A014466 -id:A014466 - cp's OEIS frontend

A326361 Number of maximal intersecting antichains of sets covering n vertices with no singletons.

1, 1, 1, 2, 12, 133, 11386, 12143511

Offset: 0

Gus Wiseman, Jul 01 2019

Covering means there are no isolated vertices. A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

			The a(4) = 12 antichains:
  {{1,2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

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, A051185, A058891, A261005, A305000, A305844, A326358, A326360, A326362, A326363.

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[Select[stableSets[Subsets[Range[n]],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],Union@@#==Range[n]&]]],{n,0,5}]
(* 2nd program *)
n = 2^6; g = CompleteGraph[n]; i = 0;
While[i < n, i++; j = i; While[j < n, j++; If[BitAnd[i, j] == 0 || BitAnd[i, j] == i || BitAnd[i, j] == j, g = EdgeDelete[g, i <-> j]]]];
sets = Select[FindClique[g, Infinity, All], BitOr @@ # == n - 1 &];
Length[sets] (* Elijah Beregovsky, May 05 2020 *)

a(6)-a(7) from Elijah Beregovsky, May 05 2020

A326881 Number of set-systems with {} that are closed under intersection and cover n vertices.

Original entry on oeis.org

1, 1, 5, 71, 4223, 2725521, 151914530499, 28175294344381108057

Offset: 0

Gus Wiseman, Jul 30 2019

			The a(2) = 5 set-systems:
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{},{2},{1,2}}
  {{},{1},{2},{1,2}}

The case also closed under union is A000798.
The connected case (i.e., with maximum) is A102894.
The same for union instead of intersection is (also) A102894.
The non-covering case is A102895.
The BII-numbers of these set-systems (without the empty set) are A326880.
The unlabeled case is A326883.
Cf. A003465, A014466, A102896, A102897, A193674, A193675, A306445, A307249, A326878.

Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&Union@@#==Range[n]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Inverse binomial transform of A102895. - Andrew Howroyd, Aug 10 2019

a(5)-a(7) from Andrew Howroyd, Aug 10 2019

A326362 Number of maximal intersecting antichains of nonempty, non-singleton subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 2, 16, 163, 11742, 12160640

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

			The a(4) = 16 maximal intersecting antichains:
  {{1,2,3,4}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,4},{2,4}}
  {{1,3},{1,4},{3,4}}
  {{2,3},{2,4},{3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

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, A051185, A058891, A261005, A305000, A305844, A326358, A326360, A326361, A326363.

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}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&]]],{n,0,5}]
(* 2nd program *)
n = 2^6; g = CompleteGraph[n]; i = 0;
While[i < n, i++; j = i; While[j < n, j++; If[BitAnd[i, j] == 0 || BitAnd[i, j] == i || BitAnd[i, j] == j, g = EdgeDelete[g, i <-> j]]]];
sets = FindClique[g, Infinity, All];
Length[sets]-Log[2,n]-1 (* Elijah Beregovsky, May 06 2020 *)

For n > 1, a(n) = A326363(n) - n - 1 = A007363(n + 1) - n.

a(7) from Elijah Beregovsky, May 06 2020

A367772 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice in more than one way.

Original entry on oeis.org

0, 0, 1, 23, 1105, 154941, 66072394, 88945612865, 396990456067403

Offset: 0

Gus Wiseman, Dec 12 2023

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			Non-isomorphic representatives of the a(3) = 23 set-systems:
  {{1,2}}
  {{1,2,3}}
  {{1},{2,3}}
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Wikipedia, Axiom of choice.

For at least one choice we have A367902.
For no choices we have A367903, no singletons A367769, ranks A367907.
For a unique choice we have A367904, ranks A367908.
These set-systems have ranks A367909.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
Cf. A059201, A102896, A133686, A283877, A306445, A323818, A355741, A367770, A367862, A367869, A367901, A367905.

Table[Length[Select[Subsets[Subsets[Range[n]]], Length[Select[Tuples[#], UnsameQ@@#&]]>1&]], {n,0,3}]

A367903(n) + A367904(n) + a(n) = A058891(n).

a(5)-a(8) from Christian Sievers, Jul 26 2024

A306505 Number of non-isomorphic antichains of nonempty subsets of {1,...,n}.

Original entry on oeis.org

1, 2, 4, 9, 29, 209, 16352, 490013147, 1392195548889993357, 789204635842035040527740846300252679

Offset: 0

Gus Wiseman, Feb 20 2019

The spanning case is A006602 or A261005. The labeled case is A014466.
From Petros Hadjicostas, Apr 22 2020: (Start)
a(n) is the number of "types" of log-linear hierarchical models on n factors in the sense of Colin Mallows (see the emails to N. J. A. Sloane).
Two hierarchical models on n factors belong to the same "type" iff one can obtained from the other by a permutation of the factors.
The total number of hierarchical log-linear models on n factors (in all "types") is given by A014466(n) = A000372(n) - 1.
The name of a hierarchical log-linear model on factors is based on the collection of maximal interaction terms, which must be an antichain (by the definition of maximality).
In his example on p. 1, Colin Mallows groups the A014466(3) = 19 hierarchical log-linear models on n = 3 factors x, y, z into a(3) = 9 types. See my example below for more details. (End)
First differs from A348260(n + 1) - 1 at a(5) = 209, A348260(6) - 1 = 232. - Gus Wiseman, Nov 28 2021

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 9 antichains:
  {}  {}     {}         {}
      {{1}}  {{1}}      {{1}}
             {{1,2}}    {{1,2}}
             {{1},{2}}  {{1},{2}}
                        {{1,2,3}}
                        {{1},{2,3}}
                        {{1},{2},{3}}
                        {{1,3},{2,3}}
                        {{1,2},{1,3},{2,3}}
From _Petros Hadjicostas_, Apr 23 2020: (Start)
We expand _Colin Mallows_'s example from p. 1 of his list of 1991 emails. For n = 3, we have the following a(3) = 9 "types" of log-linear hierarchical models:
Type 1: ( ), Type 2: (x), (y), (z), Type 3: (x,y), (y,z), (z,x), Type 4: (x,y,z), Type 5: (xy), (yz), (zx), Type 6: (xy,z), (yz,x), (zx,y), Type 7: (xy,xz), (yx,yz), (zx,zy), Type 8: (xy,yz,zx), Type 9: (xyz).
For each model, the name only contains the maximal terms. See p. 36 in Wickramasinghe (2008) for the full description of the 19 models.
Strictly speaking, I should have used set notation (rather than parentheses) for the name of each model, but I follow the tradition of the theory of log-linear models. In addition, in an interaction term such as xy, the order of the factors is irrelevant.
Models in the same type essentially have similar statistical properties.
For example, models in Type 7 have the property that two factors are conditionally independent of one another given each level (= category) of the third factor.
Models in Type 6 are such that two factors are jointly independent from the third one. (End)

C. L. Mallows, Emails to N. J. A. Sloane, Jun-Jul 1991, p. 1.
R. I. P. Wickramasinghe, Topics in log-linear models, Master of Science thesis in Statistics, Texas Tech University, Lubbock, TX, 2008, p. 36.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A000372, A003182, A006126, A006602, A014466, A261005, A293606, A293993, A304996, A305000, A305001, A305857, A317674, A319721, A320449, A321679.

Cf. A007363, A306007, A307249, A326358, A326359, A326360, A326363.

a(n) = A003182(n) - 1.

Partial sums of A006602 minus 1.

a(8) from A003182. - Bartlomiej Pawelski, Nov 27 2022

a(9) from A003182. - Dmitry I. Ignatov, Nov 27 2023

A325107 Number of subsets of {1...n} with no binary containments.

