A319719 -id:A319719 - cp's OEIS frontend

A320351 Number of connected multiset partitions of integer partitions of n.

1, 1, 3, 5, 11, 18, 38, 66, 130, 237, 449, 823, 1538

Offset: 0

Gus Wiseman, Oct 11 2018

			The a(1) = 1 through a(5) = 18 multiset partitions:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{1,1}}    {{1,1,2}}          {{1,1,3}}
                    {{1},{1},{1}}  {{2},{2}}          {{1,2,2}}
                                   {{1,1,1,1}}        {{1,1,1,2}}
                                   {{1},{1,2}}        {{1},{1,3}}
                                   {{1},{1,1,1}}      {{2},{1,2}}
                                   {{1,1},{1,1}}      {{1,1,1,1,1}}
                                   {{1},{1},{1,1}}    {{1},{1,1,2}}
                                   {{1},{1},{1},{1}}  {{1,1},{1,2}}
                                                      {{1},{1,1,1,1}}
                                                      {{1,1},{1,1,1}}
                                                      {{1},{1},{1,2}}
                                                      {{1},{1},{1,1,1}}
                                                      {{1},{1,1},{1,1}}
                                                      {{1},{1},{1},{1,1}}
                                                      {{1},{1},{1},{1},{1}}

Cf. A001970, A007718, A048143, A056156, A258466, A261006, A293994, A319719, A320328, A320330, A320331, A320355, A320356.

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]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],Length[csm[#]]==1&]],{n,8}]

A320799 Number of non-isomorphic (not necessarily strict) antichains of multisets of weight n with no singletons or leaves (vertices that appear only once).

Original entry on oeis.org

1, 0, 1, 1, 5, 4, 22, 27, 107, 212, 689

Offset: 0

Gus Wiseman, Nov 02 2018

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(2) = 1 through a(7) = 27 multiset partitions:
  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}      {{1111111}}
                   {{1122}}    {{11222}}    {{111222}}      {{1112222}}
                   {{11}{11}}  {{11}{122}}  {{112222}}      {{1122222}}
                   {{11}{22}}  {{11}{222}}  {{112233}}      {{1122333}}
                   {{12}{12}}               {{111}{111}}    {{111}{1222}}
                                            {{11}{1222}}    {{11}{12222}}
                                            {{111}{222}}    {{111}{2222}}
                                            {{112}{122}}    {{11}{12233}}
                                            {{11}{2222}}    {{111}{2233}}
                                            {{112}{222}}    {{112}{1222}}
                                            {{11}{2233}}    {{11}{22222}}
                                            {{112}{233}}    {{112}{2222}}
                                            {{122}{122}}    {{11}{22333}}
                                            {{123}{123}}    {{112}{2333}}
                                            {{11}{11}{11}}  {{113}{2233}}
                                            {{11}{12}{22}}  {{122}{1233}}
                                            {{11}{22}{22}}  {{222}{1122}}
                                            {{11}{22}{33}}  {{11}{11}{122}}
                                            {{11}{23}{23}}  {{11}{11}{222}}
                                            {{12}{12}{12}}  {{11}{12}{222}}
                                            {{12}{12}{22}}  {{11}{12}{233}}
                                            {{12}{13}{23}}  {{11}{22}{233}}
                                                            {{11}{22}{333}}
                                                            {{12}{12}{222}}
                                                            {{12}{12}{233}}
                                                            {{12}{12}{333}}
                                                            {{12}{13}{233}}

Cf. A006126, A096827, A285572, A293993, A293994, A302545, A318099, A319560, A319719, A319721.

Cf. A320797, A320798, A320804, A320811, A320812, A320813.

