A001201 -id:A001201 - 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.

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.

A137348 Number of Steiner quadruple systems (SQS's) of order n.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 0, 30, 0, 2520, 0, 0, 0, 37362124800, 0, 14311959985625702400, 0, 0, 0

Offset: 1

Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), Apr 08 2008, May 13 2008

The values are calculated by utilizing the Knuth's Algorithm X. Only the number of non-isomorphic SQS's is presented in peer-reviewed literature and scientific textbooks. The algorithm was verified to be valid by seeking STS's presented in A001201.
n=14 calculated from "Mendelsohn and Hung: On Steiner Systems S(3,4,14) and S(4,5,15), Util. Math. Vol 1 (1972), pp. 5-95" with orbit-stabilizer theorem
n=15 is given in "Petteri Kaski, Patric R. J. Östergård (Patric.Ostergard(AT)hut.fi) and O. Pottonen, The Steiner quadruple systems of order 16". SQS(20) is still unknown.

			There are 2520 SQS's on 10 points.

Petteri Kaski, Patric R. J. Östergård (Patric.Ostergard(AT)hut.fi) and O. Pottonen, The Steiner quadruple systems of order 16
N. S. Mendelsohn and S. H. Y. Hung, On the Steiner Systems S(3,4,14) and S(4,5,15), Util. Math. Vol. 1, 1972, pp. 5-95

Vesa Linja-aho, Home Page.
Vesa Linja-aho, Python program
Index entries for sequences related to Steiner systems

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.

cp's OEIS Frontend

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

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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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A137348 Number of Steiner quadruple systems (SQS's) of order 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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