A319781 -id:A319781

A317757

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

+10
38

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

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

EXAMPLE

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}.

MATHEMATICA

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}]

PROG

(PARI)

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

CROSSREFS

Cf. A007716, A035310, A255906, A281116, A317073, A317533.

Cf. A317748, A317751, A317752, A317755.

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 06 2018

EXTENSIONS

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

STATUS

approved

A319759

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

+10
16

1, 0, 0, 0, 0, 0, 1, 2, 13, 49, 199

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

A multiset partition is intersecting if no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

LINKS

Table of n, a(n) for n=0..10.

EXAMPLE

Non-isomorphic representatives of the a(6) = 1 through a(8) = 13 multiset partitions:

6: {{1,2},{1,3},{2,3}}

7: {{1,2},{1,3},{2,3,3}}

{{1,3},{1,4},{2,3,4}}

8: {{1,2},{1,3},{2,2,3,3}}

{{1,2},{1,3},{2,3,3,3}}

{{1,2},{1,3},{2,3,4,4}}

{{1,2},{1,3,3},{2,3,3}}

{{1,2},{1,3,4},{2,3,4}}

{{1,3},{1,4},{2,3,4,4}}

{{1,3},{1,1,2},{2,3,3}}

{{1,3},{1,2,2},{2,3,3}}

{{1,4},{1,5},{2,3,4,5}}

{{2,3},{1,2,4},{3,4,4}}

{{2,4},{1,2,3},{3,4,4}}

{{2,4},{1,2,5},{3,4,5}}

{{1,2},{1,3},{2,3},{2,3}}

CROSSREFS

Cf. A007716, A283877, A305854, A306006, A317757, A318715, A318717.

Cf. A319752, A319762, A319763, A319764, A319765, A319779, A319781.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Sep 27 2018

STATUS

approved

A319779

Number of intersecting multiset partitions of weight n whose dual is not an intersecting multiset partition.

+10
16

1, 0, 0, 0, 1, 4, 20, 66, 226, 696, 2156

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

The dual of a multiset partition has, for each vertex, one part 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 weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

LINKS

Table of n, a(n) for n=0..10.

EXAMPLE

Non-isomorphic representatives of the a(4) = 1 through a(6) = 20 multiset partitions:

4: {{1,3},{2,3}}

5: {{1,2},{2,3,3}}

{{1,3},{2,3,3}}

{{1,4},{2,3,4}}

{{3},{1,3},{2,3}}

6: {{1,2},{2,3,3,3}}

{{1,3},{2,2,3,3}}

{{1,3},{2,3,3,3}}

{{1,3},{2,3,4,4}}

{{1,4},{2,3,4,4}}

{{1,5},{2,3,4,5}}

{{1,1,2},{2,3,3}}

{{1,2,2},{2,3,3}}

{{1,2,3},{3,4,4}}

{{1,2,4},{3,4,4}}

{{1,2,5},{3,4,5}}

{{1,3,3},{2,3,3}}

{{1,3,4},{2,3,4}}

{{2},{1,2},{2,3,3}}

{{3},{1,3},{2,3,3}}

{{4},{1,4},{2,3,4}}

{{1,3},{2,3},{2,3}}

{{1,3},{2,3},{3,3}}

{{1,4},{2,4},{3,4}}

{{3},{3},{1,3},{2,3}}

CROSSREFS

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.

Cf. A319775, A319778, A319781, A319783.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Sep 27 2018

STATUS

approved

A319778

Number of non-isomorphic set systems of weight n with empty intersection whose dual is also a set system with empty intersection.

+10
12

1, 0, 1, 1, 2, 5, 13, 28, 72, 181, 483

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

The dual of a multiset partition has, for each vertex, one part 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 dual of a multiset partition has empty intersection iff no part contains all the vertices.

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

LINKS

Table of n, a(n) for n=0..10.

EXAMPLE

Non-isomorphic representatives of the a(2) = 1 through a(6) = 13 multiset partitions:

2: {{1},{2}}

3: {{1},{2},{3}}

4: {{1},{3},{2,3}}

{{1},{2},{3},{4}}

5: {{1},{2,4},{3,4}}

{{2},{1,3},{2,3}}

{{1},{2},{3},{2,3}}

{{1},{2},{4},{3,4}}

{{1},{2},{3},{4},{5}}

6: {{3},{1,4},{2,3,4}}

{{1,2},{1,3},{2,3}}

{{1,3},{2,4},{3,4}}

{{1},{2},{1,3},{2,3}}

{{1},{2},{3,5},{4,5}}

{{1},{3},{4},{2,3,4}}

{{1},{3},{2,4},{3,4}}

{{1},{4},{2,4},{3,4}}

{{2},{3},{1,3},{2,3}}

{{2},{4},{1,2},{3,4}}

{{1},{2},{3},{4},{3,4}}

{{1},{2},{3},{5},{4,5}}

{{1},{2},{3},{4},{5},{6}}

CROSSREFS

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

Cf. A319775, A319779, A319781, A319783.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Sep 27 2018

