# Counting Designs

In this talk I will indulge in some very rough counting. I will start with an exceedingly simple upper bound for the number of Latin squares of order *n* (having a given set of entries). I will then describe a corresponding lower bound due essentially to H. J. yser [5], which is perhaps surprisingly (although crudely and relatively speaking) not that much lower.

The talk progresses to a simple upper bound for the number of Steiner triple systems of order *n* (having a given point set). An outline is then given of R. M. Wilson’s lower bound [6]. This is (again crudely and relatively speaking) not that much lower.

I will move on to discuss a representative sample of recent results of D. M. Donovan and myself concerning the numbers of designs with affine and projective parameters [2, 3]. The relevant designs will be described and a rough outline given of a recursive construction that facilitates the production of lower bounds.

These lower bounds improve and extend earlier results of other authors, including most recently those of D. C. Clark, D. Jungnickel and V. D. Tonchev [1, 4].

There remains, however, a large gap between the best known upper and lower bounds for these designs.

References

[1] D. C. Clark, D. Jungnickel and V. D.Tonchev. *Correction to: ”Exponential bounds on the number of designs with affine parameters”*. J. Combin. Des. 19 (2011), no. 2, 156-166.

[2] D. M. Donovan and M. J. Grannell. *Designs having the parameters of projective and affine spaces. *Des. Codes Cryptogr. 60 (2011), no. 3, 225240.

[3] D. M. Donovan and M. J. Grannell. *On the number of designs with affine parameters. *Des. Codes Cryptogr., to appear.

[4] D. Jungnickel and V. D. Tonchev. *The number of designs with geometric parameters grows exponentially. *Des. Codes Cryptogr. 55 (2010), no. 2-3, 131-140.

[5] H. J. Ryser. *Permanents and systems of district representatives.* In “Combinatorial Mathematics and its Applications” (Proc. Conf., Univ. North Carolina, Chapel Hill, N.C., 1967)pp. 55-70 Univ. North Carolina Press, Chapel Hill, N.C.

[6] R. M. Wilson, *Nonisomorphic Steiner triple systems.* Math. Z. 135 (1973/74), 303-313.

Image: Nuchylee