A322133 Regular triangle read by rows where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with k vertices.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 5, 8, 3, 1, 0, 7, 17, 12, 3, 1, 0, 11, 46, 45, 18, 4, 1, 0, 15, 94, 141, 76, 23, 4, 1, 0, 22, 212, 432, 333, 124, 30, 5, 1, 0, 30, 416, 1231, 1254, 622, 178, 37, 5, 1, 0, 42, 848, 3346, 4601, 2914, 1058, 252, 45, 6, 1

Offset: 0

Gus Wiseman, Nov 27 2018

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

			Triangle begins:
    1
    0    1
    0    2    1
    0    3    2    1
    0    5    8    3    1
    0    7   17   12    3    1
    0   11   46   45   18    4    1
    0   15   94  141   76   23    4    1
    0   22  212  432  333  124   30    5    1
    0   30  416 1231 1254  622  178   37    5    1
    0   42  848 3346 4601 2914 1058  252   45    6    1
Non-isomorphic representatives of the multiset partitions counted in row 4:
  {{1,1,1,1}}        {{1,1,2,2}}      {{1,2,3,3}}    {{1,2,3,4}}
  {{1},{1,1,1}}      {{1,2,2,2}}      {{1,3},{2,3}}
  {{1,1},{1,1}}      {{1},{1,2,2}}    {{3},{1,2,3}}
  {{1},{1},{1,1}}    {{1,2},{1,2}}
  {{1},{1},{1},{1}}  {{1,2},{2,2}}
                     {{2},{1,2,2}}
                     {{1},{2},{1,2}}
                     {{2},{2},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A007718.

Cf. A000664, A007716, A054923, A191646, A191970, A275421, A286520, A317672, A319719, A321155, A321254, A322114.

\\ Needs G(m,n) defined in A317533 (faster PARI).
InvEulerMTS(p)={my(n=serprec(p, x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i)}
T(n)={[Vecrev(p) | p <- Vec(1 + InvEulerMTS(y^n*G(n,n) + sum(k=0, n-1, y^k*(1 - y)*G(k,n))))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 15 2024

A319496 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.

Original entry on oeis.org

2, 3, 7, 13, 19, 37, 53, 61, 89, 91, 113, 131, 151, 223, 247, 251, 281, 299, 311, 359, 377, 427, 463, 503, 593, 611, 659, 689, 703, 719, 791, 827, 851, 863, 923, 953, 1069, 1073, 1159, 1163, 1291, 1321, 1339, 1363, 1511, 1619, 1703, 1733, 1739, 1757, 1769

Offset: 1

Gus Wiseman, Dec 16 2018

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of connected strict antichains of multisets spanning an initial interval of positive integers.

			The sequence of multisystems whose MM-numbers belong to the sequence begins:
    2: {{}}
    3: {{1}}
    7: {{1,1}}
   13: {{1,2}}
   19: {{1,1,1}}
   37: {{1,1,2}}
   53: {{1,1,1,1}}
   61: {{1,2,2}}
   89: {{1,1,1,2}}
   91: {{1,1},{1,2}}
  113: {{1,2,3}}
  131: {{1,1,1,1,1}}
  151: {{1,1,2,2}}
  223: {{1,1,1,1,2}}
  247: {{1,2},{1,1,1}}
  251: {{1,2,2,2}}
  281: {{1,1,2,3}}
  299: {{1,2},{2,2}}

Cf. A003963, A006126, A055932, A056239, A112798, A285573, A286520, A293994, A302242, A318401, A319719, A319837, A320275, A320456, A320532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[200],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],stableQ[primeMS[#],Divisible],Length[zsm[primeMS[#]]]==1]&]

A322113 Number of non-isomorphic self-dual connected antichains of multisets of weight n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 5, 10, 18, 30

Offset: 0

Gus Wiseman, Nov 26 2018

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}}. A multiset partition is self-dual if it is isomorphic to its dual. For example, {{1,1},{1,2,2},{2,3,3}} is self-dual, as it is isomorphic to its dual {{1,1,2},{2,2,3},{3,3}}.

