A263864 -id:A263864 - cp's OEIS frontend

A000608 Number of connected partially ordered sets with n unlabeled elements.

1, 1, 1, 3, 10, 44, 238, 1650, 14512, 163341, 2360719, 43944974, 1055019099, 32664984238, 1303143553205, 66900392672168, 4413439778321689

Offset: 0

N. J. A. Sloane, Simon Plouffe, J. A. Wright

K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961.
G. Melançon, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gunnar Brinkmann and Brendan D. McKay, Counting unlabeled topologies and transitive relations.
G. Brinkmann and B. D. McKay, Counting unlabeled topologies and transitive relations, J. Integer Sequences, Volume 8, 2005.
G. Brinkmann and B. D. McKay, Posets on up to 16 Points [On Brendan McKay's home page]
G. Brinkmann and B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179.
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184. [Annotated scan of pages 180 and 183 only]
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961. [Annotated scanned copy]
Henry Sharp, Jr., Quasi-orderings and topologies on finite sets, Proceedings of the American Mathematical Society 17.6 (1966): 1344-1349. [Annotated scanned copy]
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
N. J. A. Sloane, Transforms
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 10 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
N. L. White, Two letters to N. J. A. Sloane, 1970, with hand-drawn enclosure
Wikipedia, Illustration n=4, Illustration n=5, Illustration n=6" (2021)
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970. [Annotated scanned copy]
J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.
J. A. Wright, Two related abstracts, 1970 and 1972 [Annotated scanned copies]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences
Index entries for sequences related to posets

Inverse Euler transform of A000112.

Cf. A263864 (multiset transform), A342500 (refined by rank).

A000112 = Cases[Import["https://oeis.org/A000112/b000112.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
{1} ~Join~ EulerInvTransform[Rest[A000112]] (* Jean-François Alcover, Dec 04 2019, updated Mar 17 2020 *)

More terms from Christian G. Bower, who pointed out connection with A000112, Jan 21 1998 and Dec 12 2001

More terms from Vladeta Jovovic, Jan 04 2006; corrected Jan 15 2006

A342447 T(n,e) is the number of unlabeled posets of n>=0 points with e>=0 arcs in the Hasse diagram, irregular triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 8, 2, 1, 1, 4, 11, 29, 12, 5, 1, 1, 4, 12, 43, 105, 92, 45, 12, 3, 1, 1, 4, 12, 46, 156, 460, 582, 487, 204, 71, 14, 7, 1, 1, 4, 12, 47, 170, 670, 2097, 3822, 4514, 3271, 1579, 561, 186, 44, 16, 4, 1, 1, 4, 12, 47, 173, 731, 2954, 10513, 24584, 40182

Offset: 0

R. J. Mathar, Mar 12 2021

Maximal e for a given n (i.e., the length of the n-th row minus 1) is A002620(n), see Mathematics StackExchange. - Andrey Zabolotskiy, Mar 12 2021

			The table starts
1 ;
1 ;
1 1 ;
1 1 3 ;
1 1 4  8  2 ;
1 1 4 11 29  12   5 ;
1 1 4 12 43 105  92   45   12    3 ;
1 1 4 12 46 156 460  582  487  204   71   14   7 ;
1 1 4 12 47 170 670 2097 3822 4514 3271 1579 561 186 44 16 4 ;
...
T(4,0) = 1: the 4-point poset with no relations, 4 isolated points in the Hasse diagram.
T(4,1) = 1: the 4-point poset with one relation, the Hasse diagram has one vertical line and 2 isolated points.
T(4,2) = 4: the 4 posets contributing to A022016(4) = 4, extended by additional isolated point when the number of points is less than 4.
T(4,3) = 8: the 8 posets contributing to A022017(3).
T(4,4) = 2: the "dagaz rune" poset {1<3, 2<3, 1<4, 2<4}
  o o
  |X|
  o o
and the "diamond" poset {1<2, 1<3, 2<4, 3<4}
    o
   / \
  o   o
   \ /
    o

R. J. Mathar, Table of T(n,e) for n<=10
Brendan McKay, Digraphs, posets tables at the bottom of the page.
R. J. Mathar, Poset Illustrations
Mathematics StackExchange, Maximum number of comparisons required to define a partial ordering, 2021.

Cf. A000112 (row sums), A263864, A022016 (convergents down rows), A002620, A342472 (lower bound row length), A342590 (connected), A342589 (labeled), A376633 (self-dual).

T(n,0) = T(n,1) = 1.

T(n,e) = A022016(e) for n >= 2e.

T(0,0) = 1 prepended and "conjecture" removed from A022016 formula. Andrey Zabolotskiy, Mar 12 2021

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

A000608 Number of connected partially ordered sets with n unlabeled elements.

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A342447 T(n,e) is the number of unlabeled posets of n>=0 points with e>=0 arcs in the Hasse diagram, irregular triangle read by rows.

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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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