A129427 - OEIS

%I #45 Mar 07 2022 07:33:59

%S 1,2,8,31,140,722,4439,32654,289519,3054067,37584620,527968286,

%T 8308434931,144345554051,2738280739075,56245013793246,

%U 1242596591479816,29366532494796900,739033832149588904,19726887762569763453

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

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

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

%H R. de Mello Koch, S. Ramgoolam, <a href="https://doi.org/10.1103/PhysRevD.85.026007">Strings from Feynman graph counting: Without large N</a>, Phys. Rev. D 85 (2012) 026007

%H P. A. Morris, <a href="/A003175/a003175.pdf">Letter to N. J. A. Sloane, Mar 02 1971</a>.

%H R. C. Read, <a href="http://dx.doi.org/10.1112/jlms/s1-34.4.417">The enumeration of locally restricted graphs (I)</a>, J. London Math. Soc. 34 (1959) 417-436. - _Jason Kimberley_, Sep 17 2009

%F a(n)=N\{S_{2n}[S_3] * S_{3n}[S_2]\}. - _Jason Kimberley_, Sep 17 2009

%o (Sage)

%o h = SymmetricFunctions(QQ).homogeneous()

%o def A129427(n):

%o     X = h([2*n]).plethysm(h([3]))

%o     Y = h([3*n]).plethysm(h([2]))

%o     return X.scalar(Y)

%o # _Bruce Westbury_, Aug 16 2013

%Y Column k=3 of A167625.

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

%K nonn

%O 0,2

%A _Brendan McKay_, Apr 15 2007

%E Using equation (5.8) of Read 1959, new terms a(12) and a(13) were computed in MAGMA by _Jason Kimberley_, Sep 17 2009

%E Further terms a(14)-a(16) also computed by _Jason Kimberley_, announced Nov 09 2009

%E Formula corrected from n vertices to 2n vertices by _Jason Kimberley_, Nov 09 2009

%E Added a(0). - _N. J. A. Sloane_, Aug 26 2013

%E a(17)-a(19) from _Sean A. Irvine_, Oct 29 2016