Original entry on oeis.org

1, 2, 4, 5, 10, 13, 18, 19, 38, 52, 77, 83, 133, 147, 166, 167, 334, 482, 764, 848, 1465, 1680, 1987, 2007, 3699, 4413, 5488, 5572, 7264, 7412, 7579, 7580, 15160, 22573, 37251, 42824, 77387, 92863, 116453, 118461, 227502, 286775, 382573, 392246, 555661, 574113

Offset: 0

Gus Wiseman, Mar 28 2019

nonn

A pair of positive integers is a binary containment if the positions of 1's in the reversed binary expansion of the first are a subset of the positions of 1's in the reversed binary expansion of the second.

			The a(0) = 1 through a(6) = 18 subsets:
  {}  {}   {}     {}     {}       {}       {}
      {1}  {1}    {1}    {1}      {1}      {1}
           {2}    {2}    {2}      {2}      {2}
           {1,2}  {3}    {3}      {3}      {3}
                  {1,2}  {4}      {4}      {4}
                         {1,2}    {5}      {5}
                         {1,4}    {1,2}    {6}
                         {2,4}    {1,4}    {1,2}
                         {3,4}    {2,4}    {1,4}
                         {1,2,4}  {2,5}    {1,6}
                                  {3,4}    {2,4}
                                  {3,5}    {2,5}
                                  {1,2,4}  {3,4}
                                           {3,5}
                                           {3,6}
                                           {5,6}
                                           {1,2,4}
                                           {3,5,6}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..129, (terms up to a(71) from Alois P. Heinz)

Cf. A006126, A014466, A019565, A267610.

Cf. A325095, A325096, A325101, A325105, A325106, A325108, A325109, A325110.

c:= proc() option remember; local i, x, y;
      x, y:= map(n-> Bits[Split](n), [args])[];
      for i to nops(x) do
        if x[i]=1 and y[i]=0 then return false fi
      od; true
    end:
b:= proc(n, s) option remember; `if`(n=0, 1, b(n-1, s)+
     `if`(ormap(i-> c(n, i), s), 0, b(n-1, s union {n})))
    end:
a:= n-> b(n, {}):
seq(a(n), n=0..34);  # Alois P. Heinz, Mar 28 2019

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Range[n]],stableQ[#,SubsetQ[binpos[#1],binpos[#2]]&]&]],{n,0,13}]

a(2^n - 1) = A014466(n).

a(16)-a(45) from Alois P. Heinz, Mar 28 2019

A327351 Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 4, 3, 2, 0, 30, 40, 27, 17, 0, 546, 1365, 1842, 1690, 1451, 0, 41334

Offset: 0

Gus Wiseman, Sep 09 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.
The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.
If empty edges are allowed, we have T(0,0) = 2.

			Triangle begins:
    1
    1    0
    1    1    0
    4    3    2    0
   30   40   27   17    0
  546 1365 1842 1690 1451    0

Row sums are A307249, or A006126 if empty edges are allowed.
Column k = 0 is A120338, if we assume A120338(0) = A120338(1) = 1.
Column k = 1 is A327356.
Column k = n - 1 is A327020.
The unlabeled version is A327359.
The version for vertex-connectivity >= k is A327350.
The version for spanning edge-connectivity is A327352.
The version for non-spanning edge-connectivity is A327353, with covering case A327357.
Cf. A003465, A006126, A014466, A048143, A293993, A323818, A326704, A327125, A327334, A327336.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
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]]]];
vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Union@@#==Range[n]&&vertConnSys[Range[n],#]==k&]],{n,0,4},{k,0,n}]

a(21) from Robert Price, May 28 2021

A006896 a(n) is the number of hierarchical linear models on n labeled factors allowing 2-way interactions (but no higher order interactions); or the number of simple labeled graphs with nodes chosen from an n-set.

