A129427 Number of isomorphism classes of 3-regular multigraphs of order 2n, loops allowed.

1, 2, 8, 31, 140, 722, 4439, 32654, 289519, 3054067, 37584620, 527968286, 8308434931, 144345554051, 2738280739075, 56245013793246, 1242596591479816, 29366532494796900, 739033832149588904, 19726887762569763453

Offset: 0

Author

Brendan McKay, Apr 15 2007

Keywords

• nonn

Comments

a(1)..a(11) computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

References

• P. A. Morris, Letter to N. J. A. Sloane, Mar 02 1971.

Links

• R. de Mello Koch, S. Ramgoolam, Strings from Feynman graph counting: Without large N, Phys. Rev. D 85 (2012) 026007
• P. A. Morris, Letter to N. J. A. Sloane, Mar 02 1971.
• R. C. Read, The enumeration of locally restricted graphs (I), J. London Math. Soc. 34 (1959) 417-436. - _Jason Kimberley_, Sep 17 2009

Crossrefs

Column k=3 of A167625.

Cf. A005967 (connected, inv. Euler trans.), A129416, A129429, A129431, A129433, A129435, A129437, A005638.

Programs

Sage

h = SymmetricFunctions(QQ).homogeneous()
def A129427(n):
    X = h([2*n]).plethysm(h([3]))
    Y = h([3*n]).plethysm(h([2]))
    return X.scalar(Y)
# Bruce Westbury, Aug 16 2013

Formula

a(n)=N\{S_{2n}[S_3] * S_{3n}[S_2]\}. - Jason Kimberley, Sep 17 2009

Extensions

Using equation (5.8) of Read 1959, new terms a(12) and a(13) were computed in MAGMA by Jason Kimberley, Sep 17 2009

Further terms a(14)-a(16) also computed by Jason Kimberley, announced Nov 09 2009

Formula corrected from n vertices to 2n vertices by Jason Kimberley, Nov 09 2009

Added a(0). - N. J. A. Sloane, Aug 26 2013

a(17)-a(19) from Sean A. Irvine, Oct 29 2016