A326970 -id:A326970 - cp's OEIS frontend

A326978 Number of integer partitions of n such that the dual of the multiset partition obtained by factoring each part into prime numbers is a weak antichain.

1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 38, 52, 68, 91, 116, 149, 191, 249, 311, 399, 498, 622, 773, 971, 1193, 1478, 1811, 2222, 2709, 3311, 4021, 4882, 5894, 7110, 8554, 10273, 12312, 14734, 17578, 20941, 24905, 29570, 35056, 41475, 48983, 57752, 68025, 79988

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}}.

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

			The a(0) = 1 through a(7) = 15 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)
           (11)  (21)   (22)    (32)     (33)      (43)
                 (111)  (31)    (41)     (42)      (52)
                        (211)   (221)    (51)      (61)
                        (1111)  (311)    (222)     (322)
                                (2111)   (321)     (331)
                                (11111)  (411)     (421)
                                         (2211)    (511)
                                         (3111)    (2221)
                                         (21111)   (3211)
                                         (111111)  (4111)
                                                   (22111)
                                                   (31111)
                                                   (211111)
                                                   (1111111)

Set-systems whose dual is a weak antichain are A326968.
Factorizations whose dual is a weak antichain are A326975.
The version where the dual is a strict antichain is A326977.
Cf. A000041, A319564, A319728, A326966, A326969, A326970, A326971, A326973.

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],stableQ[dual[primeMS/@#],submultQ]&]],{n,0,30}]

A326969 Number of sets of subsets of {1..n} whose dual is a weak antichain.

Original entry on oeis.org

2, 4, 12, 112, 38892

Offset: 0

Gus Wiseman, Aug 10 2019

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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

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

Sets of subsets whose dual is strict are A326941.
The BII-numbers of set-systems whose dual is a weak antichain are A326966.
Sets of subsets whose dual is a (strict) antichain are A326967.
The case without empty edges is A326968.
Cf. A001146, A059052, A326951, A326970, A326971, A326975, A326978.

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]]],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

a(n) = 2 * A326968(n).

a(n) = 2 * Sum_{k = 0..n} binomial(n, k) * A326970(k).

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.

A327018 Number of non-isomorphic set-systems of weight n whose dual is a weak antichain.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 17, 24, 51, 80, 180

Offset: 0

Gus Wiseman, Aug 15 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}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 17 multiset partitions:
  {1}  {12}    {123}      {1234}        {12345}          {123456}
       {1}{2}  {1}{23}    {1}{234}      {1}{2345}        {1}{23456}
               {1}{2}{3}  {12}{34}      {12}{345}        {12}{3456}
                          {1}{2}{12}    {1}{2}{345}      {123}{456}
                          {1}{2}{34}    {1}{23}{45}      {12}{13}{23}
                          {1}{2}{3}{4}  {1}{2}{3}{23}    {1}{23}{123}
                                        {1}{2}{3}{45}    {1}{2}{3456}
                                        {1}{2}{3}{4}{5}  {1}{23}{456}
                                                         {12}{34}{56}
                                                         {1}{2}{13}{23}
                                                         {1}{2}{3}{123}
                                                         {1}{2}{3}{456}
                                                         {1}{2}{34}{56}
                                                         {3}{4}{12}{34}
                                                         {1}{2}{3}{4}{34}
                                                         {1}{2}{3}{4}{56}
                                                         {1}{2}{3}{4}{5}{6}

Cf. A007716, A283877, A293993, A319643, A319721, A326966, A326968, A326970, A326972, A326973, A326974, A326975, A326978, A327017, A327019.

A327059 Number of pairwise intersecting set-systems covering a subset of {1..n} whose dual is a weak antichain.

Original entry on oeis.org

1, 2, 4, 10, 178

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) = 10 set-systems:
  {}  {}     {}      {}
      {{1}}  {{1}}   {{1}}
             {{2}}   {{2}}
             {{12}}  {{3}}
                     {{12}}
                     {{13}}
                     {{23}}
                     {{123}}
                     {{12}{13}{23}}
                     {{12}{13}{23}{123}}

Intersecting set-systems are A051185.
The BII-numbers of these set-systems are the intersection of A326910 and A326966.
Set-systems whose dual is a weak antichain are A326968.
The covering version is A327058.
The unlabeled multiset partition version is A327060.
Cf. A000372, A305844, A326951, A326965, A326970, A327057, A327062.

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]=={}&],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

Binomial transform of A327058.

cp's OEIS Frontend

A326978 Number of integer partitions of n such that the dual of the multiset partition obtained by factoring each part into prime numbers is a weak antichain.

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A326969 Number of sets of subsets of {1..n} whose dual is a weak antichain.

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

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A327018 Number of non-isomorphic set-systems of weight n whose dual is a weak antichain.

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A327059 Number of pairwise intersecting set-systems covering a subset of {1..n} whose dual is a weak antichain.

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