A048172 Number of labeled series-parallel graphs with n edges.

1, 3, 19, 195, 2791, 51303, 1152019, 30564075, 935494831, 32447734143, 1257770533339, 53884306900515, 2528224238464471, 128934398091500823, 7101273378743303779, 420078397130637237915, 26563302733186339752511, 1788055775343964413724143, 127652707703771090396080939

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Labeled N-free posets. - Detlef Pauly (dettodet(AT)yahoo.de), Dec 27 2002

References

• Ronald C. Read, Graphical enumeration by cycle-index sums: first steps toward a unified treatment, Research Report CORR 91-19, University of Waterloo, Sept 1991.
• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.39.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..100
• F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
• Frédéric Fauvet, L. Foissy, D. Manchon, Operads of finite posets, arXiv preprint arXiv:1604.08149 [math.CO], 2016.
• S. R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
• Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.
• R. P. Stanley, Enumeration of posets generated by disjoint unions and ordinal sums, Proc. Amer. Math. Soc. 45 (1974), 295-299
• Index entries for reversions of series
• Index entries for sequences related to posets

Crossrefs

Cf. A048174, A058349, A058350.

Cf. A000112 (unlabeled posets), A001035 (labeled posets), A003430 (unlabeled analog).

Programs

Maple

with(gfun):
f := series((ln(1+x)-x^2/(1+x)), x, 21):
egf := seriestoseries(f, 'revogf'):
seriestolist(egf, 'Laplace');

Mathematica

lim = 19; Join[{1}, Drop[ CoefficientList[ InverseSeries[ Series[x + 2*(1 - Cosh[x]) , {x, 0, lim}], y] + InverseSeries[ Series[-Log[1 - x] - x^2/(1 - x),{x, 0, lim}], y], y], 2]*Range[2, lim]!] (* Jean-François Alcover, Sep 21 2011, after g.f. *)
m = 17; Rest[CoefficientList[InverseSeries[Series[Log[1+x]-x^2/(1+x), {x, 0, m}], x], x]*Table[k!,{k, 0, m}]](* Jean-François Alcover, Apr 18 2011 *)

Maxima

h(n,k):=if n=k then 0 else (-1)^(n-k)*binomial(n-k-1,k-1); a(n):=if n=1 then 1 else -sum((k!/n!*stirling1(n,k)+sum(binomial(k,j)*sum((j)!/(i)!*stirling1(i,j)*h(n-i,k-j),i,j,n-k+j),j,1,k-1)+h(n,k))*a(k),k,1,n-1); /* Vladimir Kruchinin, Sep 08 2010 */

PARI

x='x+O('x^55);
s=-log(1-x)-x^2/(1-x);
A048174=Vec(serlaplace(serreverse(s)));
t=x+2*(1-cosh(x));
A058349=Vec(serlaplace(serreverse(t)));
A048172=A048174+A058349;  A048172[1]-=1;
A048172 /* Joerg Arndt, Feb 04 2011 */

Formula

a(n) = A058349(n) + A048174(n).

a(n) = A058349(n) + A058350(n) (n>=2).

Reference (by Ronald C. Read) gives generating functions.

E.g.f. is reversion of log(1+x)-x^2/(1+x).

a(n)=if n=1 then 1 else -sum((k!/n!*stirling1(n,k)+sum(binomial(k,j)*sum((j)!/(i)!*stirling1(i,j)*h(n-i,k-j),i,j,n-k+j),j,1,k-1)+h(n,k))*a(k),k,1,n-1), h(n,k)=if n=k then 0 else (-1)^(n-k)*binomial(n-k-1,k-1), n>0. - Vladimir Kruchinin, Sep 08 2010

a(n) ~ sqrt((5+3*sqrt(5))/10) * n^(n-1) / (exp(n) * (2 - sqrt(5) + log((1+sqrt(5))/2))^(n-1/2)). - Vaclav Kotesovec, Feb 25 2014

Extensions

More terms from Joerg Arndt, Feb 04 2011