The number oft-wise balanced designs


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.

This is a preview of subscription content, access via your institution.


  1. [1]

    V. E. Aleksejev: On the number of Steiner triple systems,Math. Notes 15 (1974), 461–464.

    Google Scholar 

  2. [2]

    C. J. Colbourn, K. T. Phelps, andV. Rödl: Block sizes in pairwise balanced designs,Canadian Math. Bull. 27 (1984), 375–380.

    Google Scholar 

  3. [3]

    P. Erdős: Problems and results on block designs and set systems,Proc. Thirteenth Southeastern Conf. Combinatorics, Graph Theory and Computing,1982, pp. 3–18.

  4. [4]

    H. Lenz: On the number of Steiner quadruple systems,Mitt. Math. Sem. Giessen 169 (1985), 55–71.

    Google Scholar 

  5. [5]

    K. T. Phelps: On the number of commutative Latin squares,Ars Combinatoria 10 (1980), 311–322.

    Google Scholar 

  6. [6]

    V. Rödl: On a packing and covering problem,European J. Combinatorics,5 (1985), 69–78.

    Google Scholar 

  7. [7]

    R. M. Wilson: Nonisomorphic Steiner triple systems,Math. Z. 135 (1974), 303–313.

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Colbourn, C.J., Hoffman, D.G., Phelps, K.T. et al. The number oft-wise balanced designs. Combinatorica 11, 207–218 (1991).

Download citation

AMS subject classification (1980)

  • 05 B 30
  • 05 B 05