A058350 -id:A058350 - 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

A058349 Number of connected labeled series-parallel posets on n nodes.

Original entry on oeis.org

1, 2, 12, 122, 1740, 31922, 715932, 18978122, 580513260, 20125554242, 779832497532, 33398722757402, 1566656717322060, 79879485803841362, 4398701789915269212, 260166428897541369962, 16449181879032096013740, 1107112451498156565581282, 79030557433744270179981372

Offset: 1

N. J. A. Sloane, Dec 16 2000

Also, number of labeled blobs with n edges.

R. C. Read, Graphical enumeration by cycle-index sums: first steps toward a unified treatment, preprint, Sept. 26, 1991.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.39, page 133, g(n).

Vincenzo Librandi, Table of n, a(n) for n = 1..100
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

A053554(n) = a(n) + A058350(n) (n>=2).

(continue from A053554) t1 := log(1+EGF053554): t2 := series(t1,x,30); SERIESTOLISTMULT(t2);

Drop[ CoefficientList[ InverseSeries[ Series[x + 2*(1 - Cosh[x]) , {x, 0, 19}], y], y], 1]* Range[19]! (* Jean-François Alcover, Sep 21 2011, after g.f. *)

a(n):=if n=1 then 1 else (n-1)!*sum(binomial(n+k-1,n-1)*sum(binomial(k,j)*((sum((binomial(j,l)*((-1)^(n-l+j-1)+1)*sum(binomial(j-l,r)*2^(j-l-r-1)*(-1)^(r-j)*sum((r-2*i)^(n-l+j-1)*binomial(r,i),i,0,r),r,1,j-l))/(n-l+j-1)!,l,0,j-1))),j,1,k),k,1,n-1); /* Vladimir Kruchinin, Feb 19 2012 */

/* Joerg Arndt, Feb 04 2011 */
x='x+O('x^55); t=x+2*(1-cosh(x));
Vec(serlaplace(serreverse(t))) /* show terms */

Read (1991) reference gives generating functions (see PARI code for one example).
A048172(n) = a(n)+A048174(n), n>1.
a(n) = (n-1)!*sum(k=1..n-1, binomial(n+k-1,n-1)*sum(j=1..k, binomial(k,j)*((sum(l=0..j-1, (binomial(j,l)*((-1)^(n-l+j-1)+1)*sum(r=1..j-l, binomial(j-l,r)*2^(j-l-r-1)*(-1)^(r-j)*sum(i=0..r, (r-2*i)^(n-l+j-1)*binomial(r,i))))/(n-l+j-1)!))))), n>1, a(1)=1. - Vladimir Kruchinin, Feb 19 2012
a(n) ~ n^(n-1) / (5^(1/4)*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

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

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A048174 Number of labeled chains with n edges.

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A058349 Number of connected labeled series-parallel posets on n nodes.

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