A319719 -id:A319719 - cp's OEIS frontend

A319721 Number of non-isomorphic antichains of multisets of weight n.

1, 1, 4, 8, 24, 50, 148, 349, 1014, 2717, 8114

Offset: 0

Gus Wiseman, Sep 26 2018

In an antichain, no part is a proper submultiset of any other. The weight of an antichain 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(3) = 8 antichains:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
   {{1},{2}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{2,2}}
   {{1},{2,3}}
   {{1},{1},{1}}
   {{1},{2},{2}}
   {{1},{2},{3}}

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A001055, A001970, A007716, A096827, A253249, A285572, A285573, A293993, A318099, A319616-A319646, A319719.

A321679 Number of non-isomorphic weight-n antichains (not necessarily strict) of sets.

Original entry on oeis.org

1, 1, 3, 5, 12, 19, 45, 75, 170, 314, 713

Offset: 0

Gus Wiseman, Nov 16 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(1) = 1 through a(5) = 19 antichains:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}        {{1,2,3,4,5}}
         {{1},{1}}  {{1},{2,3}}    {{1,2},{1,2}}      {{1},{2,3,4,5}}
         {{1},{2}}  {{1},{1},{1}}  {{1},{2,3,4}}      {{1,2},{3,4,5}}
                    {{1},{2},{2}}  {{1,2},{3,4}}      {{1,4},{2,3,4}}
                    {{1},{2},{3}}  {{1,3},{2,3}}      {{1},{1},{2,3,4}}
                                   {{1},{1},{2,3}}    {{1},{2,3},{2,3}}
                                   {{1},{2},{3,4}}    {{1},{2},{3,4,5}}
                                   {{1},{1},{1},{1}}  {{1},{2,3},{4,5}}
                                   {{1},{1},{2},{2}}  {{1},{2,4},{3,4}}
                                   {{1},{2},{2},{2}}  {{1},{1},{1},{2,3}}
                                   {{1},{2},{3},{3}}  {{1},{2},{2},{3,4}}
                                   {{1},{2},{3},{4}}  {{1},{2},{3},{4,5}}
                                                      {{1},{1},{1},{1},{1}}
                                                      {{1},{1},{2},{2},{2}}
                                                      {{1},{2},{2},{2},{2}}
                                                      {{1},{2},{2},{3},{3}}
                                                      {{1},{2},{3},{3},{3}}
                                                      {{1},{2},{3},{4},{4}}
                                                      {{1},{2},{3},{4},{5}}

Cf. A006126, A049311, A096827, A285572, A293993, A293994, A318099, A319719, A319721, A320799, A321678.

A320449 Number of antichains of sets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 18, 24, 39, 58, 92, 131, 206

Offset: 0

Gus Wiseman, Oct 12 2018

			The a(1) = 1 through a(7) = 24 antichains:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1},{1}}  {{1,2}}        {{1,3}}            {{1,4}}
                    {{1},{2}}      {{1},{3}}          {{2,3}}
                    {{1},{1},{1}}  {{2},{2}}          {{1},{4}}
                                   {{1},{1},{2}}      {{2},{3}}
                                   {{1},{1},{1},{1}}  {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1},{1}}
.
  {{6}}                      {{7}}
  {{1,5}}                    {{1,6}}
  {{2,4}}                    {{2,5}}
  {{1,2,3}}                  {{3,4}}
  {{1},{5}}                  {{1,2,4}}
  {{2},{4}}                  {{1},{6}}
  {{3},{3}}                  {{2},{5}}
  {{1},{2,3}}                {{3},{4}}
  {{2},{1,3}}                {{1},{2,4}}
  {{3},{1,2}}                {{2},{1,4}}
  {{1},{1},{4}}              {{4},{1,2}}
  {{1,2},{1,2}}              {{1},{1},{5}}
  {{1},{2},{3}}              {{1,2},{1,3}}
  {{2},{2},{2}}              {{1},{2},{4}}
  {{1},{1},{1},{3}}          {{1},{3},{3}}
  {{1},{1},{2},{2}}          {{2},{2},{3}}
  {{1},{1},{1},{1},{2}}      {{1},{1},{2,3}}
  {{1},{1},{1},{1},{1},{1}}  {{1},{1},{1},{4}}
                             {{1},{1},{2},{3}}
                             {{1},{2},{2},{2}}
                             {{1},{1},{1},{1},{3}}
                             {{1},{1},{1},{2},{2}}
                             {{1},{1},{1},{1},{1},{2}}
                             {{1},{1},{1},{1},{1},{1},{1}}

