A000798 -id:A000798 - cp's OEIS frontend

A247660 Number of bi-indecomposable topologies on n labeled points.

1, 2, 15, 229, 5298, 177661, 8605831, 601894158, 60571434501, 8716575772821, 1780241773757704, 511992638746006383, 205785031866223973343, 114780183140747719353332, 88271680420025831479693219, 93054995762621287409298093049

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 [math.RA], 2014.

Cf. A000598, A247659.

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[(-2 + 3 f[x] - f[x]^2)/f[x] + O[x]^nmax, x][[2 ;; -2]] (* Jean-François Alcover, Oct 10 2018 *)

G.f.: (-2+3*f(x)-f(x)^2)/f(x), where f(x) is the g.f. of A000798.

A265847 Number of different quasi-orders with n labeled elements, modulo n.

Original entry on oeis.org

0, 0, 2, 3, 2, 1, 2, 2, 0, 3, 2, 1, 2, 6, 1, 15, 2, 1, 2

Offset: 1

Altug Alkan, Dec 21 2015

Remainder when number of different quasi-orders with n labeled elements is divided by n.
If n is an odd prime, a(n) = 2 because of the fact that A000798(p^k) == k + 1 mod p for all primes p. For k = 1, A000798(p) == 2 mod p for all primes p.
Currently, A000798 has values for n <= 18. However, thanks to A000798(p) == 2 mod p, we know that a(19) = 2.
How is the distribution of other terms such as 1 and 3 in this sequence?

			a(4) = A000798(4) mod 4 = 355 mod 4 = 3.
a(5) = A000798(5) mod 5 = 6942 mod 5 = 2.
a(6) = A000798(6) mod 6 = 209527 mod 6 = 1.

Muhammet Yasir Kizmaz, On The Number Of Topologies On A Finite Set, arXiv preprint arXiv:1503.08359 [math.NT], 2015.

Cf. A000798.

a(A000040(n)) = 2, for n > 1.

A281547 Total number of subsets of X that are both open and closed summed over all distinct topological spaces X that can be placed on an n-set.

Original entry on oeis.org

1, 2, 10, 82, 1038, 19754, 561778, 23890766, 1516425978, 142478603490, 19560464078774, 3868751287074546, 1088233853378616578, 430599111941369628326, 237480490462200909980594, 181131722604060126010422898, 189780362331001773747253412782, 271553393666987988551182068682458, 527932854364810523962111033565618786

Offset: 0

Geoffrey Critzer, Jan 23 2017

nonn

			a(2) = 10.  Let X = {a,b}.  There are four distinct topologies (A000798) that can be placed on X: {{},X}  {{},{a},X}  {{}, {b},X}  {{},{a},{b},X}.  These topologies have 2 + 2 + 2 + 4 sets respectively that are both open and closed.

Wikipedia, Clopen set.

Cf. A000798, A001035, A247232.

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}];
CoefficientList[A[Exp[x] - 1]^2 + O[x]^lg, x]*Range[0, lg - 1]! (* Jean-François Alcover, Jan 01 2020 *)

E.g.f.: A(exp(x) - 1)^2 where A(x) is the e.g.f. for A001035.

a(n) = Sum_{k=1..n} A247232(n,k)*2^k.

A326909 Number of sets of subsets of {1..n} closed under union and intersection and covering all of the vertices.

Original entry on oeis.org

2, 2, 7, 45, 500, 9053, 257151, 11161244, 725343385, 69407094565, 9639771895398, 1919182252611715, 541764452276876719, 214777343584048313318, 118575323291814379721651, 90492591258634595795504697, 94844885130660856889237907260, 135738086271526574073701454370969, 263921383510041055422284977248713291

Offset: 0

Gus Wiseman, Aug 04 2019

nonn

Differs from A326878 in having a(0) = 2 instead of 1.

			The a(0) = 2 through a(2) = 7 sets of subsets:
  {}    {{1}}     {{1,2}}
  {{}}  {{},{1}}  {{},{1,2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Covering sets of subsets are A000371.
The case without empty sets is A108798.
The case with a single covering edge is A326878.
The unlabeled version is A326898 for n > 0.
The case closed only under union is A326906.
The case closed only under intersection is (also) A326906.
Cf. A000798, A001930, A003465, A006058, A306445, A326876, A326882, A326907, A326908.

Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,3}]
(* Second program: *)
A000798 = Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {, }][[All, 2]];
A006058 = Cases[Import["https://oeis.org/A006058/b006058.txt", "Table"], {, }][[All, 2]];
a[n_] := A006058[[n + 1]] + A000798[[n + 1]];
a /@ Range[0, 18] (* Jean-François Alcover, Dec 30 2019 *)

a(n) = A000798(n) + A006058(n). - Jean-François Alcover, Dec 30 2019, after Gus Wiseman's comment in A006058.

a(18) from A000798+A006058 by Jean-François Alcover, Dec 30 2019

A360984 Triangular array read by rows. T(n,k) is the number of idempotent Boolean relation matrices on [n] with exactly k reflexive points, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 6, 4, 1, 27, 66, 29, 1, 108, 780, 1116, 355, 1, 405, 8020, 29250, 28405, 6942, 1, 1458, 76110, 649260, 1460425, 1068576, 209527

Offset: 0

Geoffrey Critzer, Feb 27 2023

			Triangle T(n,k) begins:
  1;
  1,   1;
  1,   6,    4;
  1,  27,   66,    29;
  1, 108,  780,  1116,   355;
  1, 405, 8020, 29250, 28405, 6942;
  ...

