

A003090


Number of species (or "main classes" or "paratopy classes") of Latin squares of order n.
(Formerly M0387)


12



1, 1, 1, 2, 2, 12, 147, 283657, 19270853541, 34817397894749939, 2036029552582883134196099
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OFFSET

1,4


COMMENTS

Urzua abstract: "We start by showing a one to one correspondence between arrangements of d lines in P^2 and lines in P^{d2}. Then we apply this to classify (3,q)nets on P^2 for all 2 <= q <= 6. For the new case q=6, we have a priori twelve possible cases, but we obtain that only six of them are realizable on P^2 over C. We give equations for the lines defining these nets. We also construct a three dimensional family of (3,8)nets corresponding to the multiplication table of the Quaternion group. After that, we define more general arrangements of curves and relate them, via moduli spaces of pointed stable curves of genus zero, to curves in P^{d2}. Then, we prove that there is a one to one correspondence between these more general arrangements of d curves and certain curves in P^{d2}. As a corollary, we recover the one to one correspondence for line arrangements. This more general setting not only generalizes line arrangements but also shows the ideas behind what we did in that case."  Jonathan Vos Post, Apr 05 2007


REFERENCES

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 231.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. M. Wanless, A generalization of transversals for Latin squares, Electron. J. Combin., 9 (2002), #R12.


LINKS

Table of n, a(n) for n=1..11.
A. Hulpke, P. Kaski and P. R. J. Ostergard, The number of Latin squares of order 11, Math. Comp. 80 (2011) 11971219
B. D. McKay, Latin Squares (has list of all such squares)
B. D. McKay, A. Meynert and W. Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Designs, 15 (2007), no. 2, 98119.
B. D. McKay and E. Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.
M. G. Palomo, Latin polytopes, arXiv preprint arXiv:1402.0772, 2014
Giancarlo Urzua, On line arrangements with applications to 3nets. arXiv:0704.0469 (see page 9).
M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237240]
Index entries for sequences related to Latin squares and rectangles


CROSSREFS

Cf. A000315, A002860, A040082.
Sequence in context: A032320 A032227 A032069 * A032152 A032057 A130718
Adjacent sequences: A003087 A003088 A003089 * A003091 A003092 A003093


KEYWORD

nonn,nice,hard


AUTHOR

N. J. A. Sloane.


EXTENSIONS

Two more terms (from the McKayMeynertMyrvold article) from Richard Bean (rwb(AT)eskimo.com), Feb 17 2004
There are 2036029552582883134196099 main classes of Latin squares of order 11.  Petteri Kaski (petteri.kaski(AT)cs.helsinki.fi), Sep 18 2009


STATUS

approved



