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.

Previous Showing 11-17 of 17 results.

A367917 BII-numbers of set-systems with the same number of edges as covered vertices.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 17, 19, 21, 22, 24, 26, 28, 34, 35, 37, 38, 40, 41, 44, 49, 50, 52, 56, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 97, 98, 100, 104, 112, 128, 129, 130, 131, 133, 134, 136, 137, 138, 139, 141, 142, 145, 147, 149, 150, 152
Offset: 1

Views

Author

Gus Wiseman, Dec 12 2023

Keywords

Comments

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets 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.

Examples

			The terms together with the corresponding set-systems begin:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  26: {{2},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  34: {{2},{2,3}}
  35: {{1},{2},{2,3}}
  37: {{1},{1,2},{2,3}}

Crossrefs

These set-systems are counted by A054780 and A367916, A368186.
Graphs of this type are A367862, covering A367863, unlabeled A006649.
A003465 counts set-systems covering {1..n}, unlabeled A055621.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, connected A323818, unlabeled A000612.
A070939 gives length of binary expansion.
A136556 counts set-systems on {1..n} with n edges.
Cf. A057500, A059201, A072639, A096111, A116508, A309326, A326031, A326702, A326753, A326754, A367770, A367902, A367905.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]],1];
Select[Range[0,100], Length[bpe[#]]==Length[Union@@bpe/@bpe[#]]&]

A367913 Least number k such that there are exactly n ways to choose a multiset consisting of a binary index of each binary index of k.

Original entry on oeis.org

1, 4, 64, 20, 68, 320, 52, 84, 16448, 324, 832, 116, 1104, 308, 816, 340, 836, 848, 1108, 1136, 1360, 3152, 16708, 372, 5188, 5216, 852, 880, 2884, 1364, 13376, 1392, 3184, 3424, 17220, 5204, 5220, 2868, 5728, 884, 19536, 66896, 2900, 1396, 21572, 3188, 3412
Offset: 1

Views

Author

Gus Wiseman, Dec 16 2023

Keywords

Comments

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

Examples

			The terms together with the corresponding set-systems begin:
      1: {{1}}
      4: {{1,2}}
     64: {{1,2,3}}
     20: {{1,2},{1,3}}
     68: {{1,2},{1,2,3}}
    320: {{1,2,3},{1,4}}
     52: {{1,2},{1,3},{2,3}}
     84: {{1,2},{1,3},{1,2,3}}
  16448: {{1,2,3},{1,2,3,4}}
    324: {{1,2},{1,2,3},{1,4}}
    832: {{1,2,3},{1,4},{2,4}}
    116: {{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

A version for multisets and divisors is A355734.
With distinctness we have A367910, firsts of A367905, sorted A367911.
Positions of first appearances in A367912.
The sorted version is A367915.
For sequences we have A368111, firsts of A368109, sorted A368112.
For sets we have A368184, firsts of A368183, sorted A368185.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326749, A326753, A355733, A355741, A355744.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
c=Table[Length[Union[Sort/@Tuples[bpe/@bpe[n]]]],{n,1000}];
Table[Position[c,n][[1,1]],{n,spnm[c]}]

A367915 Sorted positions of first appearances in A367912 (number of multisets that can be obtained by choosing a binary index of each binary index).

Original entry on oeis.org

1, 4, 20, 52, 64, 68, 84, 116, 308, 320, 324, 340, 372, 816, 832, 836, 848, 852, 880, 884, 1104, 1108, 1136, 1360, 1364, 1392, 1396, 1904, 1908, 2868, 2884, 2900, 2932, 3152, 3184, 3188, 3412, 3424, 3440, 3444, 3952, 3956, 5188, 5204, 5216, 5220, 5236, 5476
Offset: 1

Views

Author

Gus Wiseman, Dec 16 2023

Keywords

Comments

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

Examples

			The terms together with the corresponding set-systems begin:
     1: {{1}}
     4: {{1,2}}
    20: {{1,2},{1,3}}
    52: {{1,2},{1,3},{2,3}}
    64: {{1,2,3}}
    68: {{1,2},{1,2,3}}
    84: {{1,2},{1,3},{1,2,3}}
   116: {{1,2},{1,3},{2,3},{1,2,3}}
   308: {{1,2},{1,3},{2,3},{1,4}}
   320: {{1,2,3},{1,4}}
   324: {{1,2},{1,2,3},{1,4}}
   340: {{1,2},{1,3},{1,2,3},{1,4}}
   372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}

Crossrefs

A version for multisets and divisors is A355734.
Sorted positions of first appearances in A367912, for sequences A368109.
The unsorted version is A367913.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326749, A326753, A355733, A355744, A367905, A367906, A367911, A368112, A368185.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
c=Table[Length[Union[Sort/@Tuples[bpe/@bpe[n]]]],{n,10000}];
Select[Range[Length[c]],FreeQ[Take[c,#-1],c[[#]]]&]

A368111 Least k such that there are exactly A003586(n) ways to choose a binary index of each binary index of k.

Original entry on oeis.org

1, 4, 64, 20, 68, 52, 1088, 84, 308, 1092, 116, 5184, 820, 1108, 372, 5188, 2868, 1140, 13376, 884, 5204, 17204, 1396, 13380, 2932, 5236, 275520, 19252, 1908, 13396, 17268, 5492, 275524, 84788, 3956, 13428, 1324096, 19316, 6004, 275540, 215860, 18292, 13684
Offset: 1

Views

Author

Gus Wiseman, Dec 17 2023

Keywords

Comments

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

Examples

			The terms together with the corresponding set-systems begin:
    1: {{1}}
    4: {{1,2}}
   64: {{1,2,3}}
   20: {{1,2},{1,3}}
   68: {{1,2},{1,2,3}}
   52: {{1,2},{1,3},{2,3}}
   84: {{1,2},{1,3},{1,2,3}}
  308: {{1,2},{1,3},{2,3},{1,4}}
  116: {{1,2},{1,3},{2,3},{1,2,3}}
  820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
  372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}
  884: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4}}

Crossrefs

With distinctness we have A367910, sorted A367911, firsts of A367905.
For multisets we have A367913, sorted A367915, firsts of A367912.
Positions of first appearances in A368109.
The sorted version is A368112.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A253317, A309326, A326031, A326702, A326753, A355731, A355741, A355744, A367906.

