A030129 -id:A030129 - cp's OEIS frontend

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

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

Offset: 0

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.

A051390 Number of nonisomorphic Steiner quadruple systems (SQS's) of order n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			There are 4 nonisomorphic SQS's on 14 points.

CRC Handbook of Combinatorial Designs, 1996, circa p. 70.
A. Hartman and K. T. Phelps, Steiner quadruple systems, pp. 205-240 of Contemporary Design Theory, ed. Jeffrey H. Dinitz and D. R. Stinson, Wiley, 1992.

Petteri Kaski, Patric R. J. Östergård (Patric.Ostergard(AT)hut.fi) and O. Pottonen, The Steiner quadruple systems of order 16, Journal of Combinatorial Theory, Series A, Volume 113, Issue 8, November 2006, Pages 1764-1770.
V. A. Zinoviev and D. V. Zinoviev, Classification of Steiner Quadruple Systems of order 16 and rank 14, [English translation from Russian], Problemy Peredachi Informatsii, 42 (No. 3, 2006), 59-72.
V. A. Zinoviev and D. V. Zinoviev, Classification of Steiner Quadruple Systems of order 16 and rank 14, Problems of Information Transmission, July-September 2006, Volume 42, Issue 3, pp 217-229; from [in Russian], Problemy Peredachi Informatsii, 42 (No. 3, 2006), 59-72.
Index entries for sequences related to Steiner systems

See A124120, A124119 for other versions of this sequence. The present entry is the official version.

Cf. A030129, A001201, A030128.

a(n) = 0 unless n = 1 or n == 2 or 4 (mod 6).

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.

A187567 Number of Steiner Systems S(2,4,n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0

Offset: 1

N. J. A. Sloane, Mar 11 2011

S(2,4,n) exists if and only if n == 1 or 4 (mod 12).

a(28) >= 4653.

Colin Reid and Alex Rosa, Steiner systems S(2,4,v) - a survey, Electronic Journal of Combinatorics, Dynamic Survey DS18, Feb 01 2010.
E. Spence, The complete classification of Steiner Systems S(2,4,25), J. Combin. Designs, 4 (1996), 295-300.
Index entries for sequences related to Steiner systems

Cf. A187585, A001201, A030129.

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.

A282746 Number of 2-(n,3,lambda_min) designs.

Original entry on oeis.org

1, 1, 3077244, 1, 960

Offset: 6

N. J. A. Sloane, Mar 05 2017

The corresponding values of lambda_min are 2, 1, 6, 1, 2.

Other known terms are a(12) = 242995846, a(13) = 2, a(15) = 80, a(19) = 11084874829.

Gholamreza B. Khosrovshahi and Behruz Tayfeh-Rezaie, Classification of simple 2-(11, 3, 3) designs, Discrete Mathematics 309.3 (2009): 515-520.

Cf. A030129.

Edited by Andrei Zabolotskii, Jul 08 2025

cp's OEIS Frontend

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

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A051390 Number of nonisomorphic Steiner quadruple systems (SQS's) of order n.

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

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

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

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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.

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A187567 Number of Steiner Systems S(2,4,n).

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

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A282746 Number of 2-(n,3,lambda_min) designs.

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