cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-20 of 20 results.

A326360 Number of maximal antichains of nonempty, non-singleton subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 2, 13, 279, 29820, 123590767
Offset: 0

Views

Author

Gus Wiseman, Jul 01 2019

Keywords

Comments

A set system (set of sets) is an antichain if no element is a subset of any other.

Examples

			The a(1) = 1 through a(4) = 13 maximal antichains:
  {}  {12}  {123}         {1234}
            {12}{13}{23}  {12}{134}{234}
                          {13}{124}{234}
                          {14}{123}{234}
                          {23}{124}{134}
                          {24}{123}{134}
                          {34}{123}{124}
                          {12}{13}{14}{234}
                          {12}{23}{24}{134}
                          {13}{23}{34}{124}
                          {14}{24}{34}{123}
                          {123}{124}{134}{234}
                          {12}{13}{14}{23}{24}{34}

Links

Crossrefs

Antichains of nonempty, non-singleton sets are A307249.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A000372, A003182, A006126, A006602, A014466, A058891, A261005, A305000, A305844, A326358, A326361, A326362, A326363.

Programs

  • Mathematica
    stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n],{2,n}],SubsetQ]]],{n,0,4}]
  • Python
    # see Ignatov links
# Dmitry I. Ignatov, Oct 14 2021

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A326359(k) for n >= 2. - Andrew Howroyd, Nov 19 2021

Extensions

a(6) from Andrew Howroyd, Aug 14 2019
a(7) from Dmitry I. Ignatov, Oct 14 2021

A035347 Triangle of a(n,k) = number of minimal covers of an n-set that cover k points of that set uniquely (n >= 1, k >= 1).

Original entry on oeis.org

1, 0, 2, 0, 3, 5, 0, 6, 28, 15, 0, 10, 190, 210, 52, 0, 15, 1340, 3360, 1506, 203, 0, 21, 9065, 60270, 48321, 10871, 877, 0, 28, 57512, 1132880, 1820056, 636300, 80592, 4140, 0, 36, 344316, 21067452, 76834926, 45455676, 8081928, 618939, 21147, 0, 45
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			1; 0,2; 0,3,5; 0,6,28,15; ...

Links

Crossrefs

Cf. A056885 for unlabeled case. Row sums give A046165.

Programs

  • Mathematica
    a[n_, k_] := Binomial[n, k] * Sum[ StirlingS2[k, j]*(2^j - j - 1)^(n - k), {j, 1, k}]; a[n_, n_] := Sum[ StirlingS2[n, j], {j, 1, n}]; Flatten[ Table[a[n, k], {n, 1, 10}, {k, 1, n}]] (* Jean-François Alcover, Jun 26 2012, from formula *)

Formula

a(n, k) = C(n, k)*Sum_{j=1..k} S(k, j)*(2^j-j-1)^(n-k), where S(k, j) are Stirling numbers of the second kind.
E.g.f.: Sum_{k>=1} (exp(y*x) - 1)^k/k! * exp((2^k-k-1)x). - Geoffrey Critzer, Jun 28 2013

Extensions

More terms from Vladeta Jovovic, Sep 06 2000

A368186 Number of n-covers of an unlabeled n-set.

Original entry on oeis.org

1, 1, 2, 9, 87, 1973, 118827, 20576251, 10810818595, 17821875542809, 94589477627232498, 1651805220868992729874, 96651473179540769701281003, 19238331716776641088273777321428, 13192673305726630096303157068241728202, 31503323006770789288222386469635474844616195
Offset: 0

Views

Author

Gus Wiseman, Dec 19 2023

Keywords

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 set-systems:
  {{1}}  {{1},{2}}    {{1},{2},{3}}
         {{1},{1,2}}  {{1},{2},{1,3}}
                      {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{2},{1,2,3}}
                      {{1},{1,2},{1,2,3}}
                      {{1,2},{1,3},{2,3}}
                      {{1},{2,3},{1,2,3}}
                      {{1,2},{1,3},{1,2,3}}

Links

Crossrefs

The labeled version is A054780, ranks A367917, non-covering A367916.
The case of graphs is A006649, labeled A367863, cf. A116508, A367862.
The case of connected graphs is A001429, labeled A057500.
Covers with any number of edges are counted by A003465, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Cf. A000088, A002494, A006126, A055130, A133686, A140638, A305000, A317795, A326754, A367901, A367902, A367903.

