A319077 -id:A319077 - cp's OEIS frontend

A317757 Number of non-isomorphic multiset partitions of size n such that the blocks have empty intersection.

1, 0, 1, 4, 17, 56, 205, 690, 2446, 8506, 30429, 109449, 402486, 1501424, 5714194, 22132604, 87383864, 351373406, 1439320606, 6003166059, 25488902820, 110125079184, 483987225922, 2162799298162, 9823464989574, 45332196378784, 212459227340403, 1010898241558627, 4881398739414159

Offset: 0

Gus Wiseman, Aug 06 2018

nonn

			Non-isomorphic representatives of the a(4) = 17 multiset partitions:
  {1}{234},{2}{111},{2}{113},{11}{22},{11}{23},{12}{34},
  {1}{1}{22},{1}{1}{23},{1}{2}{11},{1}{2}{12},{1}{2}{13},{1}{2}{34},{2}{3}{11},
  {1}{1}{1}{2},{1}{1}{2}{2},{1}{1}{2}{3},{1}{2}{3}{4}.

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A035310, A255906, A281116, A317073, A317533.
Cf. A317748, A317751, A317752, A317755.
Cf. A319077, A319748, A319755, A319778, A319781, A319790.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
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/@Join@@Table[Select[mps[m],Intersection@@#=={}&],{m,strnorm[n]}]]],{n,6}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
R(q, n)={vector(n, t, x*Ser(K(q, t, n)/t))}
a(n)={my(s=0); forpart(q=n, my(f=prod(i=1, #q, 1 - x^q[i]), u=R(q,n)); s+=permcount(q)*sum(k=0, n, my(c=polcoef(f,k)); if(c, c*polcoef(exp(sum(t=1, n\(k+1), x^(t*k)*u[t], O(x*x^n) ))/if(k,1-x^k,1), n))) ); s/n!} \\ Andrew Howroyd, May 30 2023

a(8)-a(10) from Gus Wiseman, Sep 27 2018

a(0)=1 prepended and terms a(11) and beyond from Andrew Howroyd, May 30 2023

A326912 BII-numbers of pairwise intersecting set-systems with empty intersection.

Original entry on oeis.org

0, 52, 116, 772, 832, 836, 1072, 1076, 1136, 1140, 1796, 1856, 1860, 2320, 2368, 2384, 2592, 2624, 2656, 2880, 3088, 3104, 3120, 3136, 3152, 3168, 3184, 3344, 3392, 3408, 3616, 3648, 3680, 3904, 4132, 4148, 4196, 4212, 4612, 4640, 4644, 4672, 4676, 4704, 4708

Offset: 1

Gus Wiseman, Aug 04 2019

nonn

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 pairwise intersecting set-systems with empty intersection, together with their BII-numbers, begins:
     0: {}
    52: {{1,2},{1,3},{2,3}}
   116: {{1,2},{1,3},{2,3},{1,2,3}}
   772: {{1,2},{1,4},{2,4}}
   832: {{1,2,3},{1,4},{2,4}}
   836: {{1,2},{1,2,3},{1,4},{2,4}}
  1072: {{1,3},{2,3},{1,2,4}}
  1076: {{1,2},{1,3},{2,3},{1,2,4}}
  1136: {{1,3},{2,3},{1,2,3},{1,2,4}}
  1140: {{1,2},{1,3},{2,3},{1,2,3},{1,2,4}}
  1796: {{1,2},{1,4},{2,4},{1,2,4}}
  1856: {{1,2,3},{1,4},{2,4},{1,2,4}}
  1860: {{1,2},{1,2,3},{1,4},{2,4},{1,2,4}}
  2320: {{1,3},{1,4},{3,4}}
  2368: {{1,2,3},{1,4},{3,4}}
  2384: {{1,3},{1,2,3},{1,4},{3,4}}
  2592: {{2,3},{2,4},{3,4}}
  2624: {{1,2,3},{2,4},{3,4}}
  2656: {{2,3},{1,2,3},{2,4},{3,4}}
  2880: {{1,2,3},{1,4},{2,4},{3,4}}

Cf. A048793, A051185, A305843, A317752, A317755, A317757, A319077, A326031, A326910, A326911.

Cf. A318715, A319759, A319762, A319763, A319764.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],(#==0||Intersection@@bpe/@bpe[#]=={})&&stableQ[bpe/@bpe[#],Intersection[#1,#2]=={}&]&]

A319748 Number of non-isomorphic set multipartitions (multisets of sets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 1, 3, 10, 25, 72, 182, 502, 1332, 3720, 10380, 30142, 88842, 270569, 842957, 2703060, 8885029, 29990388, 103743388, 367811233, 1334925589, 4957151327, 18817501736, 72972267232, 288863499000, 1166486601571, 4802115258807, 20141268290050, 86017885573548, 373852868791639

Offset: 0

Gus Wiseman, Sep 27 2018

nonn

The weight of a set multipartition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

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

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A319616.

