A336412 Number of labeled dihedral groups with a fixed identity.

1, 1, 20, 630, 18144, 3326400, 148262400, 40864824000, 6586804224000, 3041127510220800, 464463110651904000, 538583682060103680000, 99430833611096064000000, 129629398219266097152000000, 73681349947830849621196800000, 64240926985765022013480960000000

Offset: 1

Author

Dan Eilers, Jul 20 2020

Keywords

• nonn

Comments

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.

See comments in A002618 regarding automorphisms of dihedral groups by Ola Veshta and Yaghoub Sharifi. - Dan Eilers, Jun 08 2024

Examples

			For n=3 the a(3)=20 isoplanar reduced Latin squares based on the dihedral group of order 6, in lexicographical order, are:
1)             2)             3)             4)             5)
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
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
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
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
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
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
6)             7)             8)             9)             10)
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
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
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
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
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
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
11)            12)            13)            14)            15)
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
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
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
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
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
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
16)            17)            18)            19)            20)
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
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
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
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
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
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
		

References

• Denes, J. and Keedwell, A. D. (1991) Latin Squares New Developments in the Theory and Applications. p. 98.

Links

• R. A. Bailey, Quasi-Complete Latin Squares: Construction and Randomization, Journal of the Royal Statistical Society. Series B (Methodological) 46, no. 2 (1984): 330, 323-34.
• A. T. Benjamin, C. Converse, and H. A. Krieger, Note. How do I marry thee? Let me count the ways, Discrete Appl. Math. 59 (1995) 285-292.
• C. K. Nilrat and C. E. Prager, Complete latin squares: terraces for groups, Ars Combinatoria 24 (1988), 17-29.
• Yaghoub Sharifi, Automorphisms of dihedral groups.
• E. G. Thurber, Concerning the maximum number of stable matchings in the stable marriage problem, Discrete Mathematics Volume 248, Issue 1-3, 6 April 2002, 195-219.

Crossrefs

Cf. A058163 (all groups), A058162 (Abelian groups), A058161 (cyclic groups), A069156 (stable matchings), A002618 (n*phi(n)).

Programs

GAP

A336412:=List([1..16], n->Factorial(2*n-1)/Size(AutomorphismGroup(DihedralGroup(2*n)))); # Dan Eilers, Jun 08 2024

Formula

a(1) = a(2) = 1; a(n>2) = (2*n-1)! / A002618(n). - Dan Eilers, Jun 08 2024

Extensions

a(8)-a(16) and edited by Dan Eilers, Jun 08 2024