A330226 -id:A330226 - cp's OEIS frontend

A330227 Number of non-isomorphic fully chiral multiset partitions of weight n.

1, 1, 2, 7, 16, 49, 144, 447, 1417, 4707

Offset: 0

Gus Wiseman, Dec 08 2019

A multiset partition is fully chiral if every permutation of the vertices gives a different representative. 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(4) = 16 multiset partitions:
  {1}  {11}    {111}      {1111}
       {1}{1}  {122}      {1222}
               {1}{11}    {1}{111}
               {1}{22}    {11}{11}
               {2}{12}    {1}{122}
               {1}{1}{1}  {1}{222}
               {1}{2}{2}  {12}{22}
                          {1}{233}
                          {2}{122}
                          {1}{1}{11}
                          {1}{1}{22}
                          {1}{2}{22}
                          {1}{3}{23}
                          {2}{2}{12}
                          {1}{1}{1}{1}
                          {1}{2}{2}{2}

MM-numbers of these multiset partitions are the odd terms of A330236.
Non-isomorphic costrict (or T_0) multiset partitions are A316980.
Non-isomorphic achiral multiset partitions are A330223.
BII-numbers of fully chiral set-systems are A330226.
Fully chiral partitions are counted by A330228.
Fully chiral covering set-systems are A330229.
Fully chiral factorizations are A330235.
Cf. A000612, A001055, A007716, A055621, A283877, A317533, A322847, A330098, A330232.

A330229 Number of fully chiral set-systems covering n vertices.

Original entry on oeis.org

1, 1, 2, 42, 21336

Offset: 0

Gus Wiseman, Dec 08 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the vertices gives a different representative.

			The a(3) = 42 set-systems:
  {1}{2}{13}    {1}{2}{12}{13}    {1}{2}{12}{13}{123}
  {1}{2}{23}    {1}{2}{12}{23}    {1}{2}{12}{23}{123}
  {1}{3}{12}    {1}{3}{12}{13}    {1}{3}{12}{13}{123}
  {1}{3}{23}    {1}{3}{13}{23}    {1}{3}{13}{23}{123}
  {2}{3}{12}    {2}{3}{12}{23}    {2}{3}{12}{23}{123}
  {2}{3}{13}    {2}{3}{13}{23}    {2}{3}{13}{23}{123}
  {1}{12}{23}   {1}{2}{13}{123}
  {1}{13}{23}   {1}{2}{23}{123}
  {2}{12}{13}   {1}{3}{12}{123}
  {2}{13}{23}   {1}{3}{23}{123}
  {3}{12}{13}   {2}{3}{12}{123}
  {3}{12}{23}   {2}{3}{13}{123}
  {1}{12}{123}  {1}{12}{23}{123}
  {1}{13}{123}  {1}{13}{23}{123}
  {2}{12}{123}  {2}{12}{13}{123}
  {2}{23}{123}  {2}{13}{23}{123}
  {3}{13}{123}  {3}{12}{13}{123}
  {3}{23}{123}  {3}{12}{23}{123}

The non-covering version is A330282.
Costrict (or T_0) covering set-systems are A059201.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic, fully chiral multiset partitions are A330227.
Fully chiral partitions are counted by A330228.
Fully chiral covering set-systems are A330229.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A055621, A083323, A283877, A319559, A319564, A330224, A330234.

graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&Length[graprms[#]]==n!&]],{n,0,3}]

Binomial transform is A330282.

A330236 MM-numbers of fully chiral multisets of multisets.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 53, 54, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Dec 10 2019

nonn

A multiset of multisets is fully chiral every permutation of the vertices gives a different representative.

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 all fully chiral multisets of multisets together with their MM-numbers begins:
   1:             18: {}{1}{1}      37: {112}          57: {1}{111}
   2: {}          19: {111}         38: {}{111}        59: {7}
   3: {1}         20: {}{}{2}       39: {1}{12}        61: {122}
   4: {}{}        21: {1}{11}       40: {}{}{}{2}      62: {}{5}
   5: {2}         22: {}{3}         41: {6}            63: {1}{1}{11}
   6: {}{1}       23: {22}          42: {}{1}{11}      64: {}{}{}{}{}{}
   7: {11}        24: {}{}{}{1}     44: {}{}{3}        65: {2}{12}
   8: {}{}{}      25: {2}{2}        45: {1}{1}{2}      67: {8}
   9: {1}{1}      27: {1}{1}{1}     46: {}{22}         68: {}{}{4}
  10: {}{2}       28: {}{}{11}      48: {}{}{}{}{1}    69: {1}{22}
  11: {3}         31: {5}           49: {11}{11}       70: {}{2}{11}
  12: {}{}{1}     32: {}{}{}{}{}    50: {}{2}{2}       71: {113}
  14: {}{11}      34: {}{4}         53: {1111}         72: {}{}{}{1}{1}
  16: {}{}{}{}    35: {2}{11}       54: {}{1}{1}{1}    74: {}{112}
  17: {4}         36: {}{}{1}{1}    56: {}{}{}{11}     75: {1}{2}{2}
The complement starts: {13, 15, 26, 29, 30, 33, 43, 47, 51, 52, 55, 58, 60, 66, 73, 79, 85, 86, 93, 94}.

