A316978 -id:A316978 - cp's OEIS frontend

A326941 Number of T_0 sets of subsets of {1..n}.

2, 4, 14, 224, 64210, 4294322204, 18446744009291513774, 340282366920938463075992982725615419816, 115792089237316195423570985008687907843742078391854287068939455414919611614210

Offset: 0

Gus Wiseman, Aug 07 2019

nonn

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 2 through a(2) = 14 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A001146.
The covering case is A326939.
The case without empty edges is A326940.
The unlabeled version is A326949.
Cf. A003180, A059201, A316978, A319559, A319564, A319637, A326946, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]],UnsameQ@@dual[#]&]],{n,0,3}]

a(n) = 2 * A326940(n).

Binomial transform of A326939.

a(5)-a(8) from Andrew Howroyd, Aug 14 2019

A326976 Number of factorizations of n into factors > 1 such that every prime factor of n is the GCD of some subset of the factors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Aug 13 2019

nonn

			The a(72) = 5 factorizations:
  (3*4*6)
  (2*3*12)
  (2*2*3*6)
  (2*3*3*4)
  (2*2*2*3*3)

Factorizations whose dual is a weak antichain are A326975.
T_1 factorizations (whose dual is a strict antichain) are A327012.
T_0 factorizations (whose dual is strict) are A316978.
Cf. A001055, A326947, A326965, A326972, A326974, A326977, A326979.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],n==1||Union[Select[GCD@@@Rest[Subsets[#]],PrimeQ]]==First/@FactorInteger[n]&]],
{n,100}]

A330236 MM-numbers of fully chiral multisets of multisets.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 53, 54, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Dec 10 2019

nonn

A multiset of multisets is fully chiral every permutation of the vertices gives a different representative.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of all fully chiral multisets of multisets together with their MM-numbers begins:
   1:             18: {}{1}{1}      37: {112}          57: {1}{111}
   2: {}          19: {111}         38: {}{111}        59: {7}
   3: {1}         20: {}{}{2}       39: {1}{12}        61: {122}
   4: {}{}        21: {1}{11}       40: {}{}{}{2}      62: {}{5}
   5: {2}         22: {}{3}         41: {6}            63: {1}{1}{11}
   6: {}{1}       23: {22}          42: {}{1}{11}      64: {}{}{}{}{}{}
   7: {11}        24: {}{}{}{1}     44: {}{}{3}        65: {2}{12}
   8: {}{}{}      25: {2}{2}        45: {1}{1}{2}      67: {8}
   9: {1}{1}      27: {1}{1}{1}     46: {}{22}         68: {}{}{4}
  10: {}{2}       28: {}{}{11}      48: {}{}{}{}{1}    69: {1}{22}
  11: {3}         31: {5}           49: {11}{11}       70: {}{2}{11}
  12: {}{}{1}     32: {}{}{}{}{}    50: {}{2}{2}       71: {113}
  14: {}{11}      34: {}{4}         53: {1111}         72: {}{}{}{1}{1}
  16: {}{}{}{}    35: {2}{11}       54: {}{1}{1}{1}    74: {}{112}
  17: {4}         36: {}{}{1}{1}    56: {}{}{}{11}     75: {1}{2}{2}
The complement starts: {13, 15, 26, 29, 30, 33, 43, 47, 51, 52, 55, 58, 60, 66, 73, 79, 85, 86, 93, 94}.

Costrict (or T_0) factorizations are A316978.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Full chiral partitions are A330228.
Fully chiral covering set-systems are A330229.
Fully chiral factorizations are A330235.
Cf. A001055, A007716, A083323, A112798, A303975, A317533, A330098, A330223, A330224, A330232.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Select[Range[100],Length[graprms[primeMS/@primeMS[#]]]==Length[Union@@primeMS/@primeMS[#]]!&]

Numbers n such that A330098(n) = A303975(n)!.

A245567 Number of antichain covers of a labeled n-set such that for every two distinct elements in the n-set, there is a set in the antichain cover containing one of the elements but not the other.

Original entry on oeis.org

2, 1, 1, 5, 76, 5993, 7689745, 2414465044600, 56130437141763247212112, 286386577668298408602599478477358234902247

Offset: 0

Patrick De Causmaecker, Jul 25 2014

This is the number of antichain covers such that the induced partition contains only singletons. The induced partition of {{1,2},{2,3},{1,3},{3,4}} is {{1},{2},{3},{4}}, while the induced partition of {{1,2,3},{2,3,4}} is {{1},{2,3},{4}}.
This sequence is related to A006126. See 1st formula.
The sequence is also related to Dedekind numbers through Stirling numbers of the second kind. See 2nd formula.
Sets of subsets of the described type are said to be T_0. - Gus Wiseman, Aug 14 2019

			For n = 0, a(0) = 2 by the antisets {}, {{}}.
For n = 1, a(1) = 1 by the antiset {{1}}.
For n = 2, a(2) = 1 by the antiset {{1},{2}}.
For n = 3, a(3) = 5 by the antisets {{1},{2},{3}}, {{1,2},{1,3}}, {{1,2},{2,3}}, {{1,3},{2,3}}, {{1,2},{1,3},{2,3}}.

