Computational Experiments

The work presented in this paper began with an interst in classifying diagrams of the above type according to the number of components of each type. Although facilities for carrying out such computations are absent from standard mathematical software, it is quite convenient to write short programs to do so in current high-level and widely-available languages. For example, Prolog is particularly well-suited to the type of generate-and-test procedure that is required here.

The first runs were made to generate all elements of T_n and for each 1#1 count the number of each type of component of 7#7. Among the results were the data presented in Table 1. The second column lists the maximum gravity obtained for the n in question. (This is known from [2] to be [3(n - 1) / 2]. The third column (G(n)) lists the number of elements 1#1 for which 3#3 is maximum. The pattern, which suggests that the odd and even cases should be analyzed separately, suggested the conjectures that became the main theorems of this paper. The predictions given by those theorems were then verified computationally for n = 8,9,10,11.

 

Table: Raw results

n	maximum gravity	G(n)
3	3	6
4	4	60
5	6	60
6	7	1050
7	9	630

 

Table: Divided into Odd and Even Cases

m	n = 2m	n = 2m + 1	g(n), n even	g(n), n odd	G(n), n even	G(n), n odd
2	4	5	4	6	60	60
3	6	7	7	9	1050	630