Programs

  • Mathematica
    brute[m_]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i, p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}];
Table[Length[Union[First[Sort[brute[#]]]& /@ Select[Subsets[Rest[Subsets[Range[n]]],{n}], Union@@#==Range[n]&]]], {n,0,3}]
  • PARI
    permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={2^sum(j=1, #q, gcd(t, q[j])) - 1}
G(n,m)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, m, K(q,t)*x^t/t, O(x*x^m))); s+=permcount(q)*exp(g - subst(g,x,x^2))); s/n!)}
a(n)=if(n ==0, 1, polcoef(G(n,n) - G(n-1,n), n)) \\ Andrew Howroyd, Jan 03 2024

Formula

a(n) = A055130(n, n) for n > 0. - Andrew Howroyd, Jan 03 2024

Extensions

Terms a(6) and beyond from Andrew Howroyd, Jan 03 2024

A368731 Number of non-isomorphic n-element sets of nonempty subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 10, 97, 2160, 126862, 21485262, 11105374322, 18109358131513, 95465831661532570, 1660400673336788987026, 96929369602251313489896310, 19268528295096123543660356281600, 13203875101002459910158494602665950757, 31517691852305548841992346407978317698725021
Offset: 0

Views

Author

Gus Wiseman, Jan 07 2024

Keywords

Examples

			Non-isomorphic representatives of the a(3) = 10 set-systems:
  {{1},{2},{3}}
  {{1},{2},{1,2}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{1,3}}
  {{1},{1,2},{2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Links

Crossrefs

The case of graphs is A001434, labeled A116508.
Labeled version is A136556, covering A054780, binomial transform of A367916.
The case of labeled covering graphs is A367863, binomial transform A367862.
These include the set-systems ranked by A367917.
The covering case is A368186, for graphs A006649, connected A057500.
Requiring all edges to be singletons or pairs gives A368598.
A003465 counts covers with any number of edges, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Cf. A000088, A007716, A283877, A367901, A367902, A367903, A368599.

Programs

  • Mathematica
    brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Subsets[Subsets[Range[n],{1,n}],{n}]]],{n,0,4}]
  • PARI
    a(n) = polcoef(G(n, n), n) \\ G defined in A368186. - Andrew Howroyd, Jan 11 2024

Extensions

Terms a(6) and beyond from Andrew Howroyd, Jan 11 2024

A094544 Triangle of a(n,m) = number of m-member minimal T_0-covers of an n-set (n >= 0, 0<= m <=n).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 3, 1, 0, 0, 0, 16, 1, 0, 0, 0, 120, 55, 1, 0, 0, 0, 480, 1650, 156, 1, 0, 0, 0, 840, 34650, 13650, 399, 1, 0, 0, 0, 0, 554400, 873600, 89376, 960, 1, 0, 0, 0, 0, 6985440, 45208800, 14747040, 514080, 2223, 1, 0, 0, 0, 0, 69854400, 1989187200
Offset: 0

Views

Author

Goran Kilibarda and Vladeta Jovovic, May 08 2004

Keywords

Comments

A cover of a set is a T_0-cover if for every two distinct points of the set there exists a member (block) of the cover containing one but not the other point.

Examples

			1;
0, 1;
0, 0, 1;
0, 0, 3,   1;
0, 0, 0,  16,    1;
0, 0, 0, 120,   55,   1;
0, 0, 0, 480, 1650, 156, 1;
...

Links

Crossrefs

Cf. A035348, A046165, A094545 (row sums), A094546 (column sums).

Programs

  • Mathematica
    Flatten[Table[n!/m! Binomial[2^m-m-1,n-m],{n,0,10},{m,0,n}]] (* Harvey P. Dale, Jan 16 2012 *)

Formula

a(n, m) = n!/m!*binomial(2^m-m-1, n-m).
E.g.f.: Sum_{n>=0} y^n*(1+y)^(2^n-n-1)*x^n/n!.

A094545 Number of minimal T_0-covers of an n-set.

Original entry on oeis.org

1, 1, 1, 4, 17, 176, 2287, 49540, 1518337, 67457584, 4254836111, 376795261844, 46709151254449, 8061849904932136, 1936383997541071639, 646603398091877815516, 300476951799493029958913
Offset: 0

Views

Author

Goran Kilibarda and Vladeta Jovovic, May 08 2004

Keywords

Comments

A cover of a set is a T_0-cover if for every two distinct points of the set there exists a member (block) of the cover containing one but not the other point.
Row sums of A094544.

Links

Crossrefs

Cf. A035348, A046165, A094544, A094546.

Formula

a(n) = Sum_{m=0..n} (n!/m!)*binomial(2^m-m-1, n-m).
a(n) = Sum_{m=0..n} Stirling1(n, m)*A046165(m).
E.g.f.: Sum_{n>=0} x^n*(1+x)^(2^n-n-1)/n!.

A094546 Number of n-member minimal T_0-covers.

Original entry on oeis.org

1, 1, 4, 1457, 112632827396, 158158632767281777075441633086607, 6800377846899806825426438402771408584453689087636553015800284773113817943589005365456
Offset: 0

Views

Author

Goran Kilibarda and Vladeta Jovovic, May 08 2004

Keywords

Comments

A cover of a set is a T_0-cover if for every two distinct points of the set there exists a member (block) of the cover containing one but not the other point.

Links

Crossrefs

Cf. A035348, A046165, A094545.
Column sums of A094544.

