This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A002854 M0846 N0321 #159 Feb 16 2025 08:32:27
%S A002854 1,1,2,3,7,16,54,243,2038,33120,1182004,87723296,12886193064,
%T A002854 3633057074584,1944000150734320,1967881448329407496,
%U A002854 3768516017219786199856,13670271807937483065795200,94109042015724412679233018144,1232069666043220685614640133362240
%N A002854 Number of unlabeled Euler graphs with n nodes; number of unlabeled two-graphs with n nodes; number of unlabeled switching classes of graphs with n nodes; number of switching classes of unlabeled signed complete graphs on n nodes; number of Seidel matrices of order n.
%C A002854 Also called Eulerian graphs of strength 1.
%C A002854 "Switching" a graph 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.
%C A002854 "Switching" a signed simple graph at a node negates the signs of all edges incident with that node.
%C A002854 A graph is an Euler graph iff every node has even degree. It need not be connected. (Note that some graph theorists require an Euler graph to be connected so it has an Euler circuit, and call these graphs "even" graphs.)
%C A002854 The objects being counted in this sequence are unlabeled.
%D A002854 F. Buekenhout, ed., Handbook of Incidence Geometry, 1995, p. 881.
%D A002854 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.
%D A002854 CRC Handbook of Combinatorial Designs, 1996, p. 687.
%D A002854 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 114, Eq. (4.7.1).
%D A002854 R. W. Robinson, Enumeration of Euler graphs, pp. 147-153 of F. Harary, editor, Proof Techniques in Graph Theory. Academic Press, NY, 1969.
%D A002854 R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1979.
%D A002854 J. J. Seidel, A survey of two-graphs, pp. 481-511 of Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973), Vol. I, Accademia Nazionale dei Lincei, Rome, 1976; also pp. 146-176 in Geometry and Combinatorics: Selected Works of J.J. Seidel, ed. D.G. Corneil and R. Mathon, Academic Press, Boston, 1991..
%D A002854 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002854 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002854 Max Alekseyev, <a href="/A002854/b002854.txt">Table of n, a(n) for n = 1..88</a> (terms 1..26 from R. W. Robinson).
%H A002854 P. J. Cameron, <a href="https://doi.org/10.1007/BF01215145">Cohomological aspects of two-graphs</a>, Math. Zeit., 157 (1977), 101-119.
%H A002854 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs., 3 (2000), #00.1.5.
%H A002854 P. J. Cameron and C. R. Johnson, <a href="http://dx.doi.org/10.1016/j.disc.2004.10.029">The number of equivalence patterns of symmetric sign patterns</a>, Discr. Math., 306 (2006), 3074-3077.
%H A002854 G. Greaves, J. H. Koolen, A. Munemasa, and F. Szöllősi, <a href="http://arxiv.org/abs/1403.2155">Equiangular lines in Euclidean spaces</a>, arXiv:1403.2155 [math.CO], 2014.
%H A002854 Akihiro Higashitani and Kenta Ueyama, <a href="https://arxiv.org/abs/2107.12927">Combinatorial classification of (+/-1)-skew projective spaces</a>, arXiv:2107.12927 [math.RA], 2021.
%H A002854 Akihiro Higashitani and Kenta Ueyama, <a href="https://arxiv.org/abs/2409.10904">Combinatorics of graded module categories over skew polynomial algebras at roots of unity</a>, arXiv:2409.10904 [math.RA], 2024. See p. 11.
%H A002854 T. R. Hoffman and J. P. Solazzo, <a href="http://arxiv.org/abs/1408.0334">Complex Two-Graphs via Equiangular Tight Frames</a>, arXiv:1408.0334 [math.CO], 2014.
%H A002854 Michael Hofmeister, <a href="https://dx.doi.org/10.1002/jgt.3190120316">Counting double covers of graphs</a>, Journal of Graph Theory 12.3 (1988), 437-444. (Beware of a typo!)