Patrick De Causmaecker and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.

Cf. A000372 (Dedekind numbers), A006126 (Number of antichain covers of a labeled n-set).
Sequences counting and ranking T_0 structures:
  A000112 (unlabeled topologies),
  A001035 (topologies),
  A059201 (covering set-systems),
  A245567 (antichain covers),
  A309615 (covering set-systems closed under intersection),
  A316978 (factorizations),
  A319559 (unlabeled set-systems by weight),
  A319564 (integer partitions),
  A319637 (unlabeled covering set-systems),
  A326939 (covering sets of subsets),
  A326940 (set-systems),
  A326941 (sets of subsets),
  A326943 (covering sets of subsets closed under intersection),
  A326944 (covering sets of subsets with {} and closed under intersection),
  A326945 (sets of subsets closed under intersection),
  A326946 (unlabeled set-systems),
  A326947 (BII-numbers of set-systems),
  A326948 (connected set-systems),
  A326949 (unlabeled sets of subsets),
  A326950 (antichains),
  A326959 (set-systems closed under intersection),
  A327013 (unlabeled covering set-systems closed under intersection),
  A327016 (BII-numbers of topologies).

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&stableQ[#,SubsetQ]&&UnsameQ@@dual[#]&]],{n,0,3}] (* Gus Wiseman, Aug 14 2019 *)

A000372(n) = Sum_{k=0..n} S(n+1,k+1)*a(k).
a(n) = A006126(n) - Sum_{k=1..n-1} S(n,k)*a(k).
Were n > 0 and S(n,k) is the number of ways to partition a set of n elements into k nonempty subsets.
Inverse binomial transform of A326950, if we assume a(0) = 1. - Gus Wiseman, Aug 14 2019

Definition corrected by Patrick De Causmaecker, Oct 10 2014

a(9), based on A000372, from Patrick De Causmaecker, Jun 01 2023

A326943 Number of T_0 sets of subsets of {1..n} that cover all n vertices and are closed under intersection.

Original entry on oeis.org

2, 2, 6, 70, 4078, 2704780, 151890105214, 28175292217767880450

Offset: 0

Gus Wiseman, Aug 08 2019

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 2 through a(3) = 6 sets of subsets:
  {}    {{1}}     {{1},{1,2}}
  {{}}  {{},{1}}  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A326906.
The case without empty edges is A309615.
The non-covering version is A326945.
The version not closed under intersection is A326939.
Cf. A003180, A003181, A003465, A059052, A059201, A245567, A316978, A319564, A319637, A326940, A326941, A326942, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Inverse binomial transform of A326945.

a(n) = Sum_{k=0..n} Stirling1(n,k)*A326906(k). - Andrew Howroyd, Aug 14 2019

a(5)-a(7) from Andrew Howroyd, Aug 14 2019

A326975 Number of factorizations of n into factors > 1 whose dual is a weak antichain.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 3, 2, 1, 5, 1, 7, 2, 2, 2, 9, 1, 2, 2, 3, 1, 5, 1, 2, 2, 2, 1, 5, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 2, 11, 2, 5, 1, 2, 2, 5, 1, 12, 1, 2, 2, 2, 2, 5, 1, 5, 5, 2, 1, 4, 2, 2

Offset: 1

Gus Wiseman, Aug 13 2019

nonn

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.

A weak antichain is a multiset of multisets, none of which is a proper submultiset of any other.

			The a(36) = 9 factorizations:
  (36)
  (4*9)
  (6*6)
  (2*18)
  (3*12)
  (2*2*9)
  (2*3*6)
  (3*3*4)
  (2*2*3*3)

The T_0 case (where the dual is strict) is A316978.
Set-systems whose dual is a weak antichain are A326968.
Partitions whose dual is a weak antichain are A326978.
The T_1 case (where the dual is a strict antichain) is A327012.
Cf. A001055, A059523, A316978, A319639, A326965, A326966, A326969, A326970, A326971, A326976, A326977.

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],stableQ[dual[primeMS/@#],submultQ]&]],{n,100}]

A330235 Number of fully chiral factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 0, 1, 3, 2, 0, 1, 4, 1, 0, 0, 5, 1, 4, 1, 4, 0, 0, 1, 7, 2, 0, 3, 4, 1, 0, 1, 7, 0, 0, 0, 4, 1, 0, 0, 7, 1, 0, 1, 4, 4, 0, 1, 12, 2, 4, 0, 4, 1, 7, 0, 7, 0, 0, 1, 4, 1, 0, 4, 11, 0, 0, 1, 4, 0, 0, 1, 16, 1, 0, 4, 4, 0, 0, 1, 12, 5, 0, 1, 4, 0, 0

Offset: 1

Gus Wiseman, Dec 08 2019

nonn

A multiset of multisets is fully chiral every permutation of the vertices gives a different representative. A factorization is fully chiral if taking the multiset of prime indices of each factor gives a fully chiral multiset of multisets.