Cf. A121337 (row sums), A000798 (main diagonal).

Cf. A245767, A027471 (column 1).

T(n,n) = A245767(n,n) = A000798(n).
T(n,n-1) = A245767(n,n-1).
T(n,1) = n*Sum_k Sum_j binomial(n-1,k)*binomial(n-1-k,j) = A027471(n+1).
E.g.f. for column 1 is x*exp(x)^3.
E.g.f. for column 2 is x^2/2*exp(x)^3 + x^2*exp(x)^6 + x^2/2*exp(x)^7.
E.g.f. for column 3 is x^3/3!*exp(x)^15 + x^3/3!*exp(x)^3 + x^3*exp(x)^10 + x^3*exp(x)^12 + x^3/2!*exp(x)^7 + 2*x^3/2!*exp(x)^6 + 2*x^3/2*exp(x)^12.

Rows 5 and 6 added by Geoffrey Critzer, Mar 05 2023

A018196 Possible numbers of complements of topologies on 5 points.

Original entry on oeis.org

1, 4, 8, 9, 15, 32

Offset: 0

N. J. A. Sloane

J. I. Brown and S. Watson, The number of complements of a topology on n points is at least 2^n (except for some special cases), Discr. Math., 154 (1996), 27-39.

Cf. A000798.

A101620 Decimal encoding of digraph topologies.

Original entry on oeis.org

1, 3, 9, 11, 13, 15, 129, 131, 133, 137, 139, 141, 143, 145, 153, 161, 163, 165, 171, 175, 177, 179, 187, 193, 195, 197, 205, 207, 209, 213, 221, 241, 243, 245, 255

Offset: 1

Alford Arnold, Dec 10 2004

The sequence encodes labeled digraph topologies as described by and counted in A000798.

			Let a = 2, b = 4, c = 16, d = 256, ...
a(19) = 171 because we can map { }, a, ab, ac, abc to 1 + 2 + 8 + 32 + 128
Viewed as an array the table begins
1
3
9 11 13 15
129 131 133 137 139 141 143 145 153 161 163 165 171 175 177 ...
with respectively 1 1 4 29 355 ... (A000798) entries on each row.

Cf. A000798, A001035, A001930, A074486.

A176527 Partial sums of A006058.

Original entry on oeis.org

1, 2, 5, 21, 166, 2277, 49901, 1675904, 84239935, 6231045077, 668949067432, 103005162942955, 22511886374045653, 6918461813753405930, 2965189776573865320121, 1759287329824925168339697, 1435531006280642249195752862, 1601571709194974043628781397985, 2430449338115875591262479128994073

Offset: 0

Jonathan Vos Post, Apr 19 2010

nonn

Partial sums of number of connected labeled topologies with n points. The subsequence of primes in this partial sum begins: 2, 5, and then what?

Cf. A000798, A006058.

Cases[Import["https://oeis.org/A006058/b006058.txt", "Table"], {, }][[All, 2]] // Accumulate (* Jean-François Alcover, Jan 01 2020 *)

a(n) = Sum_{i=0..n} A006058(i) = Sum_{i=0..n} Sum_{k=0..i-1} C(i, k)*A000798(k) (for i>=1).

a(16)-a(18) from Jinyuan Wang, Feb 23 2020

A178689 Partial sums of A122836 (number of topologies on n labeled elements in which at least one element belongs to some pair of noncomparable members of the topology).

Original entry on oeis.org

0, 0, 0, 10, 253, 6384, 208887, 9673189, 651633791, 63901292323, 9040801794022, 1825884406581355, 521181413335003984, 208402574279716434454, 115825454535371969786250, 88852094572776191675804592

Offset: 0

Jonathan Vos Post, Dec 25 2010

nonn

All listed (the first 16) values are nonprimes. In the underlying sequence, only A122836(5) = 6131 is prime of the listed values.

			a(4) = 0 + 0 + 0 + 10 + 243 = 253 = 11 * 23.

Cf. A000798, A122835, A122836.

a(n) = Sum_{i=0..n} A122836(i) = Sum_{i=0..n} (A000798(i) - A122835(i)) = (Sum_{i=0..n} A000798(i)) - (Sum_{i=0..n} A122835(i)).

a(13)-a(15) corrected by Georg Fischer, Dec 26 2022

A234268 Number of convex topologies on an n-point totally ordered set.

Original entry on oeis.org

1, 4, 21, 129, 876, 6376, 48829, 388771, 3191849, 26864936

Offset: 1

N. J. A. Sloane, Dec 24 2013

Tyler Clark and Tom Richmond, The Number of Convex Topologies on a Finite Totally Ordered Set, 2013, Involve, Vol. 8 (2015), No. 1, 25-32.

Cf. A000798, A007052.

cp's OEIS Frontend

A247660 Number of bi-indecomposable topologies on n labeled points.

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A265847 Number of different quasi-orders with n labeled elements, modulo n.

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A281547 Total number of subsets of X that are both open and closed summed over all distinct topological spaces X that can be placed on an n-set.

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A326909 Number of sets of subsets of {1..n} closed under union and intersection and covering all of the vertices.

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A360984 Triangular array read by rows. T(n,k) is the number of idempotent Boolean relation matrices on [n] with exactly k reflexive points, n >= 0, 0 <= k <= n.

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A018196 Possible numbers of complements of topologies on 5 points.

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A101620 Decimal encoding of digraph topologies.

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A176527 Partial sums of A006058.

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Mathematica

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A178689 Partial sums of A122836 (number of topologies on n labeled elements in which at least one element belongs to some pair of noncomparable members of the topology).

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