The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #57 Jun 30 2024 20:56:26
%S 1,1,20,630,18144,3326400,148262400,40864824000,6586804224000,
%T 3041127510220800,464463110651904000,538583682060103680000,
%U 99430833611096064000000,129629398219266097152000000,73681349947830849621196800000,64240926985765022013480960000000
%N Number of labeled dihedral groups with a fixed identity.
%C a(n) is the number of dihedral groups of order 2n with a fixed identity, or equivalently the number of reduced Latin squares of order 2n that can be viewed as the Cayley table of D_{2n}, by adding a border that matches the first row and column. The reduced Latin squares differ from each other by a permutation of their symbols. Two such Latin squares that differ by a permutation of their symbols have been called isoplanar by Bailey (1984), cited by Nilrat and Praeger (1988), cited by Denes and Keedwell (1991). Latin squares based on dihedral groups are of interest in the stable marriage problem, where Benjamin et al. (1995) exhibited such squares having many stable matchings when viewed as ranking matrices. Two isoplanar Latin squares generally produce a different number of stable matchings, so there is motivation to generate all symbol permutations to find the ones with the most stable matchings.
%C See comments in A002618 regarding automorphisms of dihedral groups by Ola Veshta and Yaghoub Sharifi. - _Dan Eilers_, Jun 08 2024
%D Denes, J. and Keedwell, A. D. (1991) Latin Squares New Developments in the Theory and Applications. p. 98.
%H R. A. Bailey, <a href="https://www.jstor.org/stable/2345518">Quasi-Complete Latin Squares: Construction and Randomization</a>, Journal of the Royal Statistical Society. Series B (Methodological) 46, no. 2 (1984): 330, 323-34.
%H A. T. Benjamin, C. Converse, and H. A. Krieger, <a href="https://doi.org/10.1016/0166-218X(95)80006-P">Note. How do I marry thee? Let me count the ways</a>, Discrete Appl. Math. 59 (1995) 285-292.
%H C. K. Nilrat and C. E. Prager, <a href="https://www.researchgate.net/publication/268840569_Complete_Latin_squares_Terraces_for_groups">Complete latin squares: terraces for groups</a>, Ars Combinatoria 24 (1988), 17-29.
%H Yaghoub Sharifi, <a href="https://ysharifi.wordpress.com/2022/09/14/automorphisms-of-dihedral-groups">Automorphisms of dihedral groups</a>.
%H E. G. Thurber, <a href="https://doi.org/10.1016/S0012-365X(01)00194-7">Concerning the maximum number of stable matchings in the stable marriage problem</a>, Discrete Mathematics Volume 248, Issue 1-3, 6 April 2002, 195-219.
%F a(1) = a(2) = 1; a(n>2) = (2*n-1)! / A002618(n). - _Dan Eilers_, Jun 08 2024
%e For n=3 the a(3)=20 isoplanar reduced Latin squares based on the dihedral group of order 6, in lexicographical order, are:
%e 1) 2) 3) 4) 5)
%e 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
%e 2 1 4 3 6 5 2 1 4 3 6 5 2 1 4 3 6 5 2 1 4 3 6 5 2 1 5 6 3 4
%e 3 5 1 6 2 4 3 5 6 2 4 1 3 6 1 5 4 2 3 6 5 2 1 4 3 4 1 2 6 5
%e 4 6 2 5 1 3 4 6 5 1 3 2 4 5 2 6 3 1 4 5 6 1 2 3 4 3 6 5 1 2
%e 5 3 6 1 4 2 5 3 2 6 1 4 5 4 6 2 1 3 5 4 1 6 3 2 5 6 2 1 4 3
%e 6 4 5 2 3 1 6 4 1 5 2 3 6 3 5 1 2 4 6 3 2 5 4 1 6 5 4 3 2 1
%e 6) 7) 8) 9) 10)
%e 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
%e 2 1 5 6 3 4 2 1 5 6 3 4 2 1 5 6 3 4 2 1 6 5 4 3 2 1 6 5 4 3
%e 3 4 6 5 2 1 3 6 1 5 4 2 3 6 4 1 2 5 3 4 1 2 6 5 3 4 5 6 1 2
%e 4 3 2 1 6 5 4 5 6 1 2 3 4 5 1 3 6 2 4 3 5 6 2 1 4 3 2 1 6 5
%e 5 6 4 3 1 2 5 4 2 3 6 1 5 4 6 2 1 3 5 6 4 3 1 2 5 6 1 2 3 4
%e 6 5 1 2 4 3 6 3 4 2 1 5 6 3 2 5 4 1 6 5 2 1 3 4 6 5 4 3 2 1
%e 11) 12) 13) 14) 15)
%e 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
%e 2 1 6 5 4 3 2 1 6 5 4 3 2 3 1 5 6 4 2 3 1 6 4 5 2 4 5 1 6 3
%e 3 5 1 6 2 4 3 5 4 1 6 2 3 1 2 6 4 5 3 1 2 5 6 4 3 6 1 5 4 2
%e 4 6 5 1 3 2 4 6 1 3 2 5 4 6 5 1 3 2 4 5 6 1 2 3 4 1 6 2 3 5
%e 5 3 4 2 6 1 5 3 2 6 1 4 5 4 6 2 1 3 5 6 4 3 1 2 5 3 2 6 1 4
%e 6 4 2 3 1 5 6 4 5 2 3 1 6 5 4 3 2 1 6 4 5 2 3 1 6 5 4 3 2 1
%e 16) 17) 18) 19) 20)
%e 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
%e 2 4 6 1 3 5 2 5 4 6 1 3 2 5 6 3 1 4 2 6 4 5 3 1 2 6 5 3 4 1
%e 3 5 1 6 2 4 3 6 1 5 4 2 3 4 1 2 6 5 3 5 1 6 2 4 3 4 1 2 6 5
%e 4 1 5 2 6 3 4 3 2 1 6 5 4 6 5 1 3 2 4 3 2 1 6 5 4 5 6 1 2 3
%e 5 6 4 3 1 2 5 1 6 3 2 4 5 1 4 6 2 3 5 4 6 2 1 3 5 3 2 6 1 4
%e 6 3 2 5 4 1 6 4 5 2 3 1 6 3 2 5 4 1 6 1 5 3 4 2 6 1 4 5 3 2
%o (GAP) A336412:=List([1..16], n->Factorial(2*n-1)/Size(AutomorphismGroup(DihedralGroup(2*n)))); # _Dan Eilers_, Jun 08 2024
%Y Cf. A058163 (all groups), A058162 (Abelian groups), A058161 (cyclic groups), A069156 (stable matchings), A002618 (n*phi(n)).
%K nonn
%O 1,3
%A _Dan Eilers_, Jul 20 2020
%E a(8)-a(16) and edited by _Dan Eilers_, Jun 08 2024