Programs

  • Mathematica
    nn=10000;
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
dd=Select[Range[nn],Max@@First/@FactorInteger[#]<=3&];
qq=Table[Length[Tuples[bpe/@bpe[n]]],{n,nn}];
kk=Select[Range[Length[dd]],SubsetQ[qq,Take[dd,#]]&]
Table[Position[qq,dd[[n]]][[1,1]],{n,kk}]

A368183 Number of sets that can be obtained by choosing a different binary index of each binary index of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 2, 1, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 1, 3, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 2, 2, 1, 3, 1, 1, 0, 2, 1, 1, 0, 1, 0, 0, 0, 3, 1, 1, 0, 1, 0, 0
Offset: 0

Views

Author

Gus Wiseman, Dec 17 2023

Keywords

Comments

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

Examples

			The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with choices (1,3,2), (2,1,3), both permutations of {1,2,3}, so a(52) = 1.

Crossrefs

For sequences we have A367905, firsts A367910, sorted A367911.
Positions of zeros are A367907.
Without distinctness we have A367912, firsts A367913, sorted A367915.
Positions of positive terms are A367906.
For sequences without distinctness: A368109, firsts A368111, sorted A368112.
Positions of first appearances are A368184, sorted A368185.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326753, A355739, A355741, A355744, A367771.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Union[Sort/@Select[Tuples[bpe/@bpe[n]],UnsameQ@@#&]]],{n,0,100}]

A368186 Number of n-covers of an unlabeled n-set.

Original entry on oeis.org

1, 1, 2, 9, 87, 1973, 118827, 20576251, 10810818595, 17821875542809, 94589477627232498, 1651805220868992729874, 96651473179540769701281003, 19238331716776641088273777321428, 13192673305726630096303157068241728202, 31503323006770789288222386469635474844616195
Offset: 0

Views

Author

Gus Wiseman, Dec 19 2023

Keywords

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 set-systems:
  {{1}}  {{1},{2}}    {{1},{2},{3}}
         {{1},{1,2}}  {{1},{2},{1,3}}
                      {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{2},{1,2,3}}
                      {{1},{1,2},{1,2,3}}
                      {{1,2},{1,3},{2,3}}
                      {{1},{2,3},{1,2,3}}
                      {{1,2},{1,3},{1,2,3}}

Links

Crossrefs

The labeled version is A054780, ranks A367917, non-covering A367916.
The case of graphs is A006649, labeled A367863, cf. A116508, A367862.
The case of connected graphs is A001429, labeled A057500.
Covers with any number of edges are counted by A003465, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Cf. A000088, A002494, A006126, A055130, A133686, A140638, A305000, A317795, A326754, A367901, A367902, A367903.

Programs

  • Mathematica
    brute[m_]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i, p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}];
Table[Length[Union[First[Sort[brute[#]]]& /@ Select[Subsets[Rest[Subsets[Range[n]]],{n}], Union@@#==Range[n]&]]], {n,0,3}]
  • PARI
    permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={2^sum(j=1, #q, gcd(t, q[j])) - 1}
G(n,m)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, m, K(q,t)*x^t/t, O(x*x^m))); s+=permcount(q)*exp(g - subst(g,x,x^2))); s/n!)}
a(n)=if(n ==0, 1, polcoef(G(n,n) - G(n-1,n), n)) \\ Andrew Howroyd, Jan 03 2024

Formula

a(n) = A055130(n, n) for n > 0. - Andrew Howroyd, Jan 03 2024

Extensions

Terms a(6) and beyond from Andrew Howroyd, Jan 03 2024

A368731 Number of non-isomorphic n-element sets of nonempty subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 10, 97, 2160, 126862, 21485262, 11105374322, 18109358131513, 95465831661532570, 1660400673336788987026, 96929369602251313489896310, 19268528295096123543660356281600, 13203875101002459910158494602665950757, 31517691852305548841992346407978317698725021
Offset: 0

Views

Author

Gus Wiseman, Jan 07 2024

Keywords

Examples

			Non-isomorphic representatives of the a(3) = 10 set-systems:
  {{1},{2},{3}}
  {{1},{2},{1,2}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{1,3}}
  {{1},{1,2},{2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Links

Crossrefs

The case of graphs is A001434, labeled A116508.
Labeled version is A136556, covering A054780, binomial transform of A367916.
The case of labeled covering graphs is A367863, binomial transform A367862.
These include the set-systems ranked by A367917.
The covering case is A368186, for graphs A006649, connected A057500.
Requiring all edges to be singletons or pairs gives A368598.
A003465 counts covers with any number of edges, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Cf. A000088, A007716, A283877, A367901, A367902, A367903, A368599.

Programs

  • Mathematica
    brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Subsets[Subsets[Range[n],{1,n}],{n}]]],{n,0,4}]
  • PARI
    a(n) = polcoef(G(n, n), n) \\ G defined in A368186. - Andrew Howroyd, Jan 11 2024

Extensions

Terms a(6) and beyond from Andrew Howroyd, Jan 11 2024
Previous Showing 11-17 of 17 results.

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