Cf. A319077, A319751, A319755, A319778, A319781, A319791.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
R(q, n)={vector(n, t, x*Ser(K(q, t, n)/t))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(u=R(q,n)); s+=permcount(q)*polcoef(exp(sum(t=1, n, u[t], O(x*x^n))) - exp(sum(t=1, n\2, x^t*u[t], O(x*x^n)))/(1-x), n)); s/n!)} \\ Andrew Howroyd, May 30 2023

Terms a(11) and beyond from Andrew Howroyd, May 30 2023

A319790 Number of non-isomorphic connected multiset partitions of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 32, 134, 588, 2335, 9335, 36506, 144263, 571238, 2291894, 9300462, 38303796, 160062325, 679333926, 2927951665, 12817221628, 56974693933, 257132512297, 1177882648846, 5475237760563, 25818721638720, 123473772356785, 598687942799298, 2942344764127039

Offset: 0

Gus Wiseman, Sep 27 2018

nonn

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(4) = 1 through a(5) = 5 connected multiset partitions:
4:  {{1},{2},{1,2}}
5: {{1},{2},{1,2,2}}
   {{1},{1,2},{2,2}}
   {{2},{3},{1,2,3}}
   {{2},{1,3},{2,3}}
  {{1},{2},{2},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A007718, A049311, A056156, A283877, A317752, A317755, A317757.

Cf. A319077, A319748, A319755, A319778, A319781, A319791.

a(n) = A007718(n) - A007716(n) + A317757(n). - Andrew Howroyd, May 31 2023

Terms a(11) and beyond from Andrew Howroyd, May 31 2023

A319791 Number of non-isomorphic connected set multipartitions (multisets of sets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 1, 3, 14, 38, 125, 360, 1107, 3297, 10292, 32134, 103759, 340566, 1148150, 3951339, 13925330, 50122316, 184365292, 692145409, 2651444318, 10356184440, 41224744182, 167150406897, 689998967755, 2898493498253, 12384852601731, 53804601888559, 237566072006014

Offset: 0

Gus Wiseman, Sep 27 2018

nonn

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(4) = 1 through a(6) = 14 set multipartitions:
4:    {{1},{2},{1,2}}
5:   {{2},{3},{1,2,3}}
     {{2},{1,3},{2,3}}
    {{1},{2},{2},{1,2}}
6:  {{1},{1,4},{2,3,4}}
    {{1},{2,3},{1,2,3}}
    {{3},{4},{1,2,3,4}}
    {{3},{1,4},{2,3,4}}
    {{1,2},{1,3},{2,3}}
    {{1,3},{2,4},{3,4}}
   {{1},{2},{3},{1,2,3}}
   {{1},{2},{1,2},{1,2}}
   {{1},{2},{1,3},{2,3}}
   {{2},{2},{1,3},{2,3}}
   {{2},{3},{3},{1,2,3}}
   {{2},{3},{1,3},{2,3}}
  {{1},{1},{2},{2},{1,2}}
  {{1},{2},{2},{2},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A007718, A049311, A056156, A281116, A283877, A317752, A317755, A317757.

Cf. A319077, A319748, A319755, A319778, A319781, A319790.

a(n) = A056156(n) - A049311(n) + A319748(n). - Andrew Howroyd, May 31 2023

Terms a(11) and beyond from Andrew Howroyd, May 31 2023

A319751 Number of non-isomorphic set systems of weight n with empty intersection.

Original entry on oeis.org

1, 0, 1, 2, 6, 13, 35, 83, 217, 556, 1504, 4103, 11715, 34137, 103155, 320217, 1025757, 3376889, 11436712, 39758152, 141817521, 518322115, 1939518461, 7422543892, 29028055198, 115908161428, 472185530376, 1961087909565, 8298093611774, 35750704171225, 156734314212418

Offset: 0

Gus Wiseman, Sep 27 2018

nonn

A set system is a finite set of finite nonempty sets. Its weight is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 set systems:
2: {{1},{2}}
3: {{1},{2,3}}
   {{1},{2},{3}}
4: {{1},{2,3,4}}
   {{1,2},{3,4}}
   {{1},{2},{1,2}}
   {{1},{2},{3,4}}
   {{1},{3},{2,3}}
   {{1},{2},{3},{4}}
5: {{1},{2,3,4,5}}
   {{1,2},{3,4,5}}
   {{1},{2},{3,4,5}}
   {{1},{4},{2,3,4}}
   {{1},{2,3},{4,5}}
   {{1},{2,4},{3,4}}
   {{2},{3},{1,2,3}}
   {{2},{1,3},{2,3}}
   {{4},{1,2},{3,4}}
   {{1},{2},{3},{2,3}}
   {{1},{2},{3},{4,5}}
   {{1},{2},{4},{3,4}}
   {{1},{2},{3},{4},{5}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A319616.

