A316978 -id:A316978 - cp's OEIS frontend

A316979 Number of strict factorizations of n into factors > 1 with no equivalent primes.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 7, 1, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 6, 1, 1, 3, 4, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 3, 3, 1, 1, 1, 7, 2, 1, 1, 6, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 18 2018

nonn

In a factorization, two primes are equivalent if each factor has in its prime factorization the same multiplicity of both primes. For example, in 60 = (2*30) the primes {3, 5} are equivalent but {2, 3} and {2, 5} are not.

			The a(24) = 5 factorizations are (2*3*4), (2*12), (3*8), (4*6), (24).
The a(36) = 4 factorizations are (2*3*6), (2*18), (3*12), (4*9).

Cf. A000009, A001055, A007716, A007717, A020555, A045778, A130091, A162247, A281116.

Cf. A316974, A316978, A316980, A316981.

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}];
Table[Length[Select[facs[n],And[UnsameQ@@#,UnsameQ@@dual[primeMS/@#]]&]],{n,100}]

a(prime^n) = A000009(n).

A326949 Number of unlabeled T_0 sets of subsets of {1..n}.

Original entry on oeis.org

2, 4, 10, 68, 3838, 37320356

Offset: 0

Gus Wiseman, Aug 08 2019

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}}. The T_0 condition means that the dual is strict (no repeated edges).

			Non-isomorphic representatives of the a(0) = 2 through a(2) = 10 sets of sets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{},{1}}
                  {{1},{2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A003180.
The labeled version is A326941.
The covering case is A326942 (first differences).
The case without empty edges is A326946.
Cf. A000371, A000612, A003181, A059052, A245567, A316978, A319559, A319564, A319637, A326939, A326940.

a(n) = 2 * A326946(n).

a(5) from Max Alekseyev, Oct 11 2023

A316981 Number of non-isomorphic strict multiset partitions of weight n with no equivalent vertices.

Original entry on oeis.org

1, 1, 2, 6, 15, 40, 121

Offset: 0

Gus Wiseman, Jul 18 2018

Also the number of nonnegative integer n X n matrices with sum of elements equal to n, under row and column permutations, with no equal rows and no equal columns.

In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second.

			Non-isomorphic representatives of the a(3) = 6 strict multiset partitions with no equivalent vertices (first column) and their duals (second column):
      (111) <-> (111)
      (122) <-> (1)(11)
    (1)(11) <-> (122)
    (1)(22) <-> (1)(22)
    (2)(12) <-> (2)(12)
  (1)(2)(3) <-> (1)(2)(3)

Cf. A000009, A001055, A007716, A007717, A020555, A045778.

Cf. A316974, A316978, A316979, A316980, A316983.

A326942 Number of unlabeled T_0 sets of subsets of {1..n} that cover all n vertices.

Original entry on oeis.org

2, 2, 6, 58, 3770

Offset: 0

Gus Wiseman, Aug 07 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, 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).

			Non-isomorphic representatives of the a(0) = 2 through a(2) = 6 sets of subsets:
  {}    {{1}}     {{1},{2}}
  {{}}  {{},{1}}  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A003181.
The case without empty edges is A319637.
The labeled version is A326939.
The non-covering version is A326949 (partial sums).
Cf. A000371, A003180, A055621, A059201, A316978, A319559, A319564, A326907, A326941, A326943, A326946.

a(n) = 2 * A319637(n).

A326959 Number of T_0 set-systems covering a subset of {1..n} that are closed under intersection.

