A330223 -id:A330223 - cp's OEIS frontend

A330233 Least MM-numbers of multisets of multisets with a given number of distinct representatives (obtainable by vertex-permutations).

1, 35, 141, 1713, 28011, 355, 34567, 4045, 54849, 64615, 15265, 95363, 126841

Offset: 1

Gus Wiseman, Dec 09 2019

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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with their corresponding multisets of multisets begins:
       1: {}
      35: {{2},{1,1}}
     141: {{1},{2,3}}
     355: {{2},{1,1,3}}
    1713: {{1},{2,3,4}}
    4045: {{2},{1,1,3,4}}
   15265: {{2},{1,4},{1,1,3}}
   28011: {{1},{2,3,4,5}}
   34567: {{1,2},{3,4,5}}
   54849: {{1},{2,3},{4,5}}
   64615: {{2},{1,1,3,4,5}}
   95363: {{2,3},{1,1,4,5}}
  126841: {{3},{1,2},{1,4,5}}

Sorted positions of first appearances in A330098.
The unsorted version is A330230.
The BII-number version is A330218.
MM-numbers of achiral multisets of multisets are A330232.
MM-numbers of fully-chiral multisets of multisets are A330236.
Cf. A001055, A003238, A007716, A056239, A112798, A302242, A303975, A322847, A330103, A330223, A330227, A330231.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Apply[Rule,Table[{p[[i]],i},{i,Length[p]}],{1}])],{p,Permutations[Union@@m]}]];
dv=Table[Length[graprms[primeMS/@primeMS[n]]],{n,1000}];
Table[Position[dv,i][[1,1]],{i,First/@Gather[dv]}]

A330234 Number of achiral factorizations of n into factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 0, 1, 2, 2, 5, 1, 0, 1, 0, 2, 2, 1, 0, 2, 2, 3, 0, 1, 2, 1, 7, 2, 2, 2, 5, 1, 2, 2, 0, 1, 2, 1, 0, 0, 2, 1, 0, 2, 0, 2, 0, 1, 0, 2, 0, 2, 2, 1, 0, 1, 2, 0, 11, 2, 2, 1, 0, 2, 2, 1, 0, 1, 2, 0, 0, 2, 2, 1, 0, 5, 2, 1, 0, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 08 2019

nonn

A multiset of multisets is achiral if it is not changed by any permutation of the vertices. A factorization is achiral if taking the multiset of prime indices of each factor gives an achiral multiset of multisets.

			The a(n) factorizations for n = 2, 6, 27, 36, 243, 216:
  (2)  (6)    (27)     (36)       (243)        (216)
       (2*3)  (3*9)    (4*9)      (3*81)       (6*36)
              (3*3*3)  (6*6)      (9*27)       (8*27)
                       (2*3*6)    (3*9*9)      (12*18)
                       (2*2*3*3)  (3*3*27)     (4*6*9)
                                  (3*3*3*9)    (6*6*6)
                                  (3*3*3*3*3)  (2*3*36)
                                               (2*3*4*9)
                                               (2*3*6*6)
                                               (2*2*3*3*6)
                                               (2*2*2*3*3*3)

The fully chiral version is A330235.
Planted achiral trees are A003238.
Achiral set-systems are counted by A083323.
BII-numbers of achiral set-systems are A330217.
Non-isomorphic achiral multiset partitions are A330223.
Achiral integer partitions are counted by A330224.
MM-numbers of achiral multisets of multisets are A330232.
Cf. A001055, A007716, A112798, A317533, A330098, A330227, A330228, A330236.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[facs[n],Length[graprms[primeMS/@#]]==1&]],{n,100}]

A330235 Number of fully chiral factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 0, 1, 3, 2, 0, 1, 4, 1, 0, 0, 5, 1, 4, 1, 4, 0, 0, 1, 7, 2, 0, 3, 4, 1, 0, 1, 7, 0, 0, 0, 4, 1, 0, 0, 7, 1, 0, 1, 4, 4, 0, 1, 12, 2, 4, 0, 4, 1, 7, 0, 7, 0, 0, 1, 4, 1, 0, 4, 11, 0, 0, 1, 4, 0, 0, 1, 16, 1, 0, 4, 4, 0, 0, 1, 12, 5, 0, 1, 4, 0, 0

Offset: 1

Gus Wiseman, Dec 08 2019

nonn

A multiset of multisets is fully chiral every permutation of the vertices gives a different representative. A factorization is fully chiral if taking the multiset of prime indices of each factor gives a fully chiral multiset of multisets.

