A336412 Number of labeled dihedral groups with a fixed identity.

1, 1, 20, 630, 18144, 3326400, 148262400

Offset

1,3

Comments

a(n) is the number of dihedral groups 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.

References

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

Links

Table of n, a(n) for n=1..7.

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.

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.

Formula

a(1) = a(2) = 1; a(n>2) = (2n - 1)!/(n*(n-1)) for prime n, otherwise (2n - 1)!/(2n) (conjectured).

a(p) = A226731(p)*2/(p-1) for prime p>2, otherwise a(n) = A226731(n) for n>2 (conjectured).

a(p) = A058163(2p) - A058162(2p) for prime p>2.

Example

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

Crossrefs

Cf. A058163 (all groups), A058162 (Abelian groups), A058161 (cyclic groups), A069156 (stable matchings), A226731.

Sequence in context: A059420 A129906 A125722 * A226731 A201724 A006410

Adjacent sequences: A336409 A336410 A336411 * A336413 A336414 A336415

Keyword

nonn,more

Author

Dan Eilers, Jul 20 2020

Status

approved