Small covers over a polytope and their Davis – Januskiewicz equivalence classes are an ongoing problem in topology. It is known that a characteristic function can be established for each small cover. These characteristic functions can be represented by the matrices and the classification problem of small covers can be transferred to operations on matrices. It is difficult to calculate the number of matrices representing characteristic functions under the equivalence relations. For this reason, a general formula is obtained by using a one-to-one correspondence with acyclic digraphs. In this correspondence, the action of the general linear group on the matrices is seen in the form of permutation of the labels of the vertices in acyclic digraphs, local complementation and sliding the edges.

In this work, we will associate the matrices representing characteristic functions with vectorweighted acyclic digraphs to enumerate the weak equivalence classes of small covers over the product space of simplicies. Here, the number of rows of the matrices is equal to the sum of the dimensions of the simplicies, while the number of columns is equal to the sum of the sum of the dimensions and the number of terms in the product. Based on this information, we will bring the matrices associated with characteristic functions to a permanent form under the action of the general linear group. This form will give a vector matrix, and we will construct a corresponding adjacency matrix of an acyclic digraph that accepts the vectors of this matrix as the weight vector on the edges. The right action of the automorphism group of simplicies will appear on digraphs as a permutation of the labels of the vertices or as a local complementation on the weight vectors of the edges. Finally, using the combinatorial structure of the Stirling numbers of the first kind and rising factorials from which these numbers are obtained as coefficients, we will give the number of weak equivalence classes of small covers over the product of three simplicies with basic functions depending on the dimensions of the simplicies.