A349488 -id:A349488 - cp's OEIS frontend

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

Salah Uddin Mohammad, Nov 12 2021

nonn

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'.

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

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

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

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 *)

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

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

A350635 Triangle read by rows: T(n,k) is the number of n-element unlabeled P-series with k connected components.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 15, 10, 4, 1, 1, 31, 28, 11, 4, 1, 1, 63, 67, 31, 11, 4, 1, 1, 127, 167, 80, 32, 11, 4, 1, 1, 255, 388, 213, 83, 32, 11, 4, 1, 1, 511, 908, 534, 226, 84, 32, 11, 4, 1, 1, 1023, 2053, 1343, 580, 229, 84, 32, 11, 4, 1, 1

Offset: 1

Salah Uddin Mohammad, Jan 09 2022

			Triangle begins:
    1;
    1,   1;
    3,   1,  1;
    7,   4,  1,  1;
   15,  10,  4,  1,  1;
   31,  28, 11,  4,  1, 1;
   63,  67, 31, 11,  4, 1, 1;
  127, 167, 80, 32, 11, 4, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums give A349276.
Column 1 is A255047(n-1).
Cf.  A263864 (all posets), A349488 (disconnected).

B(x) = x*(1 - 2*x + 2*x^2)/((1 - x)*(1 - 2*x))
T(n)=[Vecrev(p/y) | p<-Vec(-1 + exp(sum(k=1, n, y^k*B(x^k)/k + O(x*x^n))))]
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 13 2022

G.f.: -1 + exp(Sum_{k>=1} y^k*B(x^k)/k) where B(x) = x*(1 - 2*x + 2*x^2)/((1 - x)*(1 - 2*x)). - Andrew Howroyd, Jan 13 2022

A350772 Triangle read by rows: T(n,k) is the number of n-element unlabeled series-parallel posets with k connected components.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 9, 4, 1, 1, 30, 12, 4, 1, 1, 103, 45, 13, 4, 1, 1, 375, 160, 48, 13, 4, 1, 1, 1400, 613, 175, 49, 13, 4, 1, 1, 5380, 2354, 680, 178, 49, 13, 4, 1, 1

Offset: 1

Salah Uddin Mohammad, Jan 14 2022

			Triangle begins:
     1;
     1,    1;
     3,    1,   1;
     9,    4,   1,   1;
    30,   12,   4,   1,  1;
   103,   45,  13,   4,  1,  1;
   375,  160,  48,  13,  4,  1, 1;
  1400,  613, 175,  49, 13,  4, 1, 1;
  5380, 2354, 680, 178, 49, 13, 4, 1, 1;
  ...

Row sums give A003430.
Column 1 is A007453.
Cf. A263864 (all posets), A349488 (disconnected).

A350783 Triangle read by rows: T(n,k) is the number of n-element unlabeled N-free posets with k connected components.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 9, 4, 1, 1, 31, 12, 4, 1, 1, 115, 46, 13, 4, 1, 1, 474, 173, 49, 13, 4, 1, 1, 2097, 727, 188, 50, 13, 4, 1, 1, 9967, 3195, 795, 191, 50, 13, 4, 1, 1, 50315, 15017, 3502, 810, 192, 50, 13, 4, 1, 1

Offset: 1

Salah Uddin Mohammad, Jan 16 2022

			Triangle begins:
      1;
      1,     1;
      3,     1,    1;
      9,     4,    1,   1;
     31,    12,    4,   1,   1;
    115,    46,   13,   4,   1,  1;
    474,   173,   49,  13,   4,  1,  1;
   2097,   727,  188,  50,  13,  4,  1, 1;
   9967,  3195,  795, 191,  50, 13,  4, 1, 1;
  50315, 15017, 3502, 810, 192, 50, 13, 4, 1, 1;
  ...

Salah Uddin Mohammad, Md. Shah Noor, and Md. Rashed Talukder, An Exact Enumeration of the Unlabeled Disconnected Posets, J. Int. Seq., Vol. 25 (2022), Article 22.5.4.

Row sums give A202182.
Column 1 is A202180.
Cf. A263864 (all posets), A349488 (disconnected).

A356558 Triangle read by rows: T(n,k), where n, k >= 2, is the number of n-element unlabeled connected series-parallel posets with k ordinal terms that are either the singleton or disconnected posets.

Original entry on oeis.org

1, 2, 1, 5, 3, 1, 16, 9, 4, 1, 52, 31, 14, 5, 1, 188, 108, 52, 20, 6, 1, 690, 402, 193, 80, 27, 7, 1, 2638, 1523, 744, 315, 116, 35, 8, 1, 10272, 5934, 2908, 1261, 483, 161, 44, 9, 1, 40782, 23505, 11580, 5085, 2010, 707, 216, 54, 10, 1

Offset: 2

Salah Uddin Mohammad, Aug 12 2022

If a poset P is obtained by taking the ordinal sum of the posets A and B, then the posets A and B are called the ordinal terms of P.

			Triangle begins:
      1;
      2,     1;
      5,     3,     1;
     16,     9,     4,    1;
     52,    31,    14,    5,    1;
    188,   108,    52,   20,    6,   1;
    690,   402,   193,   80,   27,   7,   1;
   2638,  1523,   744,  315,  116,  35,   8,  1;
  10272,  5934,  2908, 1261,  483, 161,  44,  9,  1;
  40782, 23505, 11580, 5085, 2010, 707, 216, 54, 10, 1;
The connected posets counted in the first three rows of the triangle are shown by using the Hasse diagram as follows:
-------
  o
  |
  o
--------------------------
                  |   o
    o     o   o   |   |
   / \     \ /    |   o
  o   o     o     |   |
                  |   o
----------------------------------------------------------
    o    o o o   o o    |                           |
   /|\    \|/    |X|    |                           |   o
  o o o    o     o o    |     o     o   o     o     |   |
                        |     |      \ /     / \    |   o
    o           o       |     o       o     o   o   |   |
    |          / \      |    / \      |      \ /    |   o
    o   o     o   \     |   o   o     o       o     |   |
     \ /      |    \    |                           |   o
      o       o     o   |                           |

Row sums give A007453.

Cf. A263864 (all posets), A349488 (disconnected).

cp's OEIS Frontend

A349276 Number of unlabeled P-series with n elements.

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A350635 Triangle read by rows: T(n,k) is the number of n-element unlabeled P-series with k connected components.

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A350772 Triangle read by rows: T(n,k) is the number of n-element unlabeled series-parallel posets with k connected components.

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A350783 Triangle read by rows: T(n,k) is the number of n-element unlabeled N-free posets with k connected components.

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A356558 Triangle read by rows: T(n,k), where n, k >= 2, is the number of n-element unlabeled connected series-parallel posets with k ordinal terms that are either the singleton or disconnected posets.

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