Question

In PHP, given

• the final string length
• the range of characters it can use
• min consecutive repetition count possible

how can you calculate the number of matches that fits these criteria?
To draw a better picture…

$range = array('a','b','c');
$length = 2; // looking for 2 digit results
$minRep = 2; // with >=2 consecutive characters
// aa,bb,cc = 3 possibilities

another one:

$range = array('a','b','c');
$length = 3; // looking for 3 digit results
$minRep = 2; // with >=2 consecutive characters
// aaa,aab,aac,baa,caa
// bbb,bba,bbc,abb,cbb
// ccc,cca,ccb,acc,bcc
// 5 + 5 + 5 = 15 possibilities
// note that combos like aa,bb,cc are not included
// because their length is smaller than $length

last one:

$range = array('a','b','c');
$length = 3; // looking for 3 digit results
$minRep = 3; // with >=3 consecutive characters
// aaa,bbb,ccc = 3 possibilities

So basically, in the 2nd example the 3rd criterion made it catch e.g. [aa]b in aab because a was repeating consecutively more than once, whereas [a]b[a] wouldn’t be a match because those a‘s are separate.

Needless to say, none of the variables is static.

Answer 1

Got it. All credit to leonbloy @mathexchange.com.

/* The main function computes the number of words that do NOT contain
 * a character repetition of length $minRep (or more). */
function countStrings($rangeLength, $length, $minRep, &$results = array())
{
  if (!isset($results[$length]))
  {
    $b = 0;

    if ($length < $minRep)
      $b = pow($rangeLength, $length);
    else
    {
      for ($i = 1; $i < $minRep; $i++)
        $b += countStrings($rangeLength, $length - $i, $minRep, $results);
      $b *= $rangeLength - 1;
    }

    $results[$length] = $b;
  }

  return $results[$length];
}

/* This one answers directly the question. */
function printNumStringsRep($rangeLength, $length, $minRep)
{
  $n = (pow($rangeLength, $length) 
            - countStrings($rangeLength, $length, $minRep));
  echo  "Size of alphabet : $rangeLength<br/>"
        . "Size of string : $length<br/>"
        . "Minimal repetition : $minRep<br/>"
        . "<strong>Number of words : $n</strong>";
}

/* Prints :
 * 
   Size of alphabet : 3
   Size of string : 3
   Minimal repetition : 2
   Number of words : 15
 *
 */
printNumStringsRep(3, 3, 2);