A000798 -id:A000798 - cp's OEIS frontend

A046906 Number of connected irreducible posets with n labeled points.

1, 1, 0, 0, 24, 1080, 52440, 3281880, 277953144, 32418855000, 5239070305080, 1173944480658840, 363936227764858584, 155521768202208047640, 91218870039317505477720, 73113879800794757415243480, 79743817918540500914682249144, 117883366412734188786535902826200, 235329353612778837110901775412557560

Offset: 0

John A. Wright

nonn

J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.

S. R. Finch, Series-parallel networks
S. R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences
Index entries for sequences related to posets

A003431 gives isomorphism classes of these posets.

nn = 18; A[x_] := Total[Cases[Import["https://oeis.org/A001035/b001035.txt",
      "Table"], {, }][[All, 2]]*Table[x^(i - 1)/(i - 1)!, {i, 1, 19}]];
Range[0, nn]! CoefficientList[ Series[(1 + Log[A[x]]) - A[ x] (1 - 1/A[x])^2 , {x, 0, nn}], x] (* Geoffrey Critzer, Jul 09 2022 *)

From Geoffrey Critzer, Jul 09 2022: (Start)
E.g.f.: 1 + log(A(x)) - A(x)(1-1/A(x))^2  where A(x) is the e.g.f. for A001035.
a(n) = A001927(n) - Sum_{k>=2} A354615(n,k). (End)

a(8)-a(18) from Geoffrey Critzer, Jul 09 2022

a(0) changed to 1 by Geoffrey Critzer, Jul 10 2022

A046907 Number of isomorphism classes of irreducible posets with n labeled points.

Original entry on oeis.org

1, 1, 1, 2, 7, 31, 184, 1351, 12524, 146468, 2177570, 41374407, 1008220289, 31559446774, 1269310589336, 65562045668340, 4345161435996517

Offset: 0

John A. Wright

G. Brinkmann, B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179 (Table 1).
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, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences
Index entries for sequences related to posets

Cf. A046908.

A000112 = Cases[Import["https://oeis.org/A000112/b000112.txt", "Table"], {, }][[All, 2]];
B[x_] = A000112.x^Range[0, Length[A000112] - 1];
A[x_] = 2 - 1/B[x];
CoefficientList[A[x] + O[x]^Length[A000112], x] (* Jean-François Alcover, Jan 01 2020 *)

G.f.: A(x) = 2-1/B(x), where B(x) is g.f. of A000112. - Vladeta Jovovic, Jan 15 2006

More terms from Vladeta Jovovic, Jan 15 2006

A046909 Number of isomorphism classes of connected irreducible quasiorders with n labeled points.

Original entry on oeis.org

1, 1, 1, 1, 2, 17, 167, 1672

Offset: 0

John A. Wright

nonn

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, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences

Cf. A046910.

A046910 Number of connected irreducible quasiorders with n labeled points.

Original entry on oeis.org

1, 1, 1, 1, 25, 1321, 70201, 4542721, 384969649, 44087846545, 6926924885881, 1503058888234201, 451117640363382697, 186980881340749198561, 106678398214255092939169, 83440038893764124092029601, 89093417035281194970121062073, 129323858612953057624127147727913, 254190262374139251098507525465587609

Offset: 0

John A. Wright

nonn

John A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.
John A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences

Cf. A046909, A335987, A000798, A001929.

nn = 18; A[x_] :=Total[Cases[Import["https://oeis.org/A000798/b000798.txt",
      "Table"], {, }][[All, 2]]*Table[x^(i - 1)/(i - 1)!, {i, 1, 19}]];
Range[0, nn]! CoefficientList[Series[1 + Log[A[x]] - A[x] (1 - 1/A[x])^2, {x, 0, nn}], x] (* Geoffrey Critzer, Jul 10 2022 *)

From Geoffrey Critzer, Jul 10 2022: (Start)
E.g.f.: 1 + log(A(x)) - A(x)*(1 - 1/A(x))^2 where A(x) is the e.g.f. for A000798.
a(n) = A001929(n) - Sum_{k>=2} A335987(n,k). (End)

a(8)-a(18) from Geoffrey Critzer, Jul 10 2022

A046911 Number of isomorphism classes of irreducible quasiorders with n labeled points.

