I am trying to find a mathematical expression for the number of all possible transitive boolean matrices of order nxn. For example,
$$T_1^n = n(n - 1)^3 + \frac{1}{6}n(n - 1)^4(n - 2) + \frac{1}{6}n(n - 1)(n - 2)(4n - 1)$$
This expresion T_1 works for only those transitive boolean matrices which has 3 nonzero elements. For example in 3x3 there are 43 transitive matrices, for 4x4 there are 276 having only 3 nonzero elements.
$$T_2^n = \frac{1}{2}n(n - 1)(n^2 - n + 3)$$
T_2 works for the number of n × n transitive Boolean matrices with two nonzero elements.
Now I am looking for an expression which can give the number of transitive Boolean matrices in nxn transitive boolean matrices with 4 non-zero elements in each transitive Boolean matrix.
Here is what I know about the total numbers of transitive Boolean matriceswith 4 non-zero elements in each transitive Boolean matrix.
Is it possible to generalize this information?
Any suggestions?
Thanks