A342500 -id:A342500 - 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E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961.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E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961.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

A007776 Number of connected posets with n elements of height 1.

Original entry on oeis.org

1, 2, 4, 10, 27, 88, 328, 1460, 7799, 51196, 422521, 4483460, 62330116, 1150504224, 28434624153, 945480850638, 42417674401330, 2572198227615998, 211135833162079184, 23487811567341121158, 3545543330739039981738, 727053904070651775719646

Offset: 2

Georg Wambach (gw(AT)informatik.Uni-Koeln.de)

nonn

Inverse Euler transform of A048194 and A049312. - Detlef Pauly (dettodet(AT)yahoo.de) and Vladeta Jovovic, Jul 25 2003
Essentially the same as A318870. - Georg Fischer, Oct 02 2018
Number of connected digraphs on n unlabeled nodes where every node has indegree 0 or outdegree 0 and there are no isolated nodes.  - Andrew Howroyd, Oct 03 2018

Andrew Howroyd, Table of n, a(n) for n = 2..50 (terms 2..40 from Alois P. Heinz)
N. J. A. Sloane, Transforms
J. Textor, A. Idelberger, and M. Liskiewicz, Learning from Pairwise Marginal Independencies, arXiv:1508.00280 [cs.AI], 2015.
Index entries for sequences related to posets

Cf. A005142, A002031 (labeled case), A048194, A049312, A055192, A318870, column 1 of A342500.

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]];
b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
b[d_] := Sum[A[n, d - n], {n, 0, d}];
EULERi[Array[b, 30]] // Rest (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz in A049312 *)

Inverse Euler transform of A055192. - Andrew Howroyd, Oct 03 2018

More terms from Vladeta Jovovic, Jul 25 2003

Offset corrected by Andrew Howroyd, Oct 03 2018

A263859 Triangle read by rows: T(n,k) (n>=1, k>=0) is the number of posets with n elements and rank k (or depth k+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 6, 1, 1, 20, 31, 10, 1, 1, 55, 162, 84, 15, 1, 1, 163, 940, 734, 185, 21, 1, 1, 556, 6372, 7305, 2380, 356, 28, 1, 1, 2222, 52336, 86683, 35070, 6259, 623, 36, 1, 1, 10765, 534741, 1261371, 619489, 125597, 14258, 1016, 45, 1

Offset: 1

Christian Stump, Oct 28 2015

Row sums give A000112, n >= 1.

The rank of a poset is the number of cover relations in a maximal chain.

			Triangle begins:
1,
1,1,
1,3,1,
1,8,6,1,
1,20,31,10,1,
1,55,162,84,15,1,
1,163,940,734,185,21,1,
1,556,6372,7305,2380,356,28,1,
1,2222,52336,86683,35070,6259,623,36,1,
1,10765,534741,1261371,619489,125597,14258,1016,45,1,
...

R. J. Mathar, Table of n, a(n) for n = 1..136
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.
FindStat - Combinatorial Statistic Finder, The rank of the poset.
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.)

Cf. A000112 (row sums), A342500 (connected).

More terms from Brinkmann-McKay (2002) added by N. J. A. Sloane, Mar 18 2017

A342501 T(n,k) is the number of connected labeled posets with n elements and rank k: triangle read by rows.

Original entry on oeis.org

1, 0, 2, 0, 6, 6, 0, 38, 84, 24, 0, 390, 1710, 840, 120, 0, 6062, 49740, 36840, 8280, 720, 0, 134526, 2050566, 2184000, 646800, 85680, 5040, 0, 4172198, 118645044, 177549624, 65313360, 10735200, 947520, 40320

Offset: 1

R. J. Mathar, Mar 14 2021

This is a variant of A342587 admitting only connected posets.

			The table starts in row n=1 and shows ranks k>=0:
1: 1
2: 0 2
3: 0 6 6
4: 0 38 84 24
5: 0 390 1710 840 120
6: 0 6062 49740 36840 8280 720
7: 0 134526 2050566 2184000 646800 85680 5040
8: 0 4172198 118645044 177549624 65313360 10735200 947520 40320

Index entries for sequences related to posets

Cf. A001927 (row sums), A000142 (diagonal), A002031/A002027 (rank 1), A342500 (unlabeled).

A361954 Triangle read by rows: T(n,k) is the number of unlabeled connected weakly graded (ranked) posets with n elements and rank k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 10, 23, 8, 1, 0, 27, 107, 56, 11, 1, 0, 88, 557, 388, 98, 14, 1, 0, 328, 3271, 2888, 839, 149, 17, 1, 0, 1460, 22424, 23900, 7512, 1470, 209, 20, 1, 0, 7799, 183273, 226807, 73405, 14715, 2308, 278, 23, 1

Offset: 1

Andrew Howroyd, Mar 31 2023

Here weakly graded means that there exists a rank function rk from the poset to the integers such that whenever v covers w in the poset, we have rk(v) = rk(w) + 1.

			Triangle begins:
  1;
  0,   1;
  0,   2,    1;
  0,   4,    5,    1;
  0,  10,   23,    8,   1;
  0,  27,  107,   56,  11,   1;
  0,  88,  557,  388,  98,  14,  1;
  0, 328, 3271, 2888, 839, 149, 17, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..820 (rows 1..40)
Wikipedia, Graded poset.

Column k=2 is A007776.
Row sums are A361955.
Cf. A342500, A361953 (not necessarily connected).

\\ See PARI link in A361953 for program code.
{ my(A=A361954tabl(8)); for(i=1, #A, print(A[i, 1..i])) }

A349401 Number of unlabeled disconnected posets with n elements.

Original entry on oeis.org

0, 0, 1, 2, 6, 19, 80, 395, 2487, 19890, 206565, 2804453, 49872647, 1158843214, 35049606566, 1374685228988, 69690886873398, 4554367168841547

Offset: 0

Salah Uddin Mohammad, Nov 15 2021

Gunnar Brinkmann and Brendan D. McKay, Posets on up to 16 points.
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.

Cf. A000112, A000608, A342500.

a(n) = A000112(n) - A000608(n).

cp's OEIS Frontend

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

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A007776 Number of connected posets with n elements of height 1.

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A263859 Triangle read by rows: T(n,k) (n>=1, k>=0) is the number of posets with n elements and rank k (or depth k+1).

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A342501 T(n,k) is the number of connected labeled posets with n elements and rank k: triangle read by rows.

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A361954 Triangle read by rows: T(n,k) is the number of unlabeled connected weakly graded (ranked) posets with n elements and rank k.

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Keywords

Comments

Examples

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Programs

PARI

A349401 Number of unlabeled disconnected posets with n elements.

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