A349276 Number of unlabeled P-series with n elements.

1, 2, 5, 13, 31, 76, 178, 423, 988, 2312, 5361, 12427, 28626, 65813, 150700, 344232, 783832, 1780650, 4034591, 9121571, 20576349, 46322816, 104079338, 233421517, 522574991, 1167974002, 2606282841, 5806953923, 12919314397, 28702716868, 63682839588, 141111193270

Offset: 1

Author

Salah Uddin Mohammad, Nov 12 2021

Keywords

• nonn

Comments

The class of all P-series is a subclass of the class of series-parallel posets and it contains the class of P-graphs as a subclass.

A poset is called a P-graph if it can be expressed as the ordinal sum of the antichain posets (including the singleton poset).

A poset is called a P-series if it is either a P-graph or it can be expressed as the direct sum of the P-graphs.

For example, all the 3-element posets are P-series, where only the connected posets and the antichains are P-graphs. On the other hand, the 4-element poset <{x,y,z,w},{x<.z, z<.w, y<.w, x||y, y||z}> and its dual are both series-parallel which are not the P-series. Here, by 'x<.z' we mean 'x is covered by z'.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..3217

Crossrefs

Cf. A003430 (series-parallel posets), A255047, A349488.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
      max(1, 2^(d-1)-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Jan 05 2022

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j]*Sum[d*
     Max[1, 2^(d - 1) - 1], {d, Divisors[j]}], {j, 1, n}]/n];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={EulerT(Vec((1 -2*x +2*x^2)/((1-x)*(1-2*x)) + O(x*x^n)))} \\ Andrew Howroyd, Nov 19 2021

Formula

a(n) = A255047(n-1) + A349488(n).

G.f: -1 + exp(Sum_{k>=1} B(x^k)/k) where B(x) = x*(1 - 2*x + 2*x^2)/((1 - x)*(1 - 2*x)). - Andrew Howroyd, Jan 06 2022