Programs

  • Mathematica
    Table[Sum[(m!/n!)*Binomial[2^n - n - 1, m - n], {m, n, 2^n - 1}], {n, 0, 5}] (* G. C. Greubel, Oct 07 2017 *)
  • PARI
    for(n=0,5, print1(sum(m=n,2^n -1, (m!/n!)*binomial(2^n-n-1, m-n)), ", ")) \\ G. C. Greubel, Oct 07 2017

Formula

a(n) = Sum_{m=n..2^n-1} m!/n!*binomial(2^n-n-1, m-n).

A282575 Triangular array read by rows. T(n,k) is the number of minimal covers of an n-set with exactly k points that are in more than one set of the cover, n>=0, 0<=k<=max(0,n-2).

Original entry on oeis.org

1, 1, 2, 5, 3, 15, 28, 6, 52, 210, 190, 10, 203, 1506, 3360, 1340, 15, 877, 10871, 48321, 60270, 9065, 21, 4140, 80592, 636300, 1820056, 1132880, 57512, 28, 21147, 618939, 8081928, 45455676, 76834926, 21067452, 344316, 36, 115975, 4942070, 101684115, 1027544400, 3860929170, 3406410252, 377190240, 1966440, 45
Offset: 0

Views

Author

Geoffrey Critzer, Feb 18 2017

Keywords

Examples

			Triangle T(n,k) begins:
:    1;
:    1;
:    2;
:    5,     3;
:   15,    28,      6;
:   52,   210,    190,      10;
:  203,  1506,   3360,    1340,      15;
:  877, 10871,  48321,   60270,    9065,    21;
: 4140, 80592, 636300, 1820056, 1132880, 57512, 28;

Links

Crossrefs

Cf. A035348. Row sums A046165. Column k=0 A000110. Column k=1 A003466.
Mirrored triangle gives A035347.

Programs

  • Maple
    T:= (n, k)-> binomial(n, k)*add(Stirling2(n-k, j)*(2^j-j-1)^k, j=0..n-k):
seq(seq(T(n,k), k=0..max(0,n-2)), n=0..12);  # Alois P. Heinz, Feb 18 2017
  • Mathematica
    nn = 8; Drop[Map[Select[#, # > 0 &] &,Range[0, nn]! CoefficientList[Series[Sum[ (Exp[x] - 1)^n/n! Exp[y (2^n - n - 1) x], {n, 0,nn}], {x, 0, nn}], {x, y}]], 1] // Grid

Formula

E.g.f.: (exp(x) - 1)^n/n!*exp(y*(2^n - n - 1)*x).

A003466 Number of minimal covers of an n-set that have exactly one point which appears in more than one set in the cover.

Original entry on oeis.org

0, 3, 28, 210, 1506, 10871, 80592, 618939, 4942070, 41076508, 355372524, 3198027157, 29905143464, 290243182755, 2920041395248, 30414515081650, 327567816748638, 3643600859114439, 41809197852127240, 494367554679088923, 6017481714095327410
Offset: 2

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A046165.
Column k=1 of A282575.

Programs

  • Maple
    seq(n*add((2^k-k-1)*Stirling2(n-1,k),k=1..n-1), n = 2 .. 30); # Robert Israel, May 21 2015
  • Mathematica
    nn = 20; Range[0, nn]! CoefficientList[Series[Sum[ (Exp[x] - 1)^n/n! (2^n - n - 1) x, {n, 0, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Feb 18 2017 *)
a[2]=0;a[3]=3;a[4]=28;a[n_]:=n*Sum[(2^k-k-1)* StirlingS2[n-1,k], {k,1,n-1}];Table[a[n],{n,2,22}] (* Indranil Ghosh, Feb 20 2017 *)

Formula

a(n) = n * Sum_{k=1..n-1} (2^k-k-1) * S2(n-1,k) where S2(n,k) are the Stirling numbers of the second kind. - Sean A. Irvine, May 20 2015
a(n) = n * (A001861(n-1) - A005493(n-2) - A000110(n-1)). - Robert Israel, May 21 2015

Extensions

More terms from Sean A. Irvine, May 20 2015
Title clarified by Geoffrey Critzer, Feb 18 2017

A049056 Number of minimal ordered covers of a labeled n-set.

Original entry on oeis.org

1, 1, 3, 19, 207, 3691, 103263, 4415419, 283796607, 27094905451, 3813398797023, 786844659227419, 237151202183603007, 104128385332221915211, 66478899089080159079583, 61624041121329496987905019
Offset: 0

Views

Author

N. J. A. Sloane, Michael Somos

Keywords

Links

Crossrefs

Row sums of A049055.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[ (-1)^i*Binomial[k, i]*(2^k-1-i)^n, {k, 0, n}, {i, 0, k} ]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jan 27 2012, after Michael Somos *)
  • PARI
    {a(n)=sum(k=0, n, sum(i=0, k, (-1)^i*binomial(k, i)*(2^k-1-i)^n))} /* Michael Somos, Oct 16 2006 */

Formula

E.g.f.: Sum_{n>=0} (exp(x)-1)^n*exp(x*(2^n-n-1)), cf. A046165. - Vladeta Jovovic, Sep 01 2005
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