Original entry on oeis.org

1, 2, 5, 22, 297, 20536, 16232437, 1063231148918, 225402337742595309857

Offset: 0

Gus Wiseman, Aug 13 2019

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}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 1 through a(3) = 22 set-systems:
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{2}}        {{2}}
             {{1},{1,2}}  {{3}}
             {{2},{1,2}}  {{1},{1,2}}
                          {{1},{1,3}}
                          {{2},{1,2}}
                          {{2},{2,3}}
                          {{3},{1,3}}
                          {{3},{2,3}}
                          {{1},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1},{1,2},{1,2,3}}
                          {{1},{1,3},{1,2,3}}
                          {{2},{1,2},{1,2,3}}
                          {{2},{2,3},{1,2,3}}
                          {{3},{1,3},{1,2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{1},{1,2},{1,3},{1,2,3}}
                          {{2},{1,2},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

The covering case is A309615.
T_0 set-systems are A326940.
The version with empty edges allowed is A326945.
Cf. A051185, A058891, A059201, A316978, A319559, A309615, A319637, A326943, A326944, A326946, A326947, A326959.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],UnsameQ@@dual[#]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]

Binomial transform of A309615.

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

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

Original entry on oeis.org

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}]

A326948 Number of connected T_0 set-systems on n vertices.

Original entry on oeis.org

1, 1, 3, 86, 31302, 2146841520, 9223371978880250448, 170141183460469231408869283342774399392, 57896044618658097711785492504343953919148780260559635830120038252613826101856

Offset: 0

Gus Wiseman, Aug 08 2019

nonn

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}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(3) = 86 set-systems:
  {12}{13}         {1}{2}{13}{123}     {1}{2}{3}{13}{23}
  {12}{23}         {1}{2}{23}{123}     {1}{2}{3}{13}{123}
  {13}{23}         {1}{3}{12}{13}      {1}{2}{3}{23}{123}
  {1}{2}{123}      {1}{3}{12}{23}      {1}{2}{12}{13}{23}
  {1}{3}{123}      {1}{3}{12}{123}     {1}{2}{12}{13}{123}
  {1}{12}{13}      {1}{3}{13}{23}      {1}{2}{12}{23}{123}
  {1}{12}{23}      {1}{3}{13}{123}     {1}{2}{13}{23}{123}
  {1}{12}{123}     {1}{3}{23}{123}     {1}{3}{12}{13}{23}
  {1}{13}{23}      {1}{12}{13}{23}     {1}{3}{12}{13}{123}
  {1}{13}{123}     {1}{12}{13}{123}    {1}{3}{12}{23}{123}
  {2}{3}{123}      {1}{12}{23}{123}    {1}{3}{13}{23}{123}
  {2}{12}{13}      {1}{13}{23}{123}    {1}{12}{13}{23}{123}
  {2}{12}{23}      {2}{3}{12}{13}      {2}{3}{12}{13}{23}
  {2}{12}{123}     {2}{3}{12}{23}      {2}{3}{12}{13}{123}
  {2}{13}{23}      {2}{3}{12}{123}     {2}{3}{12}{23}{123}
  {2}{23}{123}     {2}{3}{13}{23}      {2}{3}{13}{23}{123}
  {3}{12}{13}      {2}{3}{13}{123}     {2}{12}{13}{23}{123}
  {3}{12}{23}      {2}{3}{23}{123}     {3}{12}{13}{23}{123}
  {3}{13}{23}      {2}{12}{13}{23}     {1}{2}{3}{12}{13}{23}
  {3}{13}{123}     {2}{12}{13}{123}    {1}{2}{3}{12}{13}{123}
  {3}{23}{123}     {2}{12}{23}{123}    {1}{2}{3}{12}{23}{123}
  {12}{13}{23}     {2}{13}{23}{123}    {1}{2}{3}{13}{23}{123}
  {12}{13}{123}    {3}{12}{13}{23}     {1}{2}{12}{13}{23}{123}
  {12}{23}{123}    {3}{12}{13}{123}    {1}{3}{12}{13}{23}{123}
  {13}{23}{123}    {3}{12}{23}{123}    {2}{3}{12}{13}{23}{123}
  {1}{2}{3}{123}   {3}{13}{23}{123}    {1}{2}{3}{12}{13}{23}{123}
  {1}{2}{12}{13}   {12}{13}{23}{123}
  {1}{2}{12}{23}   {1}{2}{3}{12}{13}
  {1}{2}{12}{123}  {1}{2}{3}{12}{23}
  {1}{2}{13}{23}   {1}{2}{3}{12}{123}

Andrew Howroyd, Table of n, a(n) for n = 0..11

The same with covering instead of connected is A059201, with unlabeled version A319637.
The non-T_0 version is A323818 (covering) or A326951 (not-covering).
The non-connected version is A326940, with unlabeled version A326946.
Cf. A000371, A003465, A245567, A316978, A319559, A319564, A326939, A326941, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&Length[csm[#]]<=1&&UnsameQ@@dual[#]&]],{n,0,3}]

Logarithmic transform of A059201.

