A327012 -id:A327012 - cp's OEIS frontend

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.

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}]

A326977 Number of integer partitions of n such that the dual of the multiset partition obtained by factoring each part into prime numbers is a (strict) antichain, also called T_1 integer partitions.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 49, 64, 85, 109, 141, 181, 234, 294, 375, 470, 589, 733, 917, 1131, 1401, 1720, 2113, 2581, 3153, 3833, 4655, 5631, 6801, 8192, 9849, 11816, 14148, 16899, 20153, 23990, 28503, 33815, 40038, 47330, 55858, 65841, 77475

Offset: 0

Gus Wiseman, Aug 13 2019

nonn

The dual of a multiset partition 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}}. An antichain is a set of multisets, none of which is a submultiset of any other.

			The a(0) = 1 through a(7) = 14 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (33)      (7)
           (11)  (21)   (22)    (32)     (42)      (43)
                 (111)  (31)    (41)     (51)      (52)
                        (211)   (221)    (222)     (322)
                        (1111)  (311)    (321)     (331)
                                (2111)   (411)     (421)
                                (11111)  (2211)    (511)
                                         (3111)    (2221)
                                         (21111)   (3211)
                                         (111111)  (4111)
                                                   (22111)
                                                   (31111)
                                                   (211111)
                                                   (1111111)

T_0 integer partitions are A319564.
Set-systems whose dual is a (strict) antichain are A326965.
The version where the dual is a weak antichain is A326978.
T_1 factorizations (whose dual is a strict antichain) are A327012.
Cf. A000041, A319728, A326961, A326974, A326976, A326979.

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}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@dual[primeMS/@#]&&stableQ[dual[primeMS/@#],submultQ]&]],{n,0,30}]

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}]

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.

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A326977 Number of integer partitions of n such that the dual of the multiset partition obtained by factoring each part into prime numbers is a (strict) antichain, also called T_1 integer partitions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs