Eco: The Elements of the Ars Combinatoria
“Given a number of different elements n, the number of arrangements that can be made from them, in any order whatever, is expressed by their factorial n!, calculated as 1.*2*3. . . . *n.
This is the method for calculating the possible anagrams of a word of n letters, already encountered as the art of temurah in the kabbala. The Sefer Yezirah informed us that the factorial of 5 was 120.
As n increases, the number of possible arrangements rises exponentially: the possible arrangements for 36 elements, for example, are 371, 993, 326, 789, 901, 217, 467, 999, 448, 150, 835, 200, 000, 000.
If the strings admit repetitions, then those figures grow upwards. For example, the 21 letters of the Italian alphabet can give rise to more than 51 billion billion 21-letter-long sequences (each different from the rest); when, however, it is admitted that some letters are repeated, but the sequences are shorter than the number of elements to be arranged, then the general formula for n elements taken t at a time with repetitions is n1 and the number of strings obtainable for the letters of the Italian alphabet would amount to 5 billion billion billion.
Let us suppose a different problem. There are four people, A. B, C, and D. We want to arrange these four as couples on board an aircraft in which the seats are in rows that are two across; the order is relevant because I want to know who will sit at the the window and who at the aisle.
We are thus facing a problem of permutation, that is, of arranging n elements, taken t at a time, taking the order into account. The formula for finding all the possible permutations is n!/(n-t)! In our example the persons can be disposed this way:
AB AC AD BA CA DA BC BD CD CB DB DC
Suppose, however, that the four letters represented four soldiers, and the problem is to calculate how many two-man patrols could be formed from them. In this case the order is irrelevant (AB or BA are always the same patrol). This is a problem of combination, and we solve it with the following formula: n!/t!(n-t)! In this case the possible combinations would be:
AB AC AD BC BD CD
Such calculuses are employed in the solution of many technical problems, but they can serve as discovery procedures, that is, procedures for inventing a variety of possible “scenarios.”
In semiotic terms, we are in front of an expression-system (represented both by the symbols and by the syntactic rules establishing how n elements can be arranged t at a time–and where t can coincide with n), so that the arrangement of the expression-items can automatically reveal possible content-systems.
In order to let this logic of combination or permutation work to its fullest extent, however, there should be no restrictions limiting the number of possible content-systems (or worlds) we can conceive of.
As soon as we maintain that certain universes are not possible in respect of what is given in our own past experience, or that they do not correspond to what we hold to be the laws of reason, we are, at this point, invoking external criteria not only to discriminate the results of the ars combinatoria, but also to introduce restrictions within the art itself.
We saw, for example, that, for four people, there were six possible combinations of pairs. If we specify that the pairing is of a matrimonial nature, and if A and B are men while C and D are women, then the possible combinations become four.
If A and C are brother and sister, and we take into the account the prohibition against incest, we have only three possible groupings. Yet matters such as sex, consanguinity, taboos and interdictions have nothing to do with the art itself: they are introduced from outside in order to control and limit the possibilities of the system.”
Umberto Eco, The Search for the Perfect Language, translated by James Fentress, Blackwell. Oxford, 1995, pp. 54-6.