%I A003090 M0387
%S A003090 1,1,1,2,2,12,147,283657,19270853541,34817397894749939,2036029552582883134196099
%N A003090 Number of species (or "main classes" or "paratopy classes") of Latin 
               squares of order n.
%C A003090 Urzua abstract: "We start by showing a one to one correspondence between 
               arrangements of d lines in P^2 and lines in P^{d-2}. 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^{d-2}. Then, we prove that there is 
               a one to one correspondence between these more general arrangements 
               of d curves and certain curves in P^{d-2}. 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 (jvospost3(AT)gmail.com), 
               Apr 05 2007
%D A003090 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003090 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 
               1973, p. 231.
%D A003090 I. M. Wanless, A generalization of transversals for Latin squares, Electron. 
               J. Combin., 9 (2002), #R12.
%D A003090 A. Hulpke, P. Kaski and P. R. J. Ostergard, The number of Latin squares 
               of order 11, Preprint, 2009.
%H A003090 B. D. McKay, <a href="http://cs.anu.edu.au/~bdm/data/latin.html">Latin 
               Squares</a> (has list of all such squares)
%H A003090 B. D. McKay, A. Meynert and W. Myrvold, <a href="http://cs.anu.edu.au/
               ~bdm/papers/ls_final.pdf">Small Latin Squares, Quasigroups and Loops</
               a>, J. Combin. Designs, to appear (2005).
%H A003090 B. D. McKay and E. Rogoyski, <a href="http://www.combinatorics.org/Volume_2/
               volume2.html#N3">Latin squares of order ten</a>, Electron. J. Combinatorics, 
               2 (1995) #N3.
%H A003090 <a href="Sindx_La.html#Latin">Index entries for sequences related to 
               Latin squares and rectangles</a>
%H A003090 Giancarlo Urzua, <a href="http://arXiv.org/abs/0704.0469">On line arrangements 
               with applications to 3-nets</a> (see page 9).
%Y A003090 Cf. A000315, A002860, A040082.
%Y A003090 Sequence in context: A032320 A032227 A032069 this_sequence A032152 A032057 
               A130718
%Y A003090 Adjacent sequences: A003087 A003088 A003089 this_sequence A003091 A003092 
               A003093
%K A003090 nonn,nice,hard
%O A003090 1,4
%A A003090 N. J. A. Sloane (njas(AT)research.att.com).
%E A003090 Two more terms (from the McKay-Meynert-Myrvold article) from Richard 
               Bean (rwb(AT)eskimo.com), Feb 17 2004
%E A003090 There are 2036029552582883134196099 main classes of Latin squares of 
               order 11. - Petteri Kaski (petteri.kaski(AT)cs.helsinki.fi), Sep 
               18 2009
%E A003090 arXiv URL replaced by its non-cached version - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Oct 23 2009