A327016 BII-numbers of finite T_0 topologies without their empty set.

Original entry on oeis.org

0, 1, 2, 5, 6, 7, 8, 17, 24, 25, 34, 40, 42, 69, 70, 71, 81, 85, 87, 88, 89, 93, 98, 102, 103, 104, 106, 110, 120, 121, 122, 127, 128, 257, 384, 385, 514, 640, 642, 1029, 1030, 1031, 1281, 1285, 1287, 1408, 1409, 1413, 1538, 1542, 1543, 1664, 1666, 1670, 1920

Offset: 1

Gus Wiseman, Aug 14 2019

nonn

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}}. The T_0 condition means that the dual is strict (no repeated edges).

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			The sequence of all finite T_0 topologies without their empty set together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  25: {{1},{3},{1,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  42: {{2},{3},{2,3}}
  69: {{1},{1,2},{1,2,3}}
  70: {{2},{1,2},{1,2,3}}
  71: {{1},{2},{1,2},{1,2,3}}
  81: {{1},{1,3},{1,2,3}}
  85: {{1},{1,2},{1,3},{1,2,3}}
  87: {{1},{2},{1,2},{1,3},{1,2,3}}
  88: {{3},{1,3},{1,2,3}}

Gus Wiseman, The first 100 finite T_0 topologies, represented as posets.

T_0 topologies are A001035, with unlabeled version A000112.
BII-numbers of topologies without their empty set are A326876.
BII-numbers of T_0 set-systems are A326947.
Cf. A001930, A048793, A306445, A316978, A319564, A326031, A326872, A326875, A326939, A326941, A326959.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],UnsameQ@@dual[bpe/@bpe[#]]&&SubsetQ[bpe/@bpe[#],Union[Union@@@Tuples[bpe/@bpe[#],2],DeleteCases[Intersection@@@Tuples[bpe/@bpe[#],2],{}]]]&]

A322846 Squarefree numbers whose prime indices have no equivalent primes.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 46, 51, 53, 55, 57, 59, 61, 62, 65, 66, 67, 69, 70, 71, 74, 77, 78, 82, 83, 85, 87, 89, 91, 93, 95, 97, 102, 103, 105, 106, 107, 109, 110, 111, 114, 115, 118, 119

Offset: 1

Gus Wiseman, Dec 28 2018

nonn

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 strict 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 sequence of all strict T_0 multiset multisystems together with their MM-numbers begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
  10: {{},{2}}
  11: {{3}}
  14: {{},{1,1}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  30: {{},{1},{2}}
  31: {{5}}
  33: {{1},{3}}
  34: {{},{4}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  38: {{},{1,1,1}}
  39: {{1},{1,2}}

Cf. A000009, A005117, A056239, A059201, A112798, A302242, A302505, A316978, A316979, A316983, A319558, A319564, A319728, A322847.

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],And[SquareFreeQ[#],UnsameQ@@dual[primeMS/@primeMS[#]]]&]

cp's OEIS Frontend

A316979 Number of strict factorizations of n into factors > 1 with no equivalent primes.

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A326949 Number of unlabeled T_0 sets of subsets of {1..n}.

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A316981 Number of non-isomorphic strict multiset partitions of weight n with no equivalent vertices.

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A326942 Number of unlabeled T_0 sets of subsets of {1..n} that cover all n vertices.

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A326959 Number of T_0 set-systems covering a subset of {1..n} that are closed under intersection.

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

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A326948 Number of connected T_0 set-systems on n vertices.

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A327016 BII-numbers of finite T_0 topologies without their empty set.

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A322846 Squarefree numbers whose prime indices have no equivalent primes.

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