STATUS

approved

A319077

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

+10
10

1, 0, 1, 3, 12, 37, 130, 428, 1481, 5091, 17979, 64176, 234311, 869645, 3295100, 12720494, 50083996, 200964437, 821845766, 3423694821, 14524845181, 62725701708, 275629610199, 1231863834775, 5597240308384, 25844969339979, 121224757935416, 577359833539428, 2791096628891679

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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

LINKS

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

EXAMPLE

Non-isomorphic representatives of the a(2) = 1 through a(4) = 12 strict multiset partitions with empty intersection:

2: {{1},{2}}

3: {{1},{2,2}}

{{1},{2,3}}

{{1},{2},{3}}

4: {{1},{2,2,2}}

{{1},{2,3,3}}

{{1},{2,3,4}}

{{1,1},{2,2}}

{{1,2},{3,3}}

{{1,2},{3,4}}

{{1},{2},{1,2}}

{{1},{2},{2,2}}

{{1},{2},{3,3}}

{{1},{2},{3,4}}

{{1},{3},{2,3}}

{{1},{2},{3},{4}}

PROG

(PARI)

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, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}

R(q, n)={vector(n, t, subst(x*Ser(K(q, t, n\t)/t), x, x^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] - subst(x^(t*k)*u[t] + O(x*x^(n\2)), x, x^2), O(x*x^n) ))*if(k, 1+x^k, 1), n))) ); s/n!} \\ Andrew Howroyd, May 30 2023

CROSSREFS

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A318715.

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 27 2018

EXTENSIONS

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

STATUS

approved

A319748

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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

LINKS

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

EXAMPLE

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}}

PROG

(PARI)

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

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 27 2018

EXTENSIONS

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

STATUS

approved

A319790

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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}}

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 27 2018

EXTENSIONS

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

STATUS

approved

A319791

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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}}

CROSSREFS

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

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 27 2018

EXTENSIONS

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

STATUS

approved

A319762

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

+10
4

1, 0, 0, 0, 0, 0, 1, 1, 4, 9, 24

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,9

COMMENTS

A set multipartition is intersecting if no two parts are disjoint. The weight of a set multipartition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

LINKS

Table of n, a(n) for n=0..10.

EXAMPLE

Non-isomorphic representatives of the a(6) = 1 through a(9) = 9 set multipartitions:

6: {{1,2},{1,3},{2,3}}

7: {{1,3},{1,4},{2,3,4}}

8: {{1,2},{1,3,4},{2,3,4}}

{{1,4},{1,5},{2,3,4,5}}

{{2,4},{1,2,5},{3,4,5}}

{{1,2},{1,3},{2,3},{2,3}}

9: {{1,3},{1,4,5},{2,3,4,5}}

{{1,5},{1,6},{2,3,4,5,6}}

{{2,5},{1,2,6},{3,4,5,6}}

{{1,2,3},{2,4,5},{3,4,5}}

{{1,3,5},{2,3,6},{4,5,6}}

{{1,2},{1,3},{1,4},{2,3,4}}

{{1,2},{1,3},{2,3},{1,2,3}}

{{1,3},{1,4},{1,4},{2,3,4}}

{{1,3},{1,4},{3,4},{2,3,4}}

CROSSREFS

Cf. A007716, A049311, A281116, A283877, A305854, A306006, A316980, A317757, A318715, A318717.

Cf. A319752, A319759, A319763, A319764, A319765, A319779, A319781.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Sep 27 2018

STATUS

approved

A319763

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

+10
4

1, 0, 0, 0, 0, 0, 1, 2, 12, 46, 181

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

A multiset partition is intersecting if no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

LINKS

Table of n, a(n) for n=0..10.

EXAMPLE

Non-isomorphic representatives of the a(6) = 1 through a(8) = 12 multiset partitions:

6: {{1,2},{1,3},{2,3}}

7: {{1,2},{1,3},{2,3,3}}

{{1,3},{1,4},{2,3,4}}

8: {{1,2},{1,3},{2,2,3,3}}

{{1,2},{1,3},{2,3,3,3}}

{{1,2},{1,3},{2,3,4,4}}

{{1,2},{1,3,3},{2,3,3}}

{{1,2},{1,3,4},{2,3,4}}

{{1,3},{1,4},{2,3,4,4}}

{{1,3},{1,1,2},{2,3,3}}

{{1,3},{1,2,2},{2,3,3}}

{{1,4},{1,5},{2,3,4,5}}

{{2,3},{1,2,4},{3,4,4}}

{{2,4},{1,2,3},{3,4,4}}

{{2,4},{1,2,5},{3,4,5}}

CROSSREFS

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A317757, A318715, A318717.

Cf. A319752, A319759, A319762, A319764, A319765, A319779, A319781.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Sep 27 2018

STATUS

approved