Cf. A001970, A089259, A258466, A319719, A319721, A320328, 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]]]];
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@@@#,antiQ[#]]&]],{n,10}]

A371294 Numbers whose binary indices are connected and pairwise indivisible, where two numbers are connected iff they have a common factor. A hybrid ranking sequence for connected antichains of multisets.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 40, 64, 128, 160, 256, 288, 296, 416, 512, 520, 544, 552, 640, 672, 800, 808, 928, 1024, 2048, 2176, 2304, 2432, 2560, 2688, 2816, 2944, 4096, 8192, 8200, 8224, 8232, 8320, 8352, 8480, 8488, 8608, 8704, 8712, 8736, 8744, 8832, 8864, 8992

Offset: 1

Gus Wiseman, Mar 28 2024

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.

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.

			The terms together with their prime indices of binary indices begin:
    1: {{}}
    2: {{1}}
    4: {{2}}
    8: {{1,1}}
   16: {{3}}
   32: {{1,2}}
   40: {{1,1},{1,2}}
   64: {{4}}
  128: {{1,1,1}}
  160: {{1,2},{1,1,1}}
  256: {{2,2}}
  288: {{1,2},{2,2}}
  296: {{1,1},{1,2},{2,2}}
  416: {{1,2},{1,1,1},{2,2}}
  512: {{1,3}}
  520: {{1,1},{1,3}}
  544: {{1,2},{1,3}}
  552: {{1,1},{1,2},{1,3}}
  640: {{1,1,1},{1,3}}
  672: {{1,2},{1,1,1},{1,3}}
  800: {{1,2},{2,2},{1,3}}
  808: {{1,1},{1,2},{2,2},{1,3}}
  928: {{1,2},{1,1,1},{2,2},{1,3}}

Connected case of A087086, relatively prime A328671.
For binary indices of binary indices we have A326750, non-primitive A326749.
For prime indices of prime indices we have A329559, non-primitive A305078.
Primitive case of A371291 = positions of ones in A371452.
For binary indices of prime indices we have A371445, non-primitive A325118.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326964 counts connected set-systems, covering A323818.
Cf. A001222, A051026, A285572, A303362, A304713, A305079, A316476, A319496, A319719, A326704, A371446.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],stableQ[bpe[#],Divisible]&&connectedQ[prix/@bpe[#]]&]

Intersection of A087086 and A371291.

A320353 Number of antichains of multisets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 3, 5, 11, 17, 36, 56, 107, 175, 311, 505, 887

Offset: 0

Gus Wiseman, Oct 11 2018

			The a(1) = 1 through a(5) = 17 antichains:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{2}}      {{1,1,2}}          {{1,1,3}}
                    {{1},{1},{1}}  {{1},{3}}          {{1,2,2}}
                                   {{2},{2}}          {{1},{4}}
                                   {{1,1,1,1}}        {{2},{3}}
                                   {{2},{1,1}}        {{1,1,1,2}}
                                   {{1,1},{1,1}}      {{1},{2,2}}
                                   {{1},{1},{2}}      {{3},{1,1}}
                                   {{1},{1},{1},{1}}  {{1,1,1,1,1}}
                                                      {{1,1},{1,2}}
                                                      {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{2},{1,1,1}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1},{1}}

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A001970, A006126, A050342, A089259, A258466, A261005, A261049, A293994, A319719, A319721, A320328, 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]]]];
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],antiQ]],{n,8}]

A320355 Number of connected antichains of multisets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 3, 4, 8, 9, 19, 24, 45, 71, 118, 194, 335

Offset: 0

Gus Wiseman, Oct 11 2018

			The a(1) = 1 through a(5) = 9 clutters:
  {{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}}
                                   {{2},{2}}          {{1,2,2}}
                                   {{1,1,1,1}}        {{1,1,1,2}}
                                   {{1,1},{1,1}}      {{1,1,1,1,1}}
                                   {{1},{1},{1},{1}}  {{1,1},{1,2}}
                                                      {{1},{1},{1},{1},{1}}

