A058349 -id:A058349 - cp's OEIS frontend

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

N. J. A. Sloane

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

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.

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

Cf. A048174, A058349, A058350.

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

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

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 *)

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 */

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 */

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

More terms from Joerg Arndt, Feb 04 2011

A048174 Number of labeled chains with n edges.

Original entry on oeis.org

1, 1, 7, 73, 1051, 19381, 436087, 11585953, 354981571, 12322179901, 477938035807, 20485584143113, 961567521142411, 49054912287659461, 2702571588828034567, 159911968233095867953, 10114120854154243738771, 680943323845807848142861, 48622150270026820216099567, 3670113810844512283440027673

Offset: 1

N. J. A. Sloane

Number of labeled series-parallel posets on n nodes that are not a nontrivial ordinal sum.

Let ( T, < ) and ( U, < ) be posets with T and U disjoint. Their ordinal sum is ( T union U, < ) where x

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.39, page 133, h(n).

Vincenzo Librandi, Table of n, a(n) for n = 1..100
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, Enumeration of posets generated by disjoint unions and ordinal sums, Proc. Amer. Math. Soc. 45 (1974), 295-299
Index entries for sequences related to posets

with(gfun): f := series(ln(1+x)-x^2/(1+x), x, 30):
egf := seriestoseries(f, 'revogf'):
t := series(egf/(1+egf), x, 21):
seriestolist(t, 'Laplace');

lim = 20; Drop[ CoefficientList[ InverseSeries[ Series[-Log[1 - x] - x^2/(1 - x), {x, 0, lim}], y], y], 1]*Range[lim]! (* Jean-François Alcover, Sep 21 2011, after g.f. *)
max = 18; S053554 = InverseSeries[ Series[ Log[1+x] - x^2/(1+x), {x, 0, max}], x]; Drop[ CoefficientList[ Series[ S053554 / (1+S053554), {x, 0, max}], x]* Range[0, max]!, 1] (* Jean-François Alcover, Nov 29 2011, after Maple *)

a(n):=if n=1 then 1 else (sum((n+k-1)!*sum((-1)^(j)/(k-j)!*((sum(sum((binomial(-l+i-1,l-1)*(-1)^(n-i-1)*stirling1(n+j-i-1,j-l))/(l!*(n+j-i-1)!),i,2*l,n+l-1),l,1,j))+((-1)^(n-1)*stirling1(n+j-1,j))/(n+j-1)!),j,1,k),k,1,n-1)); /* Vladimir Kruchinin, Feb 19 2012 */

x='x+O('x^66); s=serreverse(log(1+x)-x^2/(1+x)); Vec(serlaplace(s/(1+s))) \\ Joerg Arndt, Mar 11 2014

Reference gives generating functions (see PARI code for one example).
A048172(n) = A058349(n) + a(n), n>1.
A053554(n) = A058349(n) + A058350(n) (n>=2).
a(n)=sum(k=1..n-1, (n+k-1)!*sum(j=1..k, (-1)^(j)/(k-j)!*((sum(l=1..j, sum(i=2*l..n+l-1, (binomial(-l+i-1,l-1)*(-1)^(n-i-1)*stirling1(n+j-i-1,j-l))/(l!*(n+j-i-1)!))))+((-1)^(n-1)*stirling1(n+j-1,j))/(n+j-1)!))). - Vladimir Kruchinin, Feb 19 2012
a(n) ~ (5-sqrt(5)) * n^(n-1) / (2*5^(3/4)*exp(n)*(2-sqrt(5)+log((1+sqrt(5))/2))^(n-1/2)). - Vaclav Kotesovec, Mar 09 2014

More terms from Joerg Arndt, Feb 04 2011

A339299 Number of essentially series oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.

Original entry on oeis.org

0, 2, 6, 72, 840, 14040, 276360, 6494880, 175452480, 5375311200, 183962227680, 6958070380800, 288200792880000, 12974113884251520, 630742839699772800, 32933429270386444800, 1838083950894102912000, 109201772719684867622400, 6880730833827011402841600

Offset: 1

Andrew Howroyd, Dec 22 2020

nonn

See A339301 for additional details.

A058349 is the case with multiple unit elements in parallel allowed.
A058380 is the case that order is not significant in series configurations.
Cf. A339288 (unlabeled), A339300, A339301.

seq(n, Z=x)={my(p=Z+O(x^2)); for(n=2, n, p = (1 + Z)*exp(p^2/(1+p)) - 1); Vec(serlaplace(p-p/(1+p)), -n)}

E.g.f.: P(x)^2/(1 - P(x)) where P(x) is the e.g.f. of A339300.
E.g.f.: B(x)^2/(1 + B(x)) where B(x) is the e.g.f. of A339301.
E.g.f.: B(log(1+x)) where x + B(x) is the e.g.f. of A058349.

Showing 1-4 of 4 results.

cp's OEIS Frontend

A048173 Duplicate of A058349.

Views

Author

Keywords

Comments

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

Extensions

A048174 Number of labeled chains with n edges.

Views

Author

Keywords

Comments

References

Links

Programs

Maple

Mathematica

Maxima

PARI

Formula

Extensions

A339299 Number of essentially series oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula