A293606 -id:A293606 - cp's OEIS frontend

A327358 Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity >= k.

1, 1, 0, 2, 1, 0, 5, 3, 2, 0, 20, 14, 10, 6, 0, 180, 157, 128, 91, 54, 0

Offset: 0

Gus Wiseman, Sep 09 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.
The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any empty or duplicate edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.
If empty edges are allowed, we have T(0,0) = 2.

			Triangle begins:
    1
    1   0
    2   1   0
    5   3   2   0
   20  14  10   6   0
  180 157 128  91  54   0
Non-isomorphic representatives of the antichains counted in row n = 4:
  {1234}          {1234}           {1234}           {1234}
  {1}{234}        {12}{134}        {123}{124}       {12}{134}{234}
  {12}{34}        {123}{124}       {12}{13}{234}    {123}{124}{134}
  {12}{134}       {12}{13}{14}     {12}{134}{234}   {12}{13}{14}{234}
  {123}{124}      {12}{13}{24}     {123}{124}{134}  {123}{124}{134}{234}
  {1}{2}{34}      {12}{13}{234}    {12}{13}{24}{34} {12}{13}{14}{23}{24}{34}
  {2}{13}{14}     {12}{134}{234}   {12}{13}{14}{234}
  {12}{13}{14}    {123}{124}{134}  {12}{13}{14}{23}{24}
  {12}{13}{24}    {12}{13}{14}{23} {123}{124}{134}{234}
  {1}{2}{3}{4}    {12}{13}{24}{34} {12}{13}{14}{23}{24}{34}
  {12}{13}{234}   {12}{13}{14}{234}
  {12}{134}{234}  {12}{13}{14}{23}{24}
  {123}{124}{134} {123}{124}{134}{234}
  {4}{12}{13}{23} {12}{13}{14}{23}{24}{34}
  {12}{13}{14}{23}
  {12}{13}{24}{34}
  {12}{13}{14}{234}
  {12}{13}{14}{23}{24}
  {123}{124}{134}{234}
  {12}{13}{14}{23}{24}{34}

Column k = 0 is A261005, or A006602 if empty edges are allowed.
Column k = 1 is A261006 (clutters), if we assume A261006(0) = A261006(1) = 0.
Column k = 2 is A305028 (blobs), if we assume A305028(0) = A305028(2) = 0.
Column k = n - 1 is A327425 (cointersecting).
The labeled version is A327350.
Negated first differences of rows are A327359.
Cf. A006126, A055621, A120338, A293606, A293993, A327334, A327351, A327356.

A318131 Number of non-isomorphic sets of finite (possibly empty) sets with union {1,2,...,n} and intersection {}.

Original entry on oeis.org

1, 1, 6, 60, 3836, 37325360, 25626412263611792, 67516342973185974276922865448446208, 2871827610052485009904013737758920847534777143951264797898686184985092096

Offset: 0

Gus Wiseman, Aug 18 2018

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 0..12

Cf. A000371, A000612, A003465, A055621, A119563, A131288, A283877, A293606, A304997.

Cf. A318128, A318129, A318130, A318132.

sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Subsets[Range[n]]],And[Union@@#===Range[n],Intersection@@#=={}]&]]],{n,4}]

a(n) = 2*(A055621(n) - A055621(n-1)) = 2*(A000612(n) - 2*A000612(n-1) + A000612(n-2)) for n >= 2. - Andrew Howroyd, Jan 29 2024

a(5) onwards from Andrew Howroyd, Jan 29 2024

A318132 Number of non-isomorphic set-systems (finite sets of finite nonempty sets) with union {1,2,...,n} and intersection {}.

Original entry on oeis.org

1, 0, 2, 26, 1884, 18660728, 12813206113141264, 33758171486592987125648226573752576, 1435913805026242504952006868879460423733630400489039411798068453617852416

Offset: 0

Gus Wiseman, Aug 18 2018

nonn

			Non-isomorphic representatives of the a(3) = 26 set-systems:
  {{1},{2,3}}
  {{1},{2},{3}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,2},{1,3}}
  {{1},{2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,2,3}}
  {{1},{2},{1,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3}}
  {{1},{1,2},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,3},{2,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..12

Cf. A000371, A000612, A003465, A055621, A119563, A131288, A283877, A293606, A304997.

Cf. A318128, A318129, A318130, A318131.

sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Rest[Subsets[Range[n]]]],And[Union@@#===Range[n],Intersection@@#=={}]&]]],{n,4}]

a(n) = A055621(n) - 2*A055621(n-1) = A000612(n) - 3*A000612(n-1) + 2*A000612(n-2) for n >= 2. - Andrew Howroyd, Jan 29 2024

a(5) onwards from Andrew Howroyd, Jan 29 2024

A327357 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of sets covering n vertices with non-spanning edge-connectivity k.