Costrict (or T_0) factorizations are A316978.
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.
Fully chiral factorizations are A330235.
Cf. A001055, A007716, A083323, A112798, A303975, A317533, A330098, A330223, A330224, A330232.

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]}]];
Select[Range[100],Length[graprms[primeMS/@primeMS[#]]]==Length[Union@@primeMS/@primeMS[#]]!&]

Numbers n such that A330098(n) = A303975(n)!.

A330217 BII-numbers of achiral set-systems.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 9, 10, 11, 16, 25, 32, 42, 52, 63, 64, 75, 116, 127, 128, 129, 130, 131, 136, 137, 138, 139, 256, 385, 512, 642, 772, 903, 1024, 1155, 1796, 1927, 2048, 2184, 2320, 2457, 2592, 2730, 2868, 3007, 4096, 4233, 6416, 6553, 8192, 8330

Offset: 1

Gus Wiseman, Dec 06 2019

nonn

A set-system is a finite set of finite nonempty sets. It is achiral if it is not changed by any permutation of the vertices.

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 set-system 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 achiral set-systems together with their BII-numbers begins:
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   7: {{1},{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  16: {{1,3}}
  25: {{1},{3},{1,3}}
  32: {{2,3}}
  42: {{2},{3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  63: {{1},{2},{3},{1,2},{1,3},{2,3}}
  64: {{1,2,3}}
  75: {{1},{2},{3},{1,2,3}}

These are numbers n such that A330231(n) = 1.
Achiral set-systems are counted by A083323.
MG-numbers of planted achiral trees are A214577.
Non-isomorphic achiral multiset partitions are A330223.
Achiral integer partitions are counted by A330224.
BII-numbers of fully chiral set-systems are A330226.
MM-numbers of achiral multisets of multisets are A330232.
Achiral factorizations are A330234.
Cf. A000120, A003238, A016031, A048793, A070939, A326031, A326702, A327080, A327081, A330218, A330229, A330233.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Select[Range[0,1000],Length[graprms[bpe/@bpe[#]]]==1&]

A330231 Number of distinct set-systems that can be obtained by permuting the vertices of the set-system with BII-number n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 3, 6, 6, 3, 1, 2, 3, 6, 3, 3, 6, 6, 2, 1, 6, 3, 6, 6, 3, 3, 1, 3, 2, 6, 3, 6, 3, 6, 2, 6, 1, 3, 6, 3, 6, 3, 3, 6, 6, 3, 1, 3, 3, 3, 3, 6, 6, 3, 3, 3, 3, 1, 1, 3, 3, 3, 3, 6, 6, 3, 3, 3, 3, 1, 3, 6, 6, 3, 3, 6, 3, 6, 3, 3, 6

Offset: 0

Gus Wiseman, Dec 09 2019

nonn

A set-system is a finite set of finite nonempty sets.

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 set-system 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.

			30 is the MM-number of {{2},{3},{1,2},{1,3}}, with vertex permutations
  {{1},{2},{1,3},{2,3}}
  {{1},{3},{1,2},{2,3}}
  {{2},{3},{1,2},{1,3}}
so a(30) = 3.

Positions of 1's are A330217.
Positions of first appearances are A330218.
The version for MM-numbers is A330098.
Achiral set-systems are counted by A083323.
BII-numbers of fully chiral set-systems are A330226.
Cf. A000120, A003238, A007716, A016031, A048793, A055621, A070939, A214577, A326031, A326702, A330101, A330195, A330229, A330230, A330233.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[graprms[bpe/@bpe[n]]],{n,0,100}]

a(n) is a divisor of A326702(n)!.