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(1) = 1 through a(9) = 18 antichains:
  {{1}}  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}
                          {{12}{12}}  {{11}{122}}  {{112}{122}}
                                                   {{12}{13}{23}}
.
  {{1111111}}      {{11111111}}        {{111111111}}
  {{111}{1222}}    {{111}{11222}}      {{1111}{12222}}
  {{112}{1222}}    {{1112}{1222}}      {{1112}{11222}}
  {{11}{12}{233}}  {{112}{12222}}      {{1112}{12222}}
  {{12}{13}{233}}  {{1122}{1122}}      {{112}{122222}}
                   {{11}{122}{233}}    {{11}{11}{12233}}
                   {{12}{13}{2333}}    {{11}{122}{1233}}
                   {{13}{112}{233}}    {{112}{123}{233}}
                   {{13}{122}{233}}    {{113}{122}{233}}
                   {{12}{13}{24}{34}}  {{12}{111}{2333}}
                                       {{12}{13}{23333}}
                                       {{12}{133}{2233}}
                                       {{123}{123}{123}}
                                       {{13}{112}{2333}}
                                       {{22}{113}{2333}}
                                       {{12}{13}{14}{234}}
                                       {{12}{13}{22}{344}}
                                       {{12}{13}{24}{344}}

Cf. A006126, A007716, A007718, A286520, A293993, A293994, A304867, A316983, A318099, A319719, A319721, A322111, A322112.

A322138 Number of non-isomorphic weight-n blobs (2-connected weak antichains) of multisets with no singletons.

Original entry on oeis.org

1, 0, 2, 3, 7, 7, 20, 26, 78, 184, 553

Offset: 0

Gus Wiseman, Nov 27 2018

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(2) = 2 through a(7) = 26 blobs:
  {{11}}  {{111}}  {{1111}}    {{11111}}  {{111111}}      {{1111111}}
  {{12}}  {{122}}  {{1122}}    {{11222}}  {{111222}}      {{1112222}}
          {{123}}  {{1222}}    {{12222}}  {{112222}}      {{1122222}}
                   {{1233}}    {{12233}}  {{112233}}      {{1122333}}
                   {{1234}}    {{12333}}  {{122222}}      {{1222222}}
                   {{11}{11}}  {{12344}}  {{122333}}      {{1222333}}
                   {{12}{12}}  {{12345}}  {{123333}}      {{1223333}}
                                          {{123344}}      {{1223344}}
                                          {{123444}}      {{1233333}}
                                          {{123455}}      {{1233444}}
                                          {{123456}}      {{1234444}}
                                          {{111}{111}}    {{1234455}}
                                          {{112}{122}}    {{1234555}}
                                          {{122}{122}}    {{1234566}}
                                          {{123}{123}}    {{1234567}}
                                          {{123}{233}}    {{112}{1222}}
                                          {{134}{234}}    {{122}{1233}}
                                          {{11}{11}{11}}  {{123}{2233}}
                                          {{12}{12}{12}}  {{123}{2333}}
                                          {{12}{13}{23}}  {{123}{2344}}
                                                          {{134}{2344}}
                                                          {{145}{2345}}
                                                          {{223}{1233}}
                                                          {{344}{1234}}
                                                          {{12}{13}{233}}
                                                          {{13}{14}{234}}

Cf. A002218, A007718, A013922, A275307, A286520, A293994, A304118, A304887, A319719, A322110, A322117, A322118.

A318403 Number of strict connected antichains of sets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 13, 22, 31