Original entry on oeis.org

1, 0, 1, 1, 1, 4, 1, 3, 1, 30, 13, 33, 32, 6, 546, 421, 1302, 1915, 1510, 693, 316, 135, 45, 10, 1

Offset: 0

Gus Wiseman, Sep 11 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.

The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.

			Triangle begins:
    1
    0    1
    1    1
    4    1    3    1
   30   13   33   32    6
  546  421 1302 1915 1510  693  316  135   45   10    1
Row n = 3 counts the following antichains:
  {{1},{2,3}}    {{1,2,3}}  {{1,2},{1,3}}  {{1,2},{1,3},{2,3}}
  {{2},{1,3}}               {{1,2},{2,3}}
  {{3},{1,2}}               {{1,3},{2,3}}
  {{1},{2},{3}}

Row sums are A307249.
Column k = 0 is A120338.
The non-covering version is A327353.
The version for spanning edge-connectivity is A327352.
The specialization to simple graphs is A327149, with unlabeled version A327201.
Cf. A014466, A293606, A326704, A326787, A327071, A327148, A327236, A327351.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
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]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Union@@#==Range[n]&&eConn[#]==k&]],{n,0,5},{k,0,2^n}]//.{foe___,0}:>{foe}

A327437 Number of unlabeled antichains of nonempty subsets of {1..n} that are either non-connected or non-covering (spanning edge-connectivity 0).

Original entry on oeis.org

1, 1, 3, 6, 15, 52, 410, 32697

Offset: 0

Gus Wiseman, Sep 11 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.

The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 15 antichains:
  {}  {}         {}             {}
      {{1}}      {{1}}          {{1}}
      {{1},{2}}  {{1,2}}        {{1,2}}
                 {{1},{2}}      {{1},{2}}
                 {{1},{2,3}}    {{1,2,3}}
                 {{1},{2},{3}}  {{1},{2,3}}
                                {{1,2},{1,3}}
                                {{1},{2},{3}}
                                {{1},{2,3,4}}
                                {{1,2},{3,4}}
                                {{1},{2},{3,4}}
                                {{1},{2},{3},{4}}
                                {{2},{1,3},{1,4}}
                                {{1,2},{1,3},{2,3}}
                                {{4},{1,2},{1,3},{2,3}}

Column k = 0 of A327438.
The labeled version is A327355.
The covering case is A327426.
Cf. A014466 A052446, A120338, A261005, A293606, A327201, A327236, A327352, A327353, A327354, A327438.

a(n > 0) = A306505(n) - A261006(n).

A303364 Number of strict integer partitions of n with pairwise indivisible and squarefree parts.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 2, 1, 1, 3, 2, 2, 4, 3, 3, 4, 6, 5, 5, 6, 7, 8, 9, 10, 10, 11, 11, 14, 14, 17, 16, 18, 19, 23, 24, 27, 29, 30, 33, 36, 41, 41, 42, 46, 51, 56, 60, 66, 67, 71, 81, 86, 93, 96, 101, 110, 121, 129, 135, 144, 153, 159, 173, 192, 204, 207, 224

Offset: 1

Gus Wiseman, Apr 22 2018

nonn

			The a(23) = 9 strict integer partitions are (23), (13,10), (17,6), (21,2), (10,7,6), (11,7,5), (13,7,3), (11,7,3,2), (13,5,3,2).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..700 (terms 0..400 from Andrew Howroyd)

Cf. A000009, A000837, A003238, A005117, A006126, A051424, A073576, A285572, A285573, A293606, A293993, A303362, A303365.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@SquareFreeQ/@#&&Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]==={}&]],{n,60}]

lista(nn)={local(Cache=Map());
  my(excl=vector(nn, n, sumdiv(n, d, 2^(n-d))));
  my(c(n, m, b)=
     if(n==0, 1,
        while(m>n || bittest(b,0), m--; b>>=1);
        my(hk=[n, m, b], z);
        if(!mapisdefined(Cache, hk, &z),
          z = if(m, self()(n, m-1, b>>1) + self()(n-m, m, bitor(b, excl[m])), 0);
          mapput(Cache, hk, z)); z));
  my(a(n)=c(n, n, sum(i=1, n, if(!issquarefree(i), 2^(n-i)))));
  for(n=1, nn, print1(a(n), ", "))
} \\ Andrew Howroyd, Nov 02 2019

A327806 Triangle read by rows where T(n,k) is the number of antichains of sets with n vertices and vertex-connectivity >= k.

Original entry on oeis.org

1, 2, 0, 5, 1, 0, 19, 5, 2, 0, 167, 84, 44, 17, 0

Offset: 0

Gus Wiseman, Sep 26 2019

An antichain is a set of nonempty sets, none of which is a subset of any other.

