A030129 Number of nonisomorphic Steiner triple systems (STS's) S(2,3,n) on n points.

1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 80, 0, 0, 0, 11084874829, 0, 14796207517873771

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• nice
• hard
• more

Comments

a(n) also counts the following objects:

isomorphism classes of idempotent totally symmetric Latin squares of order n,

isotopism classes containing idempotent totally symmetric Latin squares of order n,

species containing idempotent totally symmetric Latin squares of order n,

isomorphism classes of totally symmetric loops of order n+1,

isomorphism classes of totally symmetric unipotent Latin squares of order n+1,

isomorphism classes containing totally symmetric reduced Latin squares of order n+1,

isotopism classes containing totally symmetric unipotent Latin squares of order n+1,

isotopism classes containing totally symmetric reduced Latin squares of order n+1,

species containing totally symmetric unipotent Latin squares of order n+1, and

species containing totally symmetric reduced Latin squares of order n+1.

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 304.
• CRC Handbook of Combinatorial Designs, 1996, p. 70.

Links

• Daniel Heinlein and Patric R. J. Östergård, Enumerating Steiner Triple Systems, arXiv:2303.01207 [math.CO], 2023.
• Petteri Kaski and Patric R. J. Östergård (Patric.Ostergard(AT)hut.fi), The Steiner triple systems of order 19.
• Petteri Kaski and Patric R. J. Östergård, The Steiner triple systems of order 19, Mathematics of Computation, Vol. 73, No. 248 (Oct., 2004), pp. 2075-2092.
• Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.
• Eric Weisstein's World of Mathematics, Steiner Triple System.
• Index entries for sequences related to Steiner systems.

Crossrefs

Cf. A001201, A030128, A051390, A124118, A124119, A076019.