Original entry on oeis.org

1, 2, 5, 18, 113, 1450, 40069, 2350602, 286192513, 71213783666, 35883905263781, 36419649682706466, 74221659280476136241, 303193505953871645562970, 2480118046704094643352358501, 40601989176407026666590990422106, 1329877330167226219547875498464516481

Offset: 0

N. J. A. Sloane, Colin Mallows, Simon Plouffe

From Petros Hadjicostas, Apr 09 2020: (Start)
Hierarchical log-linear models are by definition nonempty because they always include an "intercept" (or "overall effect").
Note that this is different from the number of graphical hierarchical log-linear models on n labeled factors as defined in the referenced Wikipedia article about log-linear models, "A log-linear model is graphical if, whenever the model contains all two-factor terms generated by a higher-order interaction, the model also contains the higher-order interaction." See also Gauraha (2016). (End) (Comment revised by N. J. A. Sloane, Apr 23 2020.)

			From _Petros Hadjicostas_, Apr 09 2020: (Start)
For n = 2, consider the pair of nodes {X, Y}. The simple labeled graphs with nodes from this set are the empty graph G1 = [], G2 = [X], G3 = [Y], G4 = [X, Y], and G5 = [X, Y, X-Y]. Thus a(2) = 5.
For n = 3, consider the three nodes {X, Y, Z}. The simple labeled graphs with nodes from this set are G1 = [], G2 = [X], G3 = [Y], G4 = [Z], G5 = [X, Y], G6 = [X, Z], G7 = [Y, Z], G8 = [X, Y, X-Y], G9 = [X, Z, X-Z], G10 = [Y, Z, Y-Z], G11 = [X, Y, Z], G12 = [X-Y Z], G13 = [X, Y,Z, X-Z], G14 = [X, Y, Z, Y-Z ], G15 = [X, Y, Z, Y-X-Z], G16 = [X, Y, Z, X-Y-Z], G17 = [Z, Y, Z, X-Z-Y], and G18 = [X, Y, Z, triangle with nodes X, Y, Z]. Thus a(3) = 18.
In Wickramasinghe (2008), for n = 2, all A014466(2) = 5 hierarchical log-linear models on two factors X and Y, which appear on p. 18, are trivially graphical; thus a(2) = 5.
For n = 3, among the A014466(3) = 19 hierarchical log-linear models on three factors X, Y, and Z, which appear on p. 36, only Model 18 is not graphical because it contains the X-Y, Y-Z, and Z-X interactions but does not contain the 3-way X-Y-Z interaction; thus a(3) = 19 - 1 = 18. (End)

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

Vincenzo Librandi, Table of n, a(n) for n = 0..80
P. De Causmaecker and S. De Wannemacker, Decomposition of Intervals in the Space of Anti-Monotonic Functions, in Manuel Ojeda-Aciego, Jan Outrata (Eds.): CLA 2013, pp. 57-67, Laboratory L3i, University of La Rochelle, 2013.
Patrick De Causmaecker and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.
Niharika Gauraha, Graphical log-linear models: Fundamental concepts and applications, arXiv:1603.04122 [stat.ME], 2016.
Y. Nardi and A. Rinaldo, The log-linear group-lasso estimator and its asymptotic properties, Bernoulli 18(3) (2012), 945-974; see footnote 1 on p. 953 and Table 2 on p. 954.
Gete Umbrey and Saifur Rahman, Application of Graph Semirings in Decision Networks, Mathematical Forum (2021) Vol. 28, 40-51.
R. I. P. Wickramasinghe, Topics in log-linear models, Master of Science thesis in Statistics, Texas Tech University, Lubbock, TX, 2008.
Wikipedia, Log-linear analysis.

Cf. A004140, A006897 (unlabeled case).

A006896 := proc(n) local k; 1+binomial(n,1) +add(binomial(n,k)*2^(1/2*k*(k-1)), k = 2 .. n) end; seq (A006896(n), n=0..20);

nn=20;g=Sum[2^Binomial[n,2]x^n/n!,{n,0,nn}];Range[0,nn]! CoefficientList[Series[Exp[x]g,{x,0,nn}],x]  (* Geoffrey Critzer, Apr 11 2012 *)

a(n)=sum(k=0,n,binomial(n,k)*2^((k^2-k)/2))

a(n) = 1 + C(n, 1) + C(n, 2)*2 + C(n, 3)*2^3 + C(n, 4)*2^6 + ... + C(n, n)*2^(n*(n-1)/2).
a(n) = 1 + A004140(n).
E.g.f.: exp(x)*A(x) where A(x) is e.g.f. for A006125. - Geoffrey Critzer, Apr 11 2012.