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A001970, A007718, A048143, A258466, A261006, A293994, A318403, A319079, A319719, A319721, A320330, A320351, A320353, 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[Length[csm[#]]==1,antiQ[#]]&]],{n,8}]

A320356 Number of strict connected antichains of multisets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 22, 35, 62, 98, 171, 277

Offset: 0

Gus Wiseman, Oct 11 2018

			The a(1) = 1 through a(6) = 13 clutters:
  {{1}}  {{2}}    {{3}}      {{4}}        {{5}}          {{6}}
         {{1,1}}  {{1,2}}    {{1,3}}      {{1,4}}        {{1,5}}
                  {{1,1,1}}  {{2,2}}      {{2,3}}        {{2,4}}
                             {{1,1,2}}    {{1,1,3}}      {{3,3}}
                             {{1,1,1,1}}  {{1,2,2}}      {{1,1,4}}
                                          {{1,1,1,2}}    {{1,2,3}}
                                          {{1,1,1,1,1}}  {{2,2,2}}
                                          {{1,1},{1,2}}  {{1,1,1,3}}
                                                         {{1,1,2,2}}
                                                         {{1,1,1,1,2}}
                                                         {{1,1},{1,3}}
                                                         {{1,1,1,1,1,1}}
                                                         {{1,2},{1,1,1}}

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A001970, A007718, A048143, A056156, A258466, A261006, A293994, A318403, A319079, A319719, A319721, A320351, A320353, A320355.

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@@#,Length[csm[#]]==1,antiQ[#]]&]],{n,8}]

A320798 Number of non-isomorphic weight-n connected antichains of non-constant multisets with multiset density -1.

Original entry on oeis.org

0, 1, 2, 5, 9, 24, 51, 134, 328, 868

Offset: 1

Gus Wiseman, Nov 29 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(2) = 1 through a(6) = 24 multiset partitions:
  {{12}}  {{122}}  {{1122}}    {{11222}}    {{111222}}
          {{123}}  {{1222}}    {{12222}}    {{112222}}
                   {{1233}}    {{12233}}    {{112233}}
                   {{1234}}    {{12333}}    {{122222}}
                   {{13}{23}}  {{12344}}    {{122333}}
                               {{12345}}    {{123333}}
                               {{12}{233}}  {{123344}}
                               {{13}{233}}  {{123444}}
                               {{14}{234}}  {{123455}}
                                            {{123456}}
                                            {{112}{233}}
                                            {{122}{233}}
                                            {{12}{2333}}
                                            {{123}{344}}
                                            {{124}{344}}
                                            {{125}{345}}
                                            {{13}{2233}}
                                            {{13}{2333}}
                                            {{13}{2344}}
                                            {{133}{233}}
                                            {{14}{2344}}
                                            {{15}{2345}}
                                            {{13}{24}{34}}
                                            {{14}{24}{34}}

Cf. A007716, A007718, A286520, A304867, A319719, A319721, A321155, A321760, A322111, A322113.

A322117 Number of non-isomorphic blobs (2-connected weak antichains) of multisets of weight n.

Original entry on oeis.org

1, 1, 3, 4, 8, 8, 21, 27, 79, 185, 554

Offset: 0

Gus Wiseman, Nov 26 2018

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 21 blobs:
  (1)  (11)    (111)      (1111)        (11111)          (111111)
       (12)    (122)      (1122)        (11222)          (111222)
       (1)(1)  (123)      (1222)        (12222)          (112222)
               (1)(1)(1)  (1233)        (12233)          (112233)
                          (1234)        (12333)          (122222)
                          (11)(11)      (12344)          (122333)
                          (12)(12)      (12345)          (123333)
                          (1)(1)(1)(1)  (1)(1)(1)(1)(1)  (123344)
                                                         (123444)
                                                         (123455)
                                                         (123456)
                                                         (111)(111)
                                                         (112)(122)
                                                         (122)(122)
                                                         (123)(123)
                                                         (123)(233)
                                                         (134)(234)
                                                         (11)(11)(11)
                                                         (12)(12)(12)
                                                         (12)(13)(23)
                                                         (1)(1)(1)(1)(1)(1)

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Cf. A002218, A007718, A013922, A275307, A286520, A293994, A304118, A304887, A319719, A319721, A322110, A322118.

