Let V={a_1,a_2,...,a_n} be a finite set with n≥2 and P_n(V)the set of all primitive binary relations on V.For Q∈P_n(V),denote by G(Q)the directed graph corresponding to Q. For positive integer d≤n,let P_n(V,d)={Q:Q∈P_n(V)and G(Q)contains exactly d loops}.In this paper,it is proved that the set of common consequent indices of binary relations in P_n(V,d) is {1,2,...,n-「d/2」}.Furthermore,the minimal extremal binary relations are described.
We discuss a family of restricted m-ary overpartition functions bm,j(n),which is the number of m-ary overpartitions of n with at most i + j copies of the non-overlined part mi allowed,and obtain a family of congruences for bm,lm-1(n).