A330228 Number of fully chiral integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 12, 18, 25, 33, 45, 61, 80, 106, 140, 176, 232, 293, 381, 476, 615, 764, 975, 1191, 1511, 1849, 2322, 2812, 3517, 4231, 5240, 6297, 7736, 9260, 11315, 13468, 16378, 19485, 23531, 27851, 33525, 39585, 47389, 55844, 66517, 78169, 92810

Offset: 0

Gus Wiseman, Dec 08 2019

nonn

A multiset partition is fully chiral if every permutation of the vertices gives a different representative. An integer partition is fully chiral if taking the multiset of prime indices of each part gives a fully chiral multiset of multisets.

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (33)      (7)
       (11)  (21)   (22)    (41)     (42)      (43)
             (111)  (31)    (221)    (51)      (322)
                    (211)   (311)    (222)     (331)
                    (1111)  (2111)   (411)     (421)
                            (11111)  (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

The Heinz numbers of these partitions are given by A330236.
Costrict (or T_0) partitions are A319564.
Achiral partitions are A330224.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic, fully chiral multiset partitions are A330227.
Fully chiral covering set-systems are A330229.
Fully chiral factorizations are A330235.
Cf. A001055, A007716, A322847, A330098, A330223.

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/@#]]==Length[Union@@primeMS/@#]!&]],{n,0,15}]

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}]

A330218 Least BII-number of a set-system with n distinct representatives obtainable by permuting the vertices.

Original entry on oeis.org

0, 5, 12, 180, 35636, 13

Offset: 1

Gus Wiseman, Dec 09 2019

nonn

A set-system is a finite set of finite nonempty sets of positive integers.

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 set-system 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 set-systems together with their BII-numbers begins:
      0: {}
      5: {{1},{1,2}}
     12: {{1,2},{3}}
    180: {{1,2},{1,3},{2,3},{4}}
  35636: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{5}}
     13: {{1},{1,2},{3}}

Positions of first appearances in A330231.
The MM-number version is A330230.
Achiral set-systems are counted by A083323.
BII-numbers of fully chiral set-systems are A330226.
Cf. A000120, A003238, A007716, A016031, A048793, A055621, A070939, A214577, A326031, A326702, A330098, A330101, A330195, A330217, A330229, A330233.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
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[bpe/@bpe[n]]],{n,0,1000}];
Table[Position[dv,i][[1,1]]-1,{i,First[Split[Union[dv],#1+1==#2&]]}]

A330282 Number of fully chiral set-systems on n vertices.

Original entry on oeis.org

1, 2, 5, 52, 21521

Offset: 0

Gus Wiseman, Dec 10 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.

			The a(0) = 1 through a(2) = 5 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{1,2}}
             {{2},{1,2}}

Costrict (or T_0) set-systems are A326940.
The covering case is A330229.
The unlabeled version is A330294, with covering case A330295.
Achiral set-systems are A083323.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Fully chiral partitions are A330228.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A016031, A319637, A330098, A330231, A330232, A330234.

graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Length[graprms[#]]==Length[Union@@#]!&]],{n,0,3}]

Binomial transform of A330229.

A330294 Number of non-isomorphic fully chiral set-systems on n vertices.

Original entry on oeis.org

1, 2, 3, 10, 899

Offset: 0

Gus Wiseman, Dec 10 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 10 set-systems:
  0  0    0        0
     {1}  {1}      {1}
          {2}{12}  {2}{12}
                   {1}{3}{23}
                   {2}{13}{23}
                   {3}{23}{123}
                   {2}{3}{13}{23}
                   {1}{3}{23}{123}
                   {2}{13}{23}{123}
                   {2}{3}{13}{23}{123}

The labeled version is A330282.
Partial sums of A330295 (the covering case).
Unlabeled costrict (or T_0) set-systems are A326946.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Fully chiral partitions are A330228.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A016031, A055621, A083323, A283877, A319637, A330098, A330231, A330232, A330234.

cp's OEIS Frontend

A330227 Number of non-isomorphic fully chiral multiset partitions of weight n.

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A330229 Number of fully chiral set-systems covering n vertices.

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A330236 MM-numbers of fully chiral multisets of multisets.

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A330217 BII-numbers of achiral set-systems.

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A330231 Number of distinct set-systems that can be obtained by permuting the vertices of the set-system with BII-number n.

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A330228 Number of fully chiral integer partitions of n.

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A330235 Number of fully chiral factorizations of n.

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A330218 Least BII-number of a set-system with n distinct representatives obtainable by permuting the vertices.

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A330282 Number of fully chiral set-systems on n vertices.

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