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Number of fixed-point-free permutation groups of degree n.
(Formerly M1730 N0685)
12

%I M1730 N0685 #54 Jan 29 2022 00:59:44

%S 1,0,1,2,7,8,37,40,200,258,1039,1501,7629,10109,54322,83975,527036,

%T 780193,5808293

%N Number of fixed-point-free permutation groups of degree n.

%C a(1) = 0 since the trivial group of degree 1 has a fixed point. One could also argue that one should set a(1) = 1 by convention.

%D G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra, 11 (1983), 863-911.

%D D. Holt, Enumerating subgroups of the symmetric group. Computational Group Theory and the Theory of Groups, II, edited by L.-C. Kappe, A. Magidin and R. Morse. AMS Contemporary Mathematics book series, vol. 511, pp. 33-37.

%D A. Hulpke, Konstruktion transitiver Permutationsgruppen, Dissertation, RWTH Aachen, 1996.

%D A. Hulpke, Constructing transitive permutation groups, J. Symbolic Comput. 39 (2005), 1-30.

%D A. Hulpke, Constructing Transitive Permutation Groups, in preparation

%D C. C. Sims, Computational methods in the study of permutation groups, pp. 169-183 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H G. Butler and J. McKay, <a href="/A000637/a000637_1.pdf">The transitive groups of degree up to eleven</a>, Comm. Algebra, 11 (1983), 863-911. [Annotated scanned copy]

%H D. Holt, <a href="/A000019/a000019_1.pdf">Enumerating subgroups of the symmetric group</a>, in Computational Group Theory and the Theory of Groups, II, edited by L.-C. Kappe, A. Magidin and R. Morse. AMS Contemporary Mathematics book series, vol. 511, pp. 33-37. [Annotated copy]

%H A. Hulpke, <a href="http://www.math.colostate.edu/~hulpke/smalldeg.html">Transitive groups of small degree</a>

%H A. C. Lunn and J. K. Senior, <a href="http://dx.doi.org/10.1021/j150301a009">Isomerism and Configuration</a>, J. Physical Chem. 33 (7) 1929, 1027-1079.

%H A. C. Lunn and J. K. Senior, <a href="/A000637/a000637.pdf">Isomerism and Configuration</a>, J. Physical Chem. 33 (7) 1929, 1027-1079. [Annotated scan of page 1069 only]

%H C. C. Sims, <a href="/A000019/a000019.pdf">Letter to N. J. A. Sloane (no date)</a>

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%F a(n) = A000638(n) - A000638(n-1). - _Christian G. Bower_, Feb 23 2006

%Y Cf. A000001, A000019, A000638, A002106, A005432, A005226.

%Y Cf. A000019, A002106. Unlabeled version of A116693.

%K nonn,hard,more,nice

%O 0,4

%A _N. J. A. Sloane_

%E More terms from _Alexander Hulpke_

%E a(2) and a(10) corrected, a(11) and a(12) added by _Christian G. Bower_, Feb 23 2006

%E Terms a(13)-a(18) were computed by Derek Holt and contributed by Alexander Hulpke. Jul 30 2010, who comments that he has verified the terms up through a(16).

%E Edited by _N. J. A. Sloane_, Jul 30 2010, at the suggestion of _Michael Somos_