Cf. A319077, A319748, A319755, A319778, A319781, A319791.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
R(q, n)={vector(n, t, x*Ser(K(q, t, n)/t))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(u=R(q,n)); s+=permcount(q)*polcoef(exp(sum(t=1, n, u[t]-subst(u[t],x,x^2), O(x*x^n))) - exp(sum(t=1, n\2, x^t*u[t] - subst(x^t*u[t],x,x^2), O(x*x^n)))*(1+x), n)); s/n!)} \\ Andrew Howroyd, May 30 2023

Terms a(11) and beyond from Andrew Howroyd, May 30 2023

A326911 BII-numbers of set-systems with empty intersection.

Original entry on oeis.org

0, 3, 7, 9, 10, 11, 12, 13, 14, 15, 18, 19, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 35, 37, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 86, 87, 89, 90, 91, 92, 93, 94, 95

Offset: 1

Gus Wiseman, Aug 04 2019

nonn

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 set-systems with empty intersection together with their BII-numbers begins:
   0: {}
   3: {{1},{2}}
   7: {{1},{2},{1,2}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  18: {{2},{1,3}}
  19: {{1},{2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  25: {{1},{3},{1,3}}
  26: {{2},{3},{1,3}}
  27: {{1},{2},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  29: {{1},{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}

Cf. A048793, A051185, A305843, A317752, A317755, A317757, A319077, A326031, A326910, A326912.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],#==0||Intersection@@bpe/@bpe[#]=={}&]

A319792 Number of non-isomorphic connected set systems of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 9, 22, 69, 190, 567, 1640, 5025, 15404, 49048, 159074, 531165, 1813627, 6352739, 22759620, 83443086, 312612543, 1196356133, 4672620842, 18615188819, 75593464871, 312729620542, 1317267618429, 5646454341658, 24618309943464, 109123789229297

Offset: 0

Gus Wiseman, Sep 27 2018

nonn

The weight of a set system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(4) = 1 through a(6) = 9 connected set systems:
4:   {{1},{2},{1,2}}
5:  {{2},{3},{1,2,3}}
    {{2},{1,3},{2,3}}
6: {{1},{1,4},{2,3,4}}
   {{1},{2,3},{1,2,3}}
   {{3},{4},{1,2,3,4}}
   {{3},{1,4},{2,3,4}}
   {{1,2},{1,3},{2,3}}
   {{1,3},{2,4},{3,4}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A007718, A049311, A056156, A281116, A283877, A300913, A305854, A316980, A317752, A317755, A317757.

Cf. A319077, A319748, A319751, A319755, A319778, A319781, A319790.

a(n) = A300913(n) - A283877(n) + A319751(n). - Andrew Howroyd, May 31 2023

Terms a(11) and beyond from Andrew Howroyd, May 31 2023

A319793 Number of non-isomorphic connected strict multiset partitions (sets of multisets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 1, 4, 24, 96, 412, 1607, 6348, 24580, 96334, 378569, 1508220, 6079720, 24879878, 103335386, 436032901, 1869019800, 8139613977, 36008825317, 161794412893, 738167013847, 3418757243139, 16068569129711, 76622168743677, 370571105669576, 1817199912384794

Offset: 0

Gus Wiseman, Sep 27 2018

nonn

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(4) = 1 through a(5) = 4 multiset partitions:
4:  {{1},{2},{1,2}}
5: {{1},{2},{1,2,2}}
   {{1},{1,2},{2,2}}
   {{2},{3},{1,2,3}}
   {{2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A007718, A056156, A281116, A283877, A316980, A317752, A317755, A317757, A319616.

Cf. A319077, A319748, A319755, A319778, A319781, A319791.

a(n) = A319557(n) - A316980(n) + A319077(n). - Andrew Howroyd, May 31 2023

Terms a(11) and beyond from Andrew Howroyd, May 31 2023

cp's OEIS Frontend

A317757 Number of non-isomorphic multiset partitions of size n such that the blocks have empty intersection.

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A326912 BII-numbers of pairwise intersecting set-systems with empty intersection.

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A319748 Number of non-isomorphic set multipartitions (multisets of sets) of weight n with empty intersection.

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A319790 Number of non-isomorphic connected multiset partitions of weight n with empty intersection.

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A319791 Number of non-isomorphic connected set multipartitions (multisets of sets) of weight n with empty intersection.

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Formula

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A319751 Number of non-isomorphic set systems of weight n with empty intersection.

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PARI

Extensions

A326911 BII-numbers of set-systems with empty intersection.

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Keywords

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Programs

Mathematica

A319792 Number of non-isomorphic connected set systems of weight n with empty intersection.

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