A conjecture on the number of SDRs of a -family

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