The number oft-wise balanced designs

Abstract

We 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.