Offset: 0

Gus Wiseman, Oct 12 2018

			The a(1) = 1 through a(10) = 13 clutters:
  {{1}}  {{2}}  {{3}}    {{4}}    {{5}}    {{6}}      {{7}}
                {{1,2}}  {{1,3}}  {{1,4}}  {{1,5}}    {{1,6}}
                                  {{2,3}}  {{2,4}}    {{2,5}}
                                           {{1,2,3}}  {{3,4}}
                                                      {{1,2,4}}
                                                      {{1,2},{1,3}}
.
  {{8}}          {{9}}          {{10}}
  {{1,7}}        {{1,8}}        {{1,9}}
  {{2,6}}        {{2,7}}        {{2,8}}
  {{3,5}}        {{3,6}}        {{3,7}}
  {{1,2,5}}      {{4,5}}        {{4,6}}
  {{1,3,4}}      {{1,2,6}}      {{1,2,7}}
  {{1,2},{1,4}}  {{1,3,5}}      {{1,3,6}}
  {{1,2},{2,3}}  {{2,3,4}}      {{1,4,5}}
                 {{1,2},{1,5}}  {{2,3,5}}
                 {{1,2},{2,4}}  {{1,2,3,4}}
                 {{1,3},{1,4}}  {{1,2},{1,6}}
                 {{1,3},{2,3}}  {{1,2},{2,5}}
                                {{1,3},{1,5}}

Cf. A001970, A007718, A048143, A050342, A056156, A089259, A261049, A293994, A319719, A320351, A320353, A320355, A320356.

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]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,And@@UnsameQ@@@#,Length[csm[#]]==1,antiQ[#]]&]],{n,8}]

A319079 Number of connected antichains of sets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 7, 12, 15, 19, 26, 43

Offset: 0

Gus Wiseman, Oct 12 2018

			The a(10) = 19 clutters:
  {{10}}
  {{1,9}}
  {{2,8}}
  {{3,7}}
  {{4,6}}
  {{1,2,7}}
  {{1,3,6}}
  {{1,4,5}}
  {{2,3,5}}
  {{1,2,3,4}}
  {{5},{5}}
  {{1,2},{1,6}}
  {{1,2},{2,5}}
  {{1,3},{1,5}}
  {{1,4},{1,4}}
  {{2,3},{2,3}}
  {{1,2},{1,2},{1,3}}
  {{2},{2},{2},{2},{2}}
  {{1},{1},{1},{1},{1},{1},{1},{1},{1},{1}}

Cf. A001970, A007718, A048143, A056156, A089259, A319719, A320351, A320353, A320355, A320356.

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]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[And@@UnsameQ@@@#,Length[csm[#]]==1,antiQ[#]]&]],{n,10}]

A321680 Number of non-isomorphic weight-n connected antichains (not necessarily strict) of multisets with multiset density -1.

Original entry on oeis.org

1, 1, 3, 4, 9, 14, 39, 80, 216, 538, 1460

Offset: 0

Gus Wiseman, Nov 16 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of 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.

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

Cf. A006126, A007718, A056156, A096827, A285572, A293993, A293994, A305052, A319719, A319721, A321194, A321585, A321681.

A321681 Number of non-isomorphic weight-n connected strict antichains of multisets with multiset density -1.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 35, 77, 205, 517, 1399

Offset: 0

Gus Wiseman, Nov 16 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of 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.

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

Cf. A006126, A007718, A056156, A096827, A285572, A293993, A293994, A305052, A319557, A319565, A319719, A319721, A321585, A321680.

cp's OEIS Frontend

A320351 Number of connected multiset partitions of integer partitions of n.

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Mathematica

A320799 Number of non-isomorphic (not necessarily strict) antichains of multisets of weight n with no singletons or leaves (vertices that appear only once).

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A322133 Regular triangle read by rows where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with k vertices.

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PARI

A319496 Numbers whose prime indices are distinct and pairwise indivisible and whose own prime indices are connected and span an initial interval of positive integers.

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Mathematica

A322113 Number of non-isomorphic self-dual connected antichains of multisets of weight n.

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A322138 Number of non-isomorphic weight-n blobs (2-connected weak antichains) of multisets with no singletons.

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A318403 Number of strict connected antichains of sets whose multiset union is an integer partition of n.

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Mathematica

A319079 Number of connected antichains of sets whose multiset union is an integer partition of n.

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Mathematica

A321680 Number of non-isomorphic weight-n connected antichains (not necessarily strict) of multisets with multiset density -1.

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A321681 Number of non-isomorphic weight-n connected strict antichains of multisets with multiset density -1.

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