OFFSET
1,3
COMMENTS
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<y if x<y and both in T or both in U, or x in T and y in U. Note ordinal sum is not commutative.
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.39, page 133, h(n).
LINKS
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
FORMULA
Reference gives generating functions (see PARI code for one example).
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
MAPLE
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');
MATHEMATICA
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 *)
PROG
(Maxima)
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 */
(PARI) x='x+O('x^66); s=serreverse(log(1+x)-x^2/(1+x)); Vec(serlaplace(s/(1+s))) \\ Joerg Arndt, Mar 11 2014
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Joerg Arndt, Feb 04 2011
STATUS
approved