A326961 -id:A326961 - cp's OEIS frontend

A327012 Number of factorizations of n into factors > 1 whose dual is a (strict) antichain.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1

Offset: 1

Gus Wiseman, Aug 13 2019

nonn

Differs from A322453 at 36, 72, 100, ...
The dual of a multiset system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The dual of a factorization is the dual of the multiset partition obtained by replacing each factor with its multiset of prime indices.
An antichain is a set of multisets, none of which is a submultiset of any other.

			The a(72) = 12 factorizations:
  (8*9)
  (3*24)
  (4*18)
  (2*4*9)
  (3*3*8)
  (3*4*6)
  (2*2*18)
  (2*3*12)
  (2*2*2*9)
  (2*2*3*6)
  (2*3*3*4)
  (2*2*2*3*3)

Set-systems whose dual is a (strict) antichain are A326965.
The version where the dual is a weak antichain is A326975.
Partitions whose dual is a (strict) antichain are A326977.
Cf. A001055, A316978, A326961, A326974, A326976, A326979.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[facs[n],UnsameQ@@dual[primeMS/@#]&&stableQ[dual[primeMS/@#],submultQ]&]],{n,100}]

A334254 Number of closure operators on a set of n elements which satisfy the T_1 separation axiom.

Original entry on oeis.org

1, 2, 1, 8, 545, 702525, 66960965307

Offset: 0

Joshua Moerman, Apr 20 2020

The T_1 axiom states that all singleton sets {x} are closed.

For n>1, this property implies strictness (meaning that the empty set is closed).

			The a(3) = 8 set-systems of closed sets:
  {{1,2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1},{2},{3},{}}
  {{1,2,3},{1,3},{1},{2},{3},{}}
  {{1,2,3},{2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{2,3},{1},{2},{3},{}}
  {{1,2,3},{1,3},{2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{2,3},{1},{2},{3},{}}

Dmitry I. Ignatov, On the Cryptomorphism between Davis' Subset Lattices, Atomic Lattices, and Closure Systems under T1 Separation Axiom, arXiv:2209.12256 [cs.DM], 2022.
Dmitry I. Ignatov, Supporting iPython code for counting closure systems w.r.t. the T_1 separation axiom, Github repository
Dmitry I. Ignatov, PDF of the supporting iPython notebook
S. Mapes, Finite atomic lattices and resolutions of monomial ideals, J. Algebra, 379 (2013), 259-276.
Eric Weisstein's World of Mathematics, Separation Axioms
Wikipedia, Separation Axiom

The number of all closure operators is given in A102896.
For T_0 closure operators, see A334252.
For strict T_1 closure operators, see A334255, the only difference is a(1).
Cf. A326960, A326961, A326979.

a(6) from Dmitry I. Ignatov, Jul 03 2022

A327058 Number of pairwise intersecting set-systems covering n vertices whose dual is a weak antichain.

Original entry on oeis.org

1, 1, 1, 3, 155

Offset: 0

Gus Wiseman, Aug 18 2019

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

			The a(0) = 1 through a(3) = 3 set-systems:
  {}  {{1}}  {{12}}  {{123}}
                     {{12}{13}{23}}
                     {{12}{13}{23}{123}}

Covering intersecting set-systems are A305843.
The BII-numbers of these set-systems are the intersection of A326910 and A326966.
Covering coantichains are A326970.
The non-covering version is A327059.
The unlabeled multiset partition version is A327060.
Cf. A006126, A051185, A059523, A305844, A319639, A326961, A326965, A326968, A327020, A327057.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
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]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],Union@@#==Range[n]&&stableQ[dual[#],SubsetQ]&]],{n,0,3}]

Inverse binomial transform of A327059.

A327011 Number of unlabeled sets of subsets covering n vertices where every vertex is the unique common element of some subset of the edges, also called unlabeled covering T_1 sets of subsets.

Original entry on oeis.org

2, 2, 4, 32, 2424

Offset: 0

Gus Wiseman, Aug 13 2019

Alternatively, these are unlabeled sets of subsets covering n vertices whose dual is a (strict) antichain. The dual of a set of subsets has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. An antichain is a set of subsets where no edge is a subset of any other.

			Non-isomorphic representatives of the a(0) = 1 through a(2) = 4 sets of subsets:
  {}    {{1}}     {{1},{2}}
  {{}}  {{},{1}}  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Unlabeled covering sets of subsets are A003181.
The same with T_0 instead of T_1 is A326942.
The non-covering version is A326951 (partial sums).
The labeled version is A326960.
The case without empty edges is A326974.
Cf. A001146, A055621, A059523, A319637, A326961, A326972, A326973, A326976, A326977, A326979.

a(n) = A326974(n) / 2.

a(n > 0) = A326951(n) - A326951(n - 1).

