A342501 -id:A342501 - cp's OEIS frontend

A001927 Number of connected partially ordered sets with n labeled points.

1, 1, 2, 12, 146, 3060, 101642, 5106612, 377403266, 40299722580, 6138497261882, 1320327172853172, 397571105288091506, 166330355795371103700, 96036130723851671469482, 76070282980382554147600692, 82226869197428315925408327266, 120722306604121583767045993825620, 239727397782668638856762574296226842

Offset: 0

N. J. A. Sloane

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.
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).

C. M. Bender et al., Combinatorics and field theory, arXiv:quant-ph/0604164, 2006.
G. Brinkmann, B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179 (Table II, up to 18 points)
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]
M. Erné, Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen, Manuscripta Math., 11 (1974), 221-259.
M. Erné, Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen, Manuscripta Math., 11 (1974), 221-259. (Annotated scanned copy)
M. Erné and K. Stege, Counting Finite Posets and Topologies, Order, 8 (1991), 247-265.
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970 [Annotated scanned copy]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences
Index entries for sequences related to posets

Cf. A000112, A001035, A000608, A066303, A342501 (refined by rank).

Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.

A001035 = {1, 1, 3, 19, 219, 4231, 130023, 6129859, 431723379, 44511042511, 6611065248783, 1396281677105899, 414864951055853499, 171850728381587059351, 98484324257128207032183, 77567171020440688353049939, 83480529785490157813844256579, 122152541250295322862941281269151, 241939392597201176602897820148085023};
max = Length[A001035]-1;
B[x_] = Sum[A001035[[k+1]]*x^k/k!, {k, 0, max}];
A[x_] = 1 + Log[B[x]];
CoefficientList[A[x] + O[x]^(max-1), x]*Range[0, max-2]! (* Jean-François Alcover, Apr 17 2014, updated Aug 30 2018 *)

E.g.f. A(x)=log(B(x)) where B(x) is e.g.f. of A001035.

More terms from Christian G. Bower, Dec 12 2001

a(17)-a(18) using data from A001035 from Alois P. Heinz, Aug 30 2018

A342587 Triangle, read by rows: T(n,k) is the number of labeled order relations on n nodes in which the longest chain has k nodes (n>=1, 1<=k<=n).

Original entry on oeis.org

1, 1, 2, 1, 12, 6, 1, 86, 108, 24, 1, 840, 2310, 960, 120, 1, 11642, 65700, 42960, 9000, 720, 1, 227892, 2583126, 2510760, 712320, 90720, 5040, 1, 6285806, 142259628, 199357704, 71310960, 11481120, 987840, 40320, 1, 243593040, 11012710470, 21774014640, 9501062760, 1781015040

Offset: 1

R. J. Mathar and Brendan McKay, Mar 16 2021

Corrects Comtet's table for k=4 and 5 in row n=8.

			Triangle T(n,k) (with n >= 1 and 1 <= k <= n) begins as follows:
  1;
  1,      2;
  1,     12,       6;
  1,     86,     108,      24;
  1,    840,    2310,     960,    120;
  1,  11642,   65700,   42960,   9000,   720;
  1, 227892, 2583126, 2510760, 712320, 90720, 5040;
  ...

Brendan McKay, Table of T(n,k) for n = 1..13
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 60.

Cf. A000142 (diagonal), A001035 (row sums), A055531 (k=2), A055532 (k=3), A055533 (subdiagonal), A055534 (subdiagonal), A081064, A342501 (connected).

A342500 T(n,k) is the number of connected unlabeled posets with n elements and rank k: triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 10, 24, 9, 1, 0, 27, 123, 73, 14, 1, 0, 88, 734, 638, 169, 20, 1, 0, 328, 5184, 6460, 2178, 334, 27, 1, 0, 1460, 44518, 78385, 32468, 5880, 594, 35, 1, 0, 7799, 472859, 1164966, 581533, 118933, 13605, 979, 44, 1

Offset: 1

R. J. Mathar, Mar 14 2021

This is a variant of A263859 admitting only connected posets.

			The table starts in row n=1 shows ranks k>=0:
1: 1
2: 0 1
3: 0 2 1
4: 0 4 5 1
5: 0 10 24 9 1
6: 0 27 123 73 14 1
7: 0 88 734 638 169 20 1
8: 0 328 5184 6460 2178 334 27 1
9: 0 1460 44518 78385 32468 5880 594 35 1
10: 0 7799 472859 1164966 581533 118933 13605 979 44 1

Index entries for sequences related to posets
R. J. Mathar, Poset Illustrations

Cf. A000608 (row sums), A007776 (rank 1), A263859, A000096 (subdiagonal), A342501 (labeled).

T(n,0) = 0 for k>0; due to the connectivity constraint.

T(n,n-1) = 1; the poset with elements in a single chain.

A008285 Erroneous version of A342587.

Original entry on oeis.org

1, 1, 2, 1, 12, 6, 1, 86, 108, 24, 1, 840, 2310, 960, 120, 1, 11642, 65700, 42960, 9000, 720, 1, 227892, 2583126, 2510760, 712320, 90720, 5040, 1, 6285806, 142259628, 199424904, 71243760, 11481120, 987840, 40320

Offset: 1

N. J. A. Sloane

dead

			Triangle T(n,k) (with n >= 1 and 1 <= k <= n) begins as follows:
  1;
  1,      2;
  1,     12,       6;
  1,     86,     108,      24;
  1,    840,    2310,     960,    120;
  1,  11642,   65700,   42960,   9000,   720;
  1, 227892, 2583126, 2510760, 712320, 90720, 5040;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 60.

Cf. A000142 (diagonal), A001035 (row sums), A055531 (k=2), A055532 (k=3), A055533 (subdiagonal), A081064, A342501 (connected).

cp's OEIS Frontend

A001927 Number of connected partially ordered sets with n labeled points.

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A342587 Triangle, read by rows: T(n,k) is the number of labeled order relations on n nodes in which the longest chain has k nodes (n>=1, 1<=k<=n).

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A342500 T(n,k) is the number of connected unlabeled posets with n elements and rank k: triangle read by rows.

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A008285 Erroneous version of A342587.

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