			The a(n) factorizations for n = 1, 4, 8, 12, 16, 24, 48:
  ()  (4)    (8)      (12)     (16)       (24)       (48)
      (2*2)  (2*4)    (2*6)    (2*8)      (3*8)      (6*8)
             (2*2*2)  (3*4)    (4*4)      (4*6)      (2*24)
                      (2*2*3)  (2*2*4)    (2*12)     (3*16)
                               (2*2*2*2)  (2*2*6)    (4*12)
                                          (2*3*4)    (2*3*8)
                                          (2*2*2*3)  (2*4*6)
                                                     (3*4*4)
                                                     (2*2*12)
                                                     (2*2*2*6)
                                                     (2*2*3*4)
                                                     (2*2*2*2*3)

The costrict (or T_0) version is A316978.
The achiral version is A330234.
Planted achiral trees are A003238.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Full chiral partitions are A330228.
Fully chiral covering set-systems are A330229.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A001055, A007716, A083323, A112798, A317533, A330098, A330223, A330224, A330232.

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]]}]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[facs[n],Length[graprms[primeMS/@#]]==Length[Union@@primeMS/@#]!&]],{n,100}]

A322847 Numbers whose prime indices have no equivalent primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75

Offset: 1

Gus Wiseman, Dec 28 2018

nonn

The complement is {13, 26, 29, 43, 47, 52, 58, 73, 79, 86, 94, ...}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
In an integer partition, two primes are equivalent if each part has in its prime factorization the same multiplicity of both primes. For example, in (6,5) the primes {2,3} are equivalent while {2,5} and {3,5} are not. In (30,6) also, the primes {2,3} are equivalent, while {2,5} and {3,5} are not.
Also MM-numbers of T_0 multiset multisystems. A multiset multisystem is a finite multiset of finite multisets. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. The dual of a multiset multisystem has, for each vertex, one block consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict (no repeated parts).

			The prime indices of 339 are {2, 30}, in which the primes {3,5} are equivalent, so 339 is not in the sequence.

Cf. A056239, A059201, A112798, A302242, A316978, A316979, A316983, A319558, A319564, A319728, A322846.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Select[Range[100],UnsameQ@@dual[primeMS/@primeMS[#]]&]

A326944 Number of T_0 sets of subsets of {1..n} that cover all n vertices, contain {}, and are closed under intersection.

Original entry on oeis.org

1, 1, 4, 58, 3846, 2685550, 151873991914, 28175291154649937052

Offset: 0

Gus Wiseman, Aug 08 2019

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 1 through a(2) = 4 sets of subsets:
  {{}}  {{},{1}}  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The version not closed under intersection is A059201.
The non-T_0 version is A326881.
The version where {} is not necessarily an edge is A326943.
Cf. A003181, A003465, A055621, A182507, A245567, A316978, A319564, A326906, A326939, A326941, A326945, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n]]],MemberQ[#,{}]&&Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

a(n) = Sum_{k=0..n} Stirling1(n,k)*A326881(k). - Andrew Howroyd, Aug 14 2019

a(5)-a(7) from Andrew Howroyd, Aug 14 2019

A326945 Number of T_0 sets of subsets of {1..n} that are closed under intersection.

Original entry on oeis.org

2, 4, 12, 96, 4404, 2725942, 151906396568, 28175293281055562650

Offset: 0

Gus Wiseman, Aug 08 2019

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 2 through a(2) = 12 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A102897.
The version not closed under intersection is A326941.
The covering case is A326943.
The case without empty edges is A326959.
Cf. A003180, A182507, A316978, A319564, A326906, A326939, A326940, A326944, A326947.

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

Binomial transform of A326943.

a(5)-a(7) from Andrew Howroyd, Aug 14 2019

cp's OEIS Frontend

A326941 Number of T_0 sets of subsets of {1..n}.

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A326976 Number of factorizations of n into factors > 1 such that every prime factor of n is the GCD of some subset of the factors.

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A330236 MM-numbers of fully chiral multisets of multisets.

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A245567 Number of antichain covers of a labeled n-set such that for every two distinct elements in the n-set, there is a set in the antichain cover containing one of the elements but not the other.

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A326943 Number of T_0 sets of subsets of {1..n} that cover all n vertices and are closed under intersection.

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A326975 Number of factorizations of n into factors > 1 whose dual is a weak antichain.

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A330235 Number of fully chiral factorizations of n.

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A322847 Numbers whose prime indices have no equivalent primes.

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