A conjecture on the number of SDRs of a (t,n)(t,n)-family

A system of distinct representatives (SDR) of a family F=(A1,…,An)F=(A1,…,An) is a sequence (x1,…,xn)(x1,…,xn) of nn distinct elements with xi∈Aixi∈Ai for 1≤i≤n1≤i≤n. Let N(F)N(F) denote the number of SDRs of a family FF; two SDRs are considered distinct if they are different in at least one component. For a nonnegative integer tt, a family F=(A1,…,An)F=(A1,…,An) is called a (t,n)(t,n)-family if the union of any k≥1k≥1 sets in the family contains at least k+tk+t elements. The famous Hall’s theorem says that N(F)≥1N(F)≥1 if and only if FF is a (0,n)(0,n)-family. Denote by M(t,n)M(t,n) the minimum number of SDRs in a (t,n)(t,n)-family. The problem of determining M(t,n)M(t,n) and those families containing exactly M(t,n)M(t,n) SDRs was first raised by Chang [G.J. Chang, On the number of SDR of a (t,n)(t,n)-family, European J. Combin. 10 (1989) 231–234]. He solved the cases when 0≤t≤20≤t≤2 and gave a conjecture for t≥3t≥3. In this paper, we solve the conjecture.