A007453 Number of unlabeled connected series-parallel posets with n nodes.

1, 1, 3, 9, 30, 103, 375, 1400, 5380, 21073, 83950, 338878, 1383576, 5702485, 23696081, 99163323, 417553252, 1767827220, 7520966100, 32135955585, 137849390424, 593407692685, 2562695780058, 11099806544050, 48206136562750, 209876865026303, 915840095739301

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• nice

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 100 terms from Jean-François Alcover)
• B. I. Bayoumi, M. H. El-Zahar and S. M. Khamis, Asymptotic enumeration of N-free partial orders, Order 6 (1989), 219-232.
• P. J. Cameron, On the probability of connectedness, Discrete Math., 167 (1997), 175-187.
• P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102.
• P. J. Cameron, Some sequences of integers, in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
• Soheir M. Khamis, Height counting of unlabeled interval and N-free posets, Discrete Math. 275 (2004), no. 1-3, 165-175.
• Index entries for sequences related to posets

Crossrefs

Cf. A003430, A007454.

Programs

Mathematica

terms = 25; A[_] = 1;
Do[A[x_] = Exp[Sum[(1/k)*(A[x^k] + 1/A[x^k] - 2 + x^k), {k, 1, terms+1}]] + O[x]^(terms+1) // Normal, terms+1];
A003430 = CoefficientList[A[x], x] // Rest;
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i-1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
EULERi[A003430] (* Jean-François Alcover, Jan 23 2020 *)

Formula

See the 1989 and 1997 papers by Cameron cited above for generating functions, and the 1997 paper for asymptotics.

Inverse Euler transform of A003430. - Sean A. Irvine, Jan 04 2018

a(n) = A003430(n) - A007454(n) for n > 1. - Sean A. Irvine, Jan 04 2018

Extensions

Name corrected by Salah Uddin Mohammad, Jun 07 2020