The number of \(t\)-wise balanced designs (Q1180422)

This paper provides asymptotically the number of \(t\)-wise balanced designs and the number of \(t\)-profiles. If these numbers are indicated by \(N_ t(n)\) and \(P_ t(n)\), respectively, then the authors show that \[ N_ t(n)=n^{[{n\choose t}/(t+1)](1+o(1))} \] and \[ \exp(c_ 1\sqrt n\log n)\leq P_ t(n)\leq\exp(c_ 2\sqrt n\log n) \] where \(o(1)\) is used to denote a quantity depending on a natural number \(n\), the value of which tends to zero as \(n\) tends to infinity.