Enumerating totally clean words

Let A be a finite alphabet and let DcA * be a finite set of words to be labelled "dirty". Let {Xa: a E A} be commuting indeterminates. To every letter a E A we assign the weight Xa and the weight of a word is the product of the weights of its letters. For example, weight (13221) = X1X3X2X2X1 = XiX~X3. Given any set S of words we let weight (S) be the sum of the weights of the members of S. For example, weight {1, 12,213, 2113} = Xl + X1X2 + X1X2X3 + XiX2X3. The significance of the formal power series weight (S) is that the coefficient of a typical term IIaEAX;" tells us the number of words in S that have lXa occurrences of the letter a, a EA. It is well known and easy to see that weight (A *) = (1 I:aEA Xa)-l. (Recall that A * is the set of all words (strings) that can be formed with the letters of A). There are three standards of cleanliness that words can have. First if we define "clean" as non-dirty, then the weight enumerator is of course