A051390 -id:A051390 - cp's OEIS frontend

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

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.

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

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.

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

A001201 Number of Steiner triple systems (STS's) on 6n+1 or 6n+3 elements.

Original entry on oeis.org

1, 1, 30, 840, 1197504000, 60281712691200, 1348410350618155344199680000

Offset: 0

N. J. A. Sloane

To be precise, for n even, a(n) is the number of STS's on 3n+1 elements, and for n odd, a(n) is the number of STS's on 3n elements. - Franklin T. Adams-Watters, Apr 10 2010

			There are 1197504000 STS's on 13 elements.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 304.
CRC Handbook of Combinatorial Designs, 1996, p. 70.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Petteri Kaski and Patric R. J. Östergård, The Steiner triple systems of order 19, Math. Comp. 73 (2004), 2075-2092.
Index entries for sequences related to Steiner systems

Indexed by A047241. Cf. A030128, A030129, A051390.

A030128 Number of Steiner triple systems (STS's) on n elements.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 30, 0, 840, 0, 0, 0, 1197504000, 0, 60281712691200, 0, 0, 0, 1348410350618155344199680000

Offset: 1

N. J. A. Sloane

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

Yue Guan, Minjia Shi, and Denis S. Krotov, The Steiner triple systems of order 21 with a transversal subdesign TD(3,6), arXiv:1905.09081 [math.CO], 2019.
Petteri Kaski and Patric R. J. Östergård, The Steiner triple systems of order 19, Math. Comp. 73 (2004), 2075-2092.
Minjia Shi, Li Xu, and Denis S. Krotov, The number of the non-full-rank Steiner triple systems, arXiv:1806.00009 [math.CO], 2018.
Index entries for sequences related to Steiner systems

Cf. A001201, A030129, A051390.

A124119 Number of nonisomorphic Steiner quadruple systems (SQS's) S(3,4,v) on v = 6n+2 or 6n+4 points.

Original entry on oeis.org

1, 1, 4, 1054163

Offset: 1

N. J. A. Sloane, Nov 30 2006

A051390 is the official version of this sequence and has all the references etc.

			There are 4 nonisomorphic SQS's on 14 points.

Eric Weisstein's World of Mathematics, Steiner Quadruple System
Index entries for sequences related to Steiner systems

Cf. A124120, A030129, A001201, A030128, A051390.

A124120 Number of nonisomorphic Steiner quadruple systems (SQS's) S(3,4,n) on n points.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1054163

Offset: 1

N. J. A. Sloane, Nov 30 2006

			There are 4 nonisomorphic SQS's on 14 points.

Index entries for sequences related to Steiner systems

A051390 is the official version of this sequence and has all the references etc.

Cf. A124119, A030129, A001201, A030128, A051390.

A066701 Triangle giving number of nonisomorphic minimal covering designs with parameters (n, k, k-1) (designs achieving the covering number C(n,k,k-1) given in A066010), for n >= 2, 2 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 5, 1, 6, 1, 1, 1, 1, 1, 77, 3, 2, 1, 1, 1, 1, 58, 1, 40, 1, 20, 1, 1, 1, 1, 2

Offset: 2

N. J. A. Sloane, Jan 11 2002

C(v,k,t) is the smallest number of k-subsets of an n-set such that every t-subset is contained in at least one of the k-subsets. This sequence says how many different solutions there are for C(n,k,k-1).

CRC Handbook of Combinatorial Designs, 1996, p. 263.
W. H. Mills and R. C. Mullin, Coverings and packings, pp. 371-399 of Jeffrey H. Dinitz and D. R. Stinson, editors, Contemporary Design Theory, Wiley, 1992.

D. Applegate, E. M. Rains and N. J. A. Sloane, On asymmetric coverings and covering numbers, J. Comb. Des. 11 (2003), 218-228.
D. Gordon, La Jolla Repository of Coverings
K. J. Nurmela and Patric R. J. Östergård, New coverings of t-sets with (t+1)-sets, J. Combinat. Designs, 7 (1999), 217-226.
K. J. Nurmela and Patric R. J. Östergård, New coverings of t-sets with (t+1)-sets (appendix), J. Combinat. Designs, 7 (1999), 217-226.
Index entries for covering numbers
Index entries for sequences related to Steiner systems

Cf. A066010. A030129 gives entries in second column in the cases when a Steiner triple system exists.

A051390 gives entries in 3rd column in the cases when a Steiner quadruple system exists.

A051391 Number of nonisomorphic Steiner triple systems (STS's) S(2,3,v) on v = 6n+1 or 6n+3 points.

Original entry on oeis.org

1, 1, 1, 1, 2, 80, 11084874829

Offset: 1

N. J. A. Sloane

			There are 2 nonisomorphic STS's on 13 points.

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

Petteri Kaski and Patric R. J. Östergård, 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.
Index entries for sequences related to Steiner systems

Indexed by A047241. Cf. A001201, A030128, A030129, A051390, A124118, A124119.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

A001201 Number of Steiner triple systems (STS's) on 6n+1 or 6n+3 elements.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A030128 Number of Steiner triple systems (STS's) on n elements.

Views

Author

Keywords

References

Links

Crossrefs

A124119 Number of nonisomorphic Steiner quadruple systems (SQS's) S(3,4,v) on v = 6n+2 or 6n+4 points.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A124120 Number of nonisomorphic Steiner quadruple systems (SQS's) S(3,4,n) on n points.

Views

Author

Keywords

Examples

Links

Crossrefs

A066701 Triangle giving number of nonisomorphic minimal covering designs with parameters (n, k, k-1) (designs achieving the covering number C(n,k,k-1) given in A066010), for n >= 2, 2 <= k <= n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A051391 Number of nonisomorphic Steiner triple systems (STS's) S(2,3,v) on v = 6n+1 or 6n+3 points.

Views

Author

Keywords

Examples

References

Links

Crossrefs