1);
  my $p2=fs($n-1,$k  ); 
  return $p1 + ($k*$p2); 
}
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use strict; use warnings; 

my $l='k '; 
$l.=sprintf "%3d ",0; 
for my $k (0..10){ 
    $l.=sprintf " %10s",$k; 
    } 
print $l."\n";
$l=~s/./-/g; 
print $l."\n";

for my $n (0..10){ 
  printf "n %3d ",$n; 
  for my $k (0..10){ 
    printf " %10s",s($n,$k); 
    }
  print "\n"; 
  }

sub s { 
  my $n=shift; 
  my $k=shift; 

  if ( $n ==0 && $k == 0 ) {return 1}; 
  if ( $k > $n ) {return 0};           # important 
  if ( $n > 0 && $k == 0 ) {return 0} 
  if ( $k == 1 )            {return 1} 
  if ($n == $k )           {return 1} 
  my $p1=s($n-1,$k-1);
  my $p2=s($n-1,$k  ); 
  return $p1 + ($k*$p2); 
}
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Global symbol "$p2" requires explicit package name at buk-2.pl line 30
+.
syntax error at buk-2.pl line 31, near "return"
  (Might be a runaway multi-line ;; string starting on line 29)
Global symbol "$p1" requires explicit package name at buk-2.pl line 31
+.
Global symbol "$p2" requires explicit package name at buk-2.pl line 31
+.
Missing right curly or square bracket at buk-2.pl line 32, at end of l
+ine
syntax error at buk-2.pl line 32, at EOF
Execution of buk-2.pl aborted due to compilation errors.
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