cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A330098 Number of distinct multisets of multisets that can be obtained by permuting the vertices of the multiset of multisets with MM-number n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2
Offset: 1

Views

Author

Gus Wiseman, Dec 09 2019

Keywords

Comments

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}}.
a(n) is a divisor of A303975(n)!.

Examples

			The vertex-permutations of {{1,2},{2,3,3}} are:
  {{1,2},{1,3,3}}
  {{1,2},{2,3,3}}
  {{1,3},{1,2,2}}
  {{1,3},{2,2,3}}
  {{2,3},{1,1,2}}
  {{2,3},{1,1,3}}
so a(4927) = 6.

Crossrefs

Positions of 1's are A330232.
Positions of first appearances are A330230 and A330233.
The BII-number version is A330231.
Cf. A001055, A003238, A007716, A055621, A056239, A112798, A302242, A303975, A322847, A330194, A330218, A330223, A330227, A330236.

Programs

  • Mathematica
    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[graprms[primeMS/@primeMS[n]]],{n,100}]

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

Views

Author

Gus Wiseman, Dec 06 2019

Keywords

Comments

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.

Examples

			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}}

Crossrefs

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.

Programs

  • Mathematica
    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&]

A330230 Least MM-number of a multiset of multisets with n distinct representatives obtainable by permuting the vertices.

Original entry on oeis.org

1, 35, 141, 1713, 28011, 355
Offset: 1

Views

Author

Gus Wiseman, Dec 09 2019

Keywords

Comments

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}}.

Examples

			The sequence of terms together with their corresponding multisets of multisets begins:
      1: {}
     35: {{2},{1,1}}
    141: {{1},{2,3}}
   1713: {{1},{2,3,4}}
  28011: {{1},{2,3,4,5}}
    355: {{2},{1,1,3}}

Crossrefs

The BII-number version is A330218.
Positions of first appearances in A330098.
The sorted version is A330233.
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, A330236.

Programs

  • Mathematica
    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[Split[Union[dv],#1+1==#2&]]}]

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

Views

Author

Gus Wiseman, Dec 09 2019

Keywords

Comments

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.

Examples

			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.

Crossrefs

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.

Programs

  • Mathematica
    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}]

Formula

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

A330226 BII-numbers of fully chiral set-systems.

Original entry on oeis.org

0, 1, 2, 5, 6, 8, 13, 14, 17, 19, 22, 23, 24, 26, 28, 29, 34, 35, 37, 39, 40, 41, 44, 46, 49, 50, 57, 58, 69, 70, 77, 78, 81, 83, 86, 87, 88, 90, 92, 93, 98, 99, 101, 103, 104, 105, 108, 110, 113, 114, 121, 122, 128, 133, 134, 145, 150, 151, 152, 156, 157, 162
Offset: 1

Views

Author

Gus Wiseman, Dec 08 2019

Keywords

Comments

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.
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.

Examples

			The sequence of all fully chiral set-systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  26: {{2},{3},{1,3}}
  28: {{3},{1,2},{1,3}}
  29: {{1},{3},{1,2},{1,3}}
  34: {{2},{2,3}}
  35: {{1},{2},{2,3}}
  37: {{1},{1,2},{2,3}}
  39: {{1},{2},{1,2},{2,3}}
For example, 28 is in the sequence because all six permutations give different representatives, namely:
  {{1},{1,2},{2,3}}
  {{1},{1,3},{2,3}}
  {{2},{1,2},{1,3}}
  {{2},{1,3},{2,3}}
  {{3},{1,2},{1,3}}
  {{3},{1,2},{2,3}}

Crossrefs

A subset of A326947.
Achiral set-systems are counted by A083323.
BII-numbers of achiral set-systems are A330217.
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. A000120, A000612, A016031, A048793, A070939, A326031, A326702, A330218, A330231, A330232.

Programs

  • Mathematica
    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[0,100],Length[graprms[bpe/@bpe[#]]]==Length[Union@@bpe/@bpe[#]]!&]

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Dec 09 2019

Keywords

Comments

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}}.

Examples

			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}}

Crossrefs

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.

Programs

  • Mathematica
    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]}]
Showing 1-6 of 6 results.

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