%H A002854 V. A. Liskovec, <a href="/A002854/a002854.pdf">Enumeration of Euler Graphs</a>, (in Russian), Akademiia Navuk BSSR, Minsk., 6 (1970), 38-46. (annotated scanned copy)
%H A002854 C. L. Mallows and N. J. A. Sloane, <a href="http://www.jstor.org/stable/2100368">Two-graphs, switching classes and Euler graphs are equal in number</a>, SIAM J. Appl. Math., 28 (1975), 876-880. (<a href="http://neilsloane.com/doc/MallowsSloane.pdf">copy</a> at N. J. A. Sloane's home page)
%H A002854 Brendan D. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/data/graphs.html">Eulerian graphs</a>
%H A002854 R. E. Peile, <a href="/A005273/a005273.pdf">Letter to N. J. A. Sloane, Feb 1989</a>.
%H A002854 R. C. Read, <a href="/A002854/a002854_1.pdf">Letter to N. J. A. Sloane, Nov. 1976</a>.
%H A002854 R. W. Robinson, <a href="/A003049/a003049.pdf">Enumeration of Euler graphs</a>, pp. 147-153 of F. Harary, editor, Proof Techniques in Graph Theory. Academic Press, NY, 1969. (Annotated scanned copy)
%H A002854 N. J. A. Sloane, <a href="/A002854/a002854.gif">Switching classes of graphs with 4 nodes</a>.
%H A002854 F. Szöllosi and Patric R. J. Östergård, <a href="https://arxiv.org/abs/1703.02943">Enumeration of Seidel matrices</a>, arXiv:1703.02943 [math.CO], 2017.
%H A002854 E. Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerianGraph.html">Eulerian Graph</a>.
%H A002854 T. Zaslavsky, <a href="https://doi.org/10.1016/0166-218X(82)90033-6">Signed graphs</a>, Discrete Appl. Math. 4 (1982), 47-74.
%F A002854 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) + i*binomial(s(i),2)) + sign(Sum_{k} s(2*k+1)). [Robinson's formula, from Mallows & Sloane, simplified.] - _M. F. Hasler_, Apr 15 2012; corrected by _Sean A. Irvine_, Nov 05 2014
%e A002854 From _Joerg Arndt_, Feb 05 2010: (Start)
%e A002854 The a(4) = 3 Euler graphs on four nodes are:
%e A002854 1) o o 2) o-o 3) o-o
%e A002854 o o |/ | |
%e A002854 o o o-o
%e A002854 (End)
%o A002854 (PARI) A002854(n)={ /* Robinson's formula, simplified */ local(s=vector(n)); my( S=0, M()=sum( j=2,n, s[j]*sum( i=1,j-1, s[i]*gcd(i,j))) + sum( i=1,n, i*binomial(s[i],2)+(i\2-1)*s[i]) + !!vecextract(s,4^round(n/2)\3), inc()=!forstep(i=n,1,-1,s[i]<n\i && s[i]++ && return; s[i]=0), t); until(inc(), t=0; for( i=1,n, if( n < t+=i*s[i], until(i++>n, s[i]=n); next(2))); t==n && S+=2^M()/prod(i=1,n,i^s[i]*s[i]!)); S} \\ _M. F. Hasler_, Apr 09 2012, adapted for current PARI version on Apr 12, 2018
%o A002854 (Python)
%o A002854 from itertools import combinations
%o A002854 from math import prod, factorial, gcd
%o A002854 from fractions import Fraction
%o A002854 from sympy.utilities.iterables import partitions
%o A002854 def A002854(n): return int(sum(Fraction(1<<sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum(((q>>1)-1)*r+(q*r*(r-1)>>1) for q, r in p.items())+any(q&1 for q in p),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # _Chai Wah Wu_, Jul 03 2024
%Y A002854 Cf. A000666, A003049, A085618, A085619, A085620, A007127, A133736.
%Y A002854 Bisections: A182012, A182055.
%Y A002854 Row sums of A341941.
%K A002854 nonn,easy,nice
%O A002854 1,3
%A A002854 _N. J. A. Sloane_
%E A002854 Terms up to a(18) confirmed by _Vladeta Jovovic_, Apr 18 2000
%E A002854 Name edited (changed "2-graph" to "two-graph" to avoid confusion with other 2-graphs) and comments on Eulerian graphs by _Thomas Zaslavsky_, Nov 21 2013
%E A002854 Name clarified by _Thomas Zaslavsky_, Apr 18 2019