The number oft-wise balanced designs

AbstractWe prove that the number oft-wise balanced designs of ordern is asymptotically $$n\left( {(_t^n )/(t + 1)} \right)(1 + o(1))$$ , provided that blocks of sizet are permitted. In the process, we prove that the number oft-profiles (multisets of block sizes) is bounded below by $$\exp \left( {c_1 = \sqrt n \log n} \right)$$ and above by $$\exp \left( {c_2 = \sqrt n \log n} \right)$$ for constants c2>c1>0.