Original entry on oeis.org

1, 1, 2, 4, 14, 62, 373, 2722

Offset: 0

John A. Wright

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, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences

Cf. A046912.

A074486 Encoding of topologies generated by classes of sets.

Original entry on oeis.org

1, 3, 9, 11, 15, 129, 131, 137, 139, 143, 153, 171, 175, 255

Offset: 0

Alford Arnold, Sep 26 2002

We map {}, a, b, c, d, ... to 1, 2, 4, 16, 256, ..., i.e., to 2^0, 2^1, 2^2, 2^4, 2^8, ... . Sets with more than 1 element are mapped to the product. So ab (a shorthand notation for {a,b}) is mapped to 2^1 * 2^2 = 2^3. The topology is represented by the sum of the representations of its components.

The sequence encodes unlabeled topologies as described in A000798.

			1+2+8 = 11 (binary 1011) encodes {}, a, ab, which is the least encoding of this topology, so 11 is in the sequence.
1+4+8 = 13 (binary 1101) encodes {}, b, ab which is topologically equivalent and larger, so it is not in the sequence. The number of equivalent cases corresponding to a(n) begins 1; 1,1,2; 1,1,3,3,6,3,3,3,6; ... and is counted by A001928 (labeled topologies).
171 (binary 1011011) is in the sequence because we map {}, a, ab, ac, abc to 1 + 2 + 8 + 32 + 128.

Cf. A000798, A001928.

Edited by Franklin T. Adams-Watters, Mar 29 2014

A173311 a(n) is the number of regular D classes in the semigroup of all binary relations on [n].

Original entry on oeis.org

1, 2, 4, 9, 25, 88, 406, 2451, 19450, 202681, 2769965, 49519392, 1154411138, 34978238590, 1373171398361, 69648249299517, 4552778914494604

Offset: 0

Jonathan Vos Post, Feb 16 2010

Previous name was: Partial sums of A000112.

K. K.-H. Butler and G. Markowsky, The Number of Maximal Subgroups of the Semigroup of Binary Relations, Kyungpook Math. J. Vol 12, June 1972.

Cf. A000112, A000798 (labeled topologies), A001035 (labeled posets), A001930 (unlabeled topologies), A006057, A079263, A079265, A007903.

Cases[Import["https://oeis.org/A000112/b000112.txt", "Table"], {, }][[All, 2]] // Accumulate (* Jean-François Alcover, Jan 01 2020 *)

a(n) = Sum_{i=0..n} A000112(i).

New name from Geoffrey Critzer, May 22 2022

A213430 The number of n X n upper triangular (0,1)-matrices M with all diagonal entries 1 such that M = f(M^2) and sum(row 1) >= sum(row 2) >= ... >= sum(row n-1) >= sum(row n) = 1 and f maps any nonzero entry to 1.

Original entry on oeis.org

1, 2, 6, 26, 159, 1347, 15593, 244173, 5131436

Offset: 1

N. J. A. Sloane, Jun 11 2012

A006455 is equivalent to this sequence without the nonincreasing condition on the row sums. - Petros Hadjicostas, Jul 20 2024

Collected papers of Professor Hansraj Gupta. Edited by R. J. Hans-Gill and Madhu Raka. Ramanujan Mathematical Society Collected Works Series, 3. See pp. 554-564.
Hansraj Gupta, Number of topologies in a finite set, Research Bulletin of the Panjab University, Vol. 19 (1968), p. 240. MR0268836.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Hansraj Gupta, Number of topologies in a finite set, Research Bulletin of the Panjab University, Vol. 19 (1968), p. 240. MR0268836.
Sean A. Irvine, Java program
Wikipedia, Finite topological space.
zbMATH, Review of Gupta article.