A327017 Number of non-isomorphic multiset partitions of weight n where every vertex, as a multiset of weight 1, is the multiset-meet of some subset of the edges.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 49, 115, 310, 830, 2383

Offset: 0

Gus Wiseman, Aug 15 2019

The multiset-meet of a collection of multisets has as underlying set the intersection of their underlying sets and as multiplicities the minima of their multiplicities.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 19 multiset partitions:
    {1}  {1}{1}  {1}{11}    {1}{111}      {1}{1111}
         {1}{2}  {1}{1}{1}  {1}{1}{11}    {1}{1}{111}
                 {1}{2}{2}  {1}{2}{12}    {1}{11}{11}
                 {1}{2}{3}  {1}{2}{22}    {1}{12}{22}
                            {1}{1}{1}{1}  {1}{2}{122}
                            {1}{1}{2}{2}  {1}{2}{222}
                            {1}{2}{2}{2}  {1}{1}{1}{11}
                            {1}{2}{3}{3}  {1}{1}{2}{22}
                            {1}{2}{3}{4}  {1}{2}{2}{12}
                                          {1}{2}{2}{22}
                                          {1}{2}{3}{23}
                                          {1}{2}{3}{33}
                                          {1}{1}{1}{1}{1}
                                          {1}{1}{2}{2}{2}
                                          {1}{2}{2}{2}{2}
                                          {1}{2}{2}{3}{3}
                                          {1}{2}{3}{3}{3}
                                          {1}{2}{3}{4}{4}
                                          {1}{2}{3}{4}{5}

Cf. A007716, A059523, A326961, A326965, A326967, A326972, A326974, A326976, A326977, A326979, A327012.

A327019 Number of non-isomorphic set-systems of weight n whose dual is a (strict) antichain.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 7, 15, 26, 61

Offset: 0

Gus Wiseman, Aug 15 2019

Also the number of non-isomorphic set-systems where every vertex is the unique common element of some subset of the edges, also called non-isomorphic T_1 set-systems.
A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}.
An antichain is a set of sets, none of which is a subset of any other.

			Non-isomorphic representatives of the a(1) = 1 through a(8) = 15 multiset partitions:
  {1}  {1}{2}  {1}{2}{3}  {1}{2}{12}    {1}{2}{3}{23}    {12}{13}{23}
                          {1}{2}{3}{4}  {1}{2}{3}{4}{5}  {1}{2}{13}{23}
                                                         {1}{2}{3}{123}
                                                         {1}{2}{3}{4}{34}
                                                         {1}{2}{3}{4}{5}{6}
.
  {1}{23}{24}{34}        {12}{13}{24}{34}
  {3}{12}{13}{23}        {2}{13}{14}{234}
  {1}{2}{3}{13}{23}      {1}{2}{13}{24}{34}
  {1}{2}{3}{24}{34}      {1}{2}{3}{14}{234}
  {1}{2}{3}{4}{234}      {1}{2}{3}{23}{123}
  {1}{2}{3}{4}{5}{45}    {1}{2}{3}{4}{1234}
  {1}{2}{3}{4}{5}{6}{7}  {1}{2}{34}{35}{45}
                         {1}{4}{23}{24}{34}
                         {2}{3}{12}{13}{23}
                         {1}{2}{3}{4}{12}{34}
                         {1}{2}{3}{4}{24}{34}
                         {1}{2}{3}{4}{35}{45}
                         {1}{2}{3}{4}{5}{345}
                         {1}{2}{3}{4}{5}{6}{56}
                         {1}{2}{3}{4}{5}{6}{7}{8}

Cf. A007716, A059523, A283877, A293993, A326961, A326965, A326974, A326976, A326977, A326979, A327012, A327018.

A355517 Number of nonisomorphic systems enumerated by A334254; that is, the number of inequivalent closure operators on a set of n elements where all singletons are closed.

Original entry on oeis.org

1, 2, 1, 4, 50, 7443, 95239971

Offset: 0

Dmitry I. Ignatov, Jul 05 2022

The T_1 axiom states that all singleton sets {x} are closed.

For n>1, this property implies strictness (meaning that the empty set is closed).

			a(0) = 1 counts the empty set, while a(1) = 2 counts {{1}} and {{},{1}}.
For a(2) = 1 the closure system is as follows:  {{1,2},{1},{2},{}}.
The a(3) = 4 inequivalent set-systems of closed sets are:
  {{1,2,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{1},{2},{3},{}}
  {{1,2,3},{1,2},{1,3},{2,3},{1},{2},{3},{}}.

Dmitry I. Ignatov, Supporting iPython code for counting nonequivalent closure systems w.r.t. the T_1 separation axiom, Github repository
Eric Weisstein's World of Mathematics, Separation Axioms
Wikipedia, Separation Axiom

The number of all closure operators is given in A102896, while A193674 contains the number of all nonisomorphic ones.
For T_1 closure operators and their strict counterparts, see A334254 and A334255, respectively; the only difference is a(1).
Cf. A326960, A326961, A326979.

cp's OEIS Frontend

A327012 Number of factorizations of n into factors > 1 whose dual is a (strict) antichain.

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A334254 Number of closure operators on a set of n elements which satisfy the T_1 separation axiom.

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A327058 Number of pairwise intersecting set-systems covering n vertices whose dual is a weak antichain.

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A327011 Number of unlabeled sets of subsets covering n vertices where every vertex is the unique common element of some subset of the edges, also called unlabeled covering T_1 sets of subsets.

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A327017 Number of non-isomorphic multiset partitions of weight n where every vertex, as a multiset of weight 1, is the multiset-meet of some subset of the edges.

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A327019 Number of non-isomorphic set-systems of weight n whose dual is a (strict) antichain.

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A355517 Number of nonisomorphic systems enumerated by A334254; that is, the number of inequivalent closure operators on a set of n elements where all singletons are closed.

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