In this section, we give an almost complete classification of the factorial functions and the structure of Eulerian Sheffer posets.

1 Characterization of the factorial functions and structure of Eulerian Sheffer posets of rank n [greater than or equal to] 5 for which B (3) = 3

1 Characterization of the factorial functions of Eulerian Sheffer posets of rank n [greater than or equal to] 5 for which B(3) = 3

In this subsection, we study the factorial functions of Eulerian Sheffer posets of rank n [greater than or equal to] 5 for which B(3) = 3

Let P and P' be two Eulerian Sheffer posets of rank 2m + 2, m [greater than or equal to] 2, such that their binomial factorial functions and coatom functions agree up to rank n [less than or equal to] 2m.

By substituting the values of the factorial functions, we have

Thus, the poset P has the same factorial functions as [C.

Classification of the factorial functions of Eulerian Sheffer posets of odd rank n = 2m + 1 [greater than or equal to] 5 with B(3) = 6 and D(3) = 8 remains open.

Is there a positive integer [alpha] such that the poset P has the same factorial functions as poset [Q.

Consequently the poset P has the following binomial and Sheffer factorial functions.

2 Characterization of the structure and factorial functions of Eulerian Sheffer posets of rank n [greater than or equal to] 5 with B (3) = 4.