The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

			Triangle begins:
    1
    2   0
    5   1   0
   19   5   2   0
  167  84  44  17   0

Except for the first column, same as the covering case A327350.
Column k = 0 is A014466 (antichains).
Column k = 1 is A048143 (clutters), if we assume A048143(0) = A048143(1) = 0.
Column k = 2 is A275307 (blobs), if we assume A275307(1) = A275307(2) = 0.
The unlabeled version is A327807.
The case for vertex connectivity exactly k is A327351.
Cf. A014466, A293606, A326704, A327057, A327062, A327125, A327358.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
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]]]];
vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],vertConnSys[Range[n],#]>=k&]],{n,0,4},{k,0,n}]

A336736 Number of factorizations of n whose distinct factors have disjoint prime signatures.

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, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 5, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Aug 06 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The a(n) factorizations for n = 36, 360, 720, 192, 288:
  (36)     (360)    (720)     (192)      (288)
  (6*6)    (5*72)   (8*90)    (3*64)     (8*36)
  (2*2*9)  (8*45)   (9*80)    (4*48)     (9*32)
  (3*3*4)  (9*40)   (10*72)   (6*32)     (16*18)
           (10*36)  (16*45)   (12*16)    (2*144)
           (5*8*9)  (5*144)   (3*8*8)    (6*6*8)
                    (5*9*16)  (4*6*8)    (2*2*72)
                    (8*9*10)  (3*4*16)   (2*9*16)
                              (3*4*4*4)  (3*3*32)
                                         (2*2*8*9)
                                         (3*3*4*8)
                                         (2*2*2*36)
                                         (2*2*2*2*2*9)

A001055 counts factorizations.
A118914 is sorted prime signature.
A124010 is prime signature.
A336737 counts factorizations with intersecting signatures.
Cf. A000372, A003182, A006126, A109298, A112798, A293606, A294068, A305844, A321469, A336424.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
Table[Length[Select[facs[n],stableQ[#,Intersection[prisig[#1],prisig[#2]]!={}&]&]],{n,100}]

A327424 Number of unlabeled, non-connected or empty antichains of nonempty subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 4, 10, 33, 234, 16579

Offset: 0

Gus Wiseman, Sep 26 2019

An antichain is a set of nonempty sets, none of which is a subset of any other. A singleton is considered to be connected.

			Non-isomorphic representatives of the a(0) = 1 through a(4) = 10 antichains:
  {}  {}  {}         {}             {}
          {{1},{2}}  {{1},{2}}      {{1},{2}}
                     {{1},{2,3}}    {{1},{2,3}}
                     {{1},{2},{3}}  {{1},{2},{3}}
                                    {{1},{2,3,4}}
                                    {{1,2},{3,4}}
                                    {{1},{2},{3,4}}
                                    {{1},{2},{3},{4}}
                                    {{1},{2,4},{3,4}}
                                    {{1},{2,3},{2,4},{3,4}}

Partial sums of the positive-index terms of A327426.
The covering case is A327426.
The labeled version is A327354.
The labeled covering case is A120338.
Unlabeled antichains that are either not connected or not covering are A327437.
The case without empty antichains is A327808.
Cf. A014466, A293606, A326704, A327062, A327355.

A327425 Number of unlabeled antichains of nonempty sets covering n vertices where every two vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 1, 1, 2, 6, 54

Offset: 0

Gus Wiseman, Sep 11 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 6 antichains:
    {1}  {12}  {123}         {1234}
               {12}{13}{23}  {12}{134}{234}
                             {124}{134}{234}
                             {12}{13}{14}{234}
                             {123}{124}{134}{234}
                             {12}{13}{14}{23}{24}{34}

The labeled version is A327020.
Unlabeled covering antichains are A261005.
The weighted version is A327060.
Cf. A006126, A014466, A055621, A293606, A293993, A305844, A307249, A319639, A326704, A327057, A327058, A327358, A327359.

cp's OEIS Frontend

A327358 Triangle read by rows where T(n,k) is the number of unlabeled antichains of nonempty sets covering n vertices with vertex-connectivity >= k.

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A318131 Number of non-isomorphic sets of finite (possibly empty) sets with union {1,2,...,n} and intersection {}.

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A318132 Number of non-isomorphic set-systems (finite sets of finite nonempty sets) with union {1,2,...,n} and intersection {}.

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A327357 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of sets covering n vertices with non-spanning edge-connectivity k.

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A327437 Number of unlabeled antichains of nonempty subsets of {1..n} that are either non-connected or non-covering (spanning edge-connectivity 0).

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A303364 Number of strict integer partitions of n with pairwise indivisible and squarefree parts.

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PARI

A327806 Triangle read by rows where T(n,k) is the number of antichains of sets with n vertices and vertex-connectivity >= k.

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A336736 Number of factorizations of n whose distinct factors have disjoint prime signatures.

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A327424 Number of unlabeled, non-connected or empty antichains of nonempty subsets of {1..n}.

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