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

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

Offset: 1

Author

N. J. A. Sloane, Dec 16 2000

Keywords

• nonn
• easy
• nice

Comments

Also, number of labeled blobs with n edges.

References

• 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).

Links

• 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

Crossrefs

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

Programs

Maple

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

Mathematica

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. *)

Maxima

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 */

PARI

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

Formula

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

Extensions

More terms from Joerg Arndt, Feb 04 2011