Counting conjugacy classes for unitriangular matrices

Let GnGn be the subgroup of GLn(q)GLn(q) consisting of upper unitriangular matrices. A longstanding conjecture, attributed to G. Higman, states that the number of conjugacy classes of GnGn is given by a polynomial in q   with integer coefficients. Recent results show that the resolution of this conjecture is linked to the classification of the canonical primitive connected matrices of GnGn. In this paper, we calculate the number of canonical primitive connected matrices A   of GnGn satisfying that ΓAΓA, the graph associated to A, has two pivot lines and there are, at most, two edges connecting the pivot lines. We prove that this number is a polynomial in q and it is connected with the combinatory of Catalan numbers.