			The a(n) factorizations for n = 1, 4, 8, 12, 16, 24, 48:
  ()  (4)    (8)      (12)     (16)       (24)       (48)
      (2*2)  (2*4)    (2*6)    (2*8)      (3*8)      (6*8)
             (2*2*2)  (3*4)    (4*4)      (4*6)      (2*24)
                      (2*2*3)  (2*2*4)    (2*12)     (3*16)
                               (2*2*2*2)  (2*2*6)    (4*12)
                                          (2*3*4)    (2*3*8)
                                          (2*2*2*3)  (2*4*6)
                                                     (3*4*4)
                                                     (2*2*12)
                                                     (2*2*2*6)
                                                     (2*2*3*4)
                                                     (2*2*2*2*3)

The costrict (or T_0) version is A316978.
The achiral version is A330234.
Planted achiral trees are A003238.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Full chiral partitions are A330228.
Fully chiral covering set-systems are A330229.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A001055, A007716, A083323, A112798, A317533, A330098, A330223, A330224, A330232.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[facs[n],Length[graprms[primeMS/@#]]==Length[Union@@primeMS/@#]!&]],{n,100}]

A330224 Number of achiral integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 13, 18, 21, 30, 32, 43, 46, 57, 64, 79, 83, 103, 107, 130, 141, 162, 171, 205, 214, 245, 258, 297, 307, 357, 373, 423, 441, 493, 513, 591, 607, 674, 702, 790, 817, 917, 938, 1040, 1078, 1186, 1216, 1362, 1395, 1534, 1580, 1738, 1779, 1956

Offset: 0

Gus Wiseman, Dec 08 2019

nonn

A multiset of multisets is achiral if it is not changed by any permutation of the vertices. An integer partition is achiral if taking the multiset of prime indices of each part gives an achiral multiset of multisets.

			The a(1) = 1 through a(7) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (52)
             (111)  (31)    (41)     (42)      (61)
                    (211)   (221)    (51)      (331)
                    (1111)  (311)    (222)     (421)
                            (2111)   (321)     (511)
                            (11111)  (411)     (2221)
                                     (2211)    (3211)
                                     (3111)    (4111)
                                     (21111)   (22111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)

The fully-chiral version is A330228.
The Heinz numbers of these partitions are given by A330232.
Achiral set-systems are counted by A083323.
BII-numbers of achiral set-systems are A330217.
Non-isomorphic achiral multiset partitions are A330223.
Achiral factorizations are A330234.
Cf. A001055, A056239, A112798, A319564, A330098, A330230.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[IntegerPartitions[n],Length[graprms[primeMS/@#]]==1&]],{n,0,30}]

More terms from Jinyuan Wang, Jun 26 2020

A370646 Number of non-isomorphic multiset partitions of weight n such that only one set can be obtained by choosing a different element of each block.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 62, 165, 475, 1400, 4334

Offset: 0

Gus Wiseman, Mar 12 2024

A multiset partition is a finite multiset of finite nonempty multisets. The weight of a multiset partition is the sum of cardinalities of its elements.

			The multiset partition {{3},{1,3},{2,3}} has unique choice (3,1,2) so is counted under a(5).
Representatives of the a(1) = 1 through a(5) = 23 multiset partitions:
  {1}  {11}    {111}      {1111}        {11111}
       {1}{2}  {1}{22}    {1}{122}      {11}{122}
               {2}{12}    {11}{22}      {1}{1222}
               {1}{2}{3}  {12}{12}      {11}{222}
                          {1}{222}      {12}{122}
                          {12}{22}      {1}{2222}
                          {2}{122}      {12}{222}
                          {1}{2}{33}    {2}{1122}
                          {1}{3}{23}    {2}{1222}
                          {1}{2}{3}{4}  {22}{122}
                                        {1}{2}{233}
                                        {1}{22}{33}
                                        {1}{23}{23}
                                        {1}{2}{333}
                                        {1}{23}{33}
                                        {1}{3}{233}
                                        {2}{12}{33}
                                        {2}{13}{23}
                                        {2}{3}{123}
                                        {3}{13}{23}
                                        {1}{2}{3}{44}
                                        {1}{2}{4}{34}
                                        {1}{2}{3}{4}{5}

For existence we have A368098, complement A368097.
Multisets of this type are ranked by A368101, see also A368100, A355529.
Subsets of this type are counted by A370584, see also A370582, A370583.
Maximal sets of this type are counted by A370585.
Partitions of this type are counted by A370594, see also A370592, A370593.
Subsets of this type are also counted by A370638, see also A370636, A370637.
Factorizations of this type are A370645, see also A368414, A368413.
Set-systems of this type are A370818, see also A367902, A367903.
A000110 counts set partitions, non-isomorphic A000041.
A001055 counts factorizations, strict A045778.
A007716 counts non-isomorphic multiset partitions, connected A007718.
Cf. A000612, A055621, A283877, A300913, A302545, A316983, A319616, A330223, A368095, A368412, A368422.

cp's OEIS Frontend

A330233 Least MM-numbers of multisets of multisets with a given number of distinct representatives (obtainable by vertex-permutations).

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A330234 Number of achiral factorizations of n into factors > 1.

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A330235 Number of fully chiral factorizations of n.

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A330224 Number of achiral integer partitions of n.

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A370646 Number of non-isomorphic multiset partitions of weight n such that only one set can be obtained by choosing a different element of each block.

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