

A006024


Number of labeled mating graphs with n nodes. Also called pointdetermining graphs.
(Formerly M3650)


27



1, 1, 1, 4, 32, 588, 21476, 1551368, 218608712, 60071657408, 32307552561088, 34179798520396032, 71474651351939175424, 296572048493274368856832, 2448649084251501449508762880, 40306353989748719650902623919616
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OFFSET

0,4


COMMENTS

A mating graph is one in which no two vertices have identical adjacencies with the other vertices.  R. C. Read (rcread(AT)math.uwaterloo.ca) and Vladeta Jovovic, Feb 10 2003
Also number of (n1)node labeled mating graphs allowing loops and without isolated nodes.  Vladeta Jovovic, Mar 08 2008


REFERENCES

R. C. Read, The Enumeration of MatingType Graphs. Report CORR 8938, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..50
I. M. Gessel and J. Li, Enumeration of PointDetermining Graphs, arXiv:0705.0042 [math.CO], 20072009.
I. M. Gessel and J. Li, Enumeration of pointdetermining graphs, J. Combinatorial Theory Ser. A 118 (2011), 591612.
R. C. Read, The Enumeration of MatingType Graphs, Dept. Combinatorics and Optimization, Univ. Waterloo, Oct 1989. (Annotated scanned copy)
D. Sumner, Point determination in graphs, Discrete Mathematics 5 (1973), 179187.


FORMULA

a(n) = Sum_{k=0..n} Stirling1(n, k)*2^binomial(k, 2).  R. C. Read (rcread(AT)math.uwaterloo.ca) and Vladeta Jovovic, Feb 10 2003
E.g.f.: Sum_{n>=0} 2^(n(n1)/2)*log(1+x)^n/n!.  Paul D. Hanna, May 20 2009


EXAMPLE

Consider the square (cycle of length 4) on vertices 1, 2, 3 and 4 in that order. Join a fifth vertex (5) to vertices 1, 3 and 4. The resulting graph is not a mating graph since vertices 1 and 3 both have the set {2, 4, 5} as neighbors. If we delete the edge (1,5) then the resulting graph is a mating graph: the neighborhood sets for vertices 1, 2, 3, 4 and 5 are respectively {2,4}, {1,3}, {2,4,5}, {1,3,5} and {3,4}  all different.


MATHEMATICA

a[n_] := Sum[StirlingS1[n, k] 2^Binomial[k, 2], {k, 0, n}];
Array[a, 15] (* JeanFrançois Alcover, Jul 25 2018 *)


PROG

(PARI) a(n)=n!*polcoeff(sum(k=0, n, 2^(k*(k1)/2)*log(1+x+x*O(x^n))^k/k!), n) \\ Paul D. Hanna, May 20 2009


CROSSREFS

Cf. A006025.
Cf. bipointdetermining graphs: labeled A129583, unlabeled A129584; connected bipointdetermining graphs: labeled A129585, unlabeled A129586; phylogenetic trees: labeled A000311, unlabeled A000669.
Cf. A007833, A079306 (connected)
Sequence in context: A086899 A219149 A013041 * A118995 A222829 A134087
Adjacent sequences: A006021 A006022 A006023 * A006025 A006026 A006027


KEYWORD

nonn


AUTHOR

Simon Plouffe


EXTENSIONS

More terms from R. C. Read (rcread(AT)math.uwaterloo.ca) and Vladeta Jovovic, Feb 10 2003
a(0)=1 prepended by Andrew Howroyd, Sep 09 2018


STATUS

approved