Error in formula line corrected Sep 15 1997 (thanks to R. K. Guy for pointing this out)
Name expanded by Petros Hadjicostas, Apr 08 2020
Edited by N. J. A. Sloane, Apr 23 2020

A111059 a(n) = Product_{k=1..n} A005117(k), the product of the first n squarefree positive integers.

Original entry on oeis.org

1, 2, 6, 30, 180, 1260, 12600, 138600, 1801800, 25225200, 378378000, 6432426000, 122216094000, 2566537974000, 56463835428000, 1298668214844000, 33765373585944000, 979195833992376000, 29375875019771280000

Offset: 1

Leroy Quet, Oct 07 2005

nonn

Do all terms belong to A242031 (weakly decreasing prime signature)? - Gus Wiseman, May 14 2021

			Since the first 6 squarefree positive integers are 1, 2, 3, 5, 6, 7, the 6th term of the sequence is 1*2*3*5*6*7 = 1260.
From _Gus Wiseman_, May 14 2021: (Start)
The sequence of terms together with their prime signatures begins:
             1: ()
             2: (1)
             6: (1,1)
            30: (1,1,1)
           180: (2,2,1)
          1260: (2,2,1,1)
         12600: (3,2,2,1)
        138600: (3,2,2,1,1)
       1801800: (3,2,2,1,1,1)
      25225200: (4,2,2,2,1,1)
     378378000: (4,3,3,2,1,1)
    6432426000: (4,3,3,2,1,1,1)
  122216094000: (4,3,3,2,1,1,1,1)
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..417

A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A072047 applies Omega to each squarefree number.
A246867 groups squarefree numbers by Heinz weight (row sums: A147655).
A261144 groups squarefree numbers by smoothness (row sums: A054640).
A319246 gives the sum of prime indices of each squarefree number.
A329631 lists prime indices of squarefree numbers (reversed: A319247).
Cf. A001221, A002110, A014466, A070826, A338901, A339360.

Rest[FoldList[Times,1,Select[Range[40],SquareFreeQ]]] (* Harvey P. Dale, Jun 14 2011 *)

m=30;k=1;for(n=1,m,if(issquarefree(n),print1(k=k*n,",")))

More terms from Klaus Brockhaus, Oct 08 2005

A304985 Number of labeled clutters (connected antichains) spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 4, 40, 1344, 203136, 495598592, 309065330371840, 14369391920653644779049472

Offset: 0

Gus Wiseman, May 23 2018

nonn

Only the non-singleton edges are required to form an antichain.

			The a(2) = 4 clutters:
{{1,2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}

Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Cf. A048143, A198085, A261006, A304912, A304981, A304982, A304983, A304984, A304985, A304986.

Cf. A000372, A006126, A014466, A304997, A304999, A306505, A317674, A319721.

For n > 1, a(n) = A048143(n) * 2^n.

cp's OEIS Frontend

A326361 Number of maximal intersecting antichains of sets covering n vertices with no singletons.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A326881 Number of set-systems with {} that are closed under intersection and cover n vertices.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A326362 Number of maximal intersecting antichains of nonempty, non-singleton subsets of {1..n}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A367772 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice in more than one way.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A306505 Number of non-isomorphic antichains of nonempty subsets of {1,...,n}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A325107 Number of subsets of {1...n} with no binary containments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A327351 Triangle read by rows where T(n,k) is the number of antichains of nonempty sets covering n vertices with vertex-connectivity exactly k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A006896 a(n) is the number of hierarchical linear models on n labeled factors allowing 2-way interactions (but no higher order interactions); or the number of simple labeled graphs with nodes chosen from an n-set.

Views

Author