A322118 Number of non-isomorphic connected multiset partitions of weight n with no singletons that cannot be capped by a tree.

Original entry on oeis.org

1, 1, 2, 3, 7, 11, 29, 55, 155, 386, 1171

Offset: 0

Gus Wiseman, Nov 26 2018

The density of a multiset partition is defined to be the sum of numbers of distinct elements in each part, minus the number of parts, minus the total number of distinct elements in the whole partition. A multiset partition is a tree if it has more than one part, is connected, and has density -1. A cap is a certain kind of non-transitive coarsening of a multiset partition. For example, the four caps of {{1,1},{1,2},{2,2}} are {{1,1},{1,2},{2,2}}, {{1,1},{1,2,2}}, {{1,1,2},{2,2}}, {{1,1,2,2}}. - Gus Wiseman, Feb 05 2021

			The multiset partition C = {{1,1},{1,2,3},{2,3,3}} is not a tree but has the cap {{1,1},{1,2,3,3}} which is a tree, so C is not counted under a(8).
Non-isomorphic representatives of the a(2) = 2 through a(6) = 29 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{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}}    {{1,1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}    {{1,1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}    {{1,1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}    {{1,2,2,2,2,2}}
                      {{1,1},{1,1}}  {{1,2,3,4,4}}    {{1,2,2,3,3,3}}
                      {{1,2},{1,2}}  {{1,2,3,4,5}}    {{1,2,3,3,3,3}}
                                     {{1,1},{1,1,1}}  {{1,2,3,3,4,4}}
                                     {{1,2},{1,2,2}}  {{1,2,3,4,4,4}}
                                     {{2,2},{1,2,2}}  {{1,2,3,4,5,5}}
                                     {{2,3},{1,2,3}}  {{1,2,3,4,5,6}}
                                                      {{1,1},{1,1,1,1}}
                                                      {{1,1,1},{1,1,1}}
                                                      {{1,1,2},{1,2,2}}
                                                      {{1,2},{1,1,2,2}}
                                                      {{1,2},{1,2,2,2}}
                                                      {{1,2},{1,2,3,3}}
                                                      {{1,2,2},{1,2,2}}
                                                      {{1,2,3},{1,2,3}}
                                                      {{1,2,3},{2,3,3}}
                                                      {{1,3,4},{2,3,4}}
                                                      {{2,2},{1,1,2,2}}
                                                      {{2,2},{1,2,2,2}}
                                                      {{2,3},{1,2,3,3}}
                                                      {{3,3},{1,2,3,3}}
                                                      {{3,4},{1,2,3,4}}
                                                      {{1,1},{1,1},{1,1}}
                                                      {{1,2},{1,2},{1,2}}
                                                      {{1,2},{1,3},{2,3}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Non-isomorphic tree multiset partitions are counted by A321229, or A321231 without singletons.
The version with singletons is A322110.
The weak-antichain case is counted by A322138, or A322117 with singletons.
Cf. A002218, A007718, A013922, A030019, A275307, A293994, A304118, A304382, A304887, A305079, A319719, A319721, A321194.

Definition corrected by Gus Wiseman, Feb 05 2021

cp's OEIS Frontend

A319721 Number of non-isomorphic antichains of multisets of weight n.

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A321679 Number of non-isomorphic weight-n antichains (not necessarily strict) of sets.

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A320449 Number of antichains of sets whose multiset union is an integer partition of n.

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A371294 Numbers whose binary indices are connected and pairwise indivisible, where two numbers are connected iff they have a common factor. A hybrid ranking sequence for connected antichains of multisets.

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A320353 Number of antichains of multisets whose multiset union is an integer partition of n.

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A320355 Number of connected antichains of multisets whose multiset union is an integer partition of n.

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A320356 Number of strict connected antichains of multisets whose multiset union is an integer partition of n.

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A320798 Number of non-isomorphic weight-n connected antichains of non-constant multisets with multiset density -1.

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A322117 Number of non-isomorphic blobs (2-connected weak antichains) of multisets of weight n.

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A322118 Number of non-isomorphic connected multiset partitions of weight n with no singletons that cannot be capped by a tree.

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