Cf. A000798, A001035, A006455.

a(7) and new name from Petros Hadjicostas, Jul 20 2024

a(8)-a(9) from Sean A. Irvine, Jul 20 2024

A247232 Triangular array read by rows: T(n,k) is the number of pre-orders on an n-set with exactly k connected components in its digraph representation, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 3, 1, 19, 9, 1, 233, 103, 18, 1, 4851, 1735, 325, 30, 1, 158175, 43201, 7320, 785, 45, 1, 7724333, 1567783, 218491, 22960, 1610, 63, 1, 550898367, 82142943, 8856974, 818461, 59570, 2954, 84, 1, 56536880923, 6187176225, 496368181, 37205658, 2518131, 135198, 4998, 108, 1

Offset: 1

Geoffrey Critzer, Nov 27 2014

The Bell transform of A001929(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			1;
3,         1;
19,        9,        1;
233,       103,      18,      1;
4851,      1735,     325,     30,     1;
158175,    43201,    7320,    785,    45,    1;
7724333,   1567783,  218491,  22960,  1610,  63,   1;
550898367, 82142943, 8856974, 818461, 59570, 2954, 84, 1;

Column 1 = A001929.

Row sums = A000798.

A001035 = Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {, }][[All, 2]];
lg = Length[A001035];
A[x_] = Sum[A001035[[n + 1]] x^n/n!, {n, 0, lg - 1}];
Rest[CoefficientList[#, y]]& /@ (CoefficientList[A[Exp[x] - 1]^y + O[x]^lg, x]*Range[0, lg - 1]!) // Flatten (* Jean-François Alcover, Jan 01 2020 *)

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
topo = oeis('A001929')  # Fetch the data via Internet.
A001929List = topo.first_terms()
A001929 = lambda n: A001929List[n]
bell_matrix(lambda n: A001929(n+1), 10) # Peter Luschny, Jan 18 2016, updated Mar 25 2020

E.g.f.: A(exp(x)-1)^y where A(x) is the e.g.f. for A001035.

A247659 Number of down-arrow-indecomposable topologies on n labeled points.

Original entry on oeis.org

1, 3, 22, 292, 6120, 193594, 9070536, 622336756, 61915861962, 8846814822932, 1798543906246948, 515674104905890202, 206833212761446463192, 115198617558900993580396, 88503974769306037986089170, 93233054587165663487254293572

Offset: 1

N. J. A. Sloane, Oct 05 2014

nonn

L. Foissy and C. Malvenuto, The Hopf algebra of finite topologies and T-partitions, arXiv preprint arXiv:1407.0476, 2014

Cf. A000598, A247660.

A000798 = {1, 1, 4, 29, 355, 6942, 209527, 9535241, 642779354, 63260289423, 8977053873043, 1816846038736192, 519355571065774021, 207881393656668953041, 115617051977054267807460, 88736269118586244492485121, 93411113411710039565210494095, 134137950093337880672321868725846, 261492535743634374805066126901117203};
f[x_] = Sum[A000798[[n+1]] x^n, {n, 0, nmax = Length[A000798]-1}];
CoefficientList[(f[x]-1)/f[x] + O[x]^nmax, x][[2 ;; -2]] (* Jean-François Alcover, Oct 10 2018 *)

G.f.: (A000798(x)-1)/A000798(x), where A000798(x) is the g.f. of A000798.

cp's OEIS Frontend

A046906 Number of connected irreducible posets with n labeled points.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A046907 Number of isomorphism classes of irreducible posets with n labeled points.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A046909 Number of isomorphism classes of connected irreducible quasiorders with n labeled points.

Views

Author

Keywords

References

Links

Crossrefs

A046910 Number of connected irreducible quasiorders with n labeled points.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A046911 Number of isomorphism classes of irreducible quasiorders with n labeled points.

Views

Author

Keywords

References

Links

Crossrefs

A074486 Encoding of topologies generated by classes of sets.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A173311 a(n) is the number of regular D classes in the semigroup of all binary relations on [n].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A213430 The number of n X n upper triangular (0,1)-matrices M with all diagonal entries 1 such that M = f(M^2) and sum(row 1) >= sum(row 2) >= ... >= sum(row n-1) >= sum(row n) = 1 and f maps any nonzero entry to 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A247232 Triangular array read by rows: T(n,k) is the number of pre-orders on an n-set with exactly k connected components in its digraph representation, n>=1, 1<=k<=n.

Views

Author

Keywords

Comments

Examples