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A002854
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Number of Euler graphs with n nodes; number of 2-graphs with n nodes; and number of switching classes of graphs with n nodes.
(Formerly M0846 N0321)
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16
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1, 1, 2, 3, 7, 16, 54, 243, 2038, 33120, 1182004, 87723296, 12886193064, 3633057074584, 1944000150734320, 1967881448329407496, 3768516017219786199856, 13670271807937483065795200
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OFFSET
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1,3
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COMMENTS
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Also called Eulerian graphs of strength 1.
"Switching" at a node complements all the edges incident with that node. The illustration (see link) shows the 3 switching classes on 4 nodes. Switching at any node is the equivalence relation.
A graph is an Euler graph iff every node has even degree. It need not be connected. (Note that some web sites give a different definition of Euler graph, and call these graphs "even" graphs.)
The objects being counted in this sequence are unlabeled.
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REFERENCES
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F. Buekenhout, ed., Handbook of Incidence Geometry, 1995, p. 881.
F. C. Bussemaker, R. A. Mathon and J. J. Seidel, Tables of two-graphs, T.H.-Report 79-WSK-05, Technological University Eindhoven, Dept. Mathematics, 1979; also pp. 71-112 of "Combinatorics and Graph Theory (Calcutta, 1980)", Lect. Notes Math. 885, 1981.
P. J. Cameron, Cohomological aspects of two-graphs, Math. Zeit., 157 (1977), 101-119.
P. J. Cameron and C. R. Johnson, The number of equivalence patterns of symmetric sign patterns, Discr. Math., 306 (2006), 3074-3077.
CRC Handbook of Combinatorial Designs, 1996, p. 687.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 114, (4.7.1).
Liskovec, V. A., Enumeration of Euler graphs. (Russian) Vesc Akad. Navuk BSSR Ser. Fz.-Mat. Navuk 1970 1970 no. 6, 38-46.
R. W. Robinson, Enumeration of Euler graphs, pp. 147-153 of F. Harary, editor, Proof Techniques in Graph Theory. Academic Press, NY, 1969.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1979.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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R. W. Robinson, Table of n, a(n) for n = 1..26
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
C. L. Mallows and N. J. A. Sloane, Two-graphs, switching classes and Euler graphs are equal in number, SIAM J. Appl. Math., 28 (1975), 876-880.
C. L. Mallows and N. J. A. Sloane, Two-graphs, switching classes and Euler graphs are equal in number, SIAM J. Appl. Math., 28 (1975), 876-880. [Copy on N. J. A. Sloane's Home Page]
N. J. A. Sloane, Switching classes of graphs with 4 nodes.
Eric Weisstein's World of Mathematics, Eulerian Graph.
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FORMULA
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a(n) = sum_{s} 2^M(s)/product_{i} i^s(i)*s(i)!, where the sum is over n-tuples s in [0..n]^n such that n=sum i*s(i), M(s) = sum_{i<j} s(i)*s(j)*gcd(i,j) + sum_{i} (s(i)*(floor[i/2]-1)+binomial(s(i),2)) + sign(sum_{k} s(2k+1)). [Robinson's formula, from Mallows & Sloane, simplified.] - M. F. Hasler, Apr 15 2012
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EXAMPLE
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Comment from Joerg Arndt, Feb 05 2010: The a(4) = 3 Euler graphs on four nodes are:
1)
o o
o o
2)
o-o
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o o
3)
o-o
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o-o
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PROG
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(PARI) A002854(n)={ /* Robinson's formula, simplified */
my( s=0, N(S)=sum( j=2, #S, S[j]*sum( i=1, j-1, S[i]*gcd(i, j))) + sum( i=1, #S, i*binomial(S[i], 2)+(i\2-1)*S[i]) + !!vecextract(S, 4^round(#S/2)\3)); forvec( S=vector(n, i, [0, n\i]), my(t=0); for( i=1, n, if( n < t+=i*S[i], until(i++>n, S[i]=n); next(2))); t==n & s+=2^N(S)/prod(i=1, n, i^S[i]*S[i]!)); s} \\ - M. F. Hasler, Apr 09 2012
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CROSSREFS
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Cf. A003049, A085618, A085619, A085620, A007127, A133736.
Bisections: A182012, A182055.
Sequence in context: A143884 A122031 A089125 * A036356 A034732 A000278
Adjacent sequences: A002851 A002852 A002853 * A002855 A002856 A002857
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Vladeta Jovovic, Apr 18 2000
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STATUS
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approved
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