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 1-8 of 8 results.

A318393 Regular tetrangle where T(n,k,i) is the number of pairs of set partitions of {1,...,n} with meet of length k and join of length i.

Original entry on oeis.org

1, 1, 2, 1, 1, 6, 3, 8, 6, 1, 1, 14, 7, 48, 36, 6, 56, 44, 12, 1, 1, 30, 15, 200, 150, 25, 560, 440, 120, 10, 552, 440, 140, 20, 1, 1, 62, 31, 720, 540, 90, 3640, 2860, 780, 65, 8280, 6600, 2100, 300, 15, 7202, 5632, 1920, 340, 30, 1, 1, 126, 63, 2408, 1806
Offset: 1

Views

Author

Gus Wiseman, Aug 25 2018

Keywords

Examples

			The T(3,3,1) = 8 pairs of set partitions:
  {{1},{2},{3}}  {{1,2,3}}
   {{1},{2,3}}  {{1,2},{3}}
   {{1},{2,3}}  {{1,3},{2}}
   {{1,2},{3}}  {{1},{2,3}}
   {{1,2},{3}}  {{1,3},{2}}
   {{1,3},{2}}  {{1},{2,3}}
   {{1,3},{2}}  {{1,2},{3}}
    {{1,2,3}}  {{1},{2},{3}}
Tetrangle begins:
   1   1     1       1            1
       2 1   6 3     14 7         30  15
             8 6 1   48 36 6      200 150 25
                     56 44 12 1   560 440 120 10
                                  552 440 140 20  1

Crossrefs

Cf. A000110, A000258, A001247, A008277, A048994, A059849, A060639, A181939, A318389, A318390, A318391, A318392.

Programs

  • Mathematica
    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]]]]]]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
spmeet[a_,b_]:=DeleteCases[Union@@Outer[Intersection,a,b,1],{}];spmeet[a_,b_,c__]:=spmeet[spmeet[a,b],c];
Table[Length[Select[Tuples[sps[Range[n]],2],And[Length[spmeet@@#]==k,Length[csm[Union@@#]]==j]&]],{n,6},{k,n},{j,k}]

A318391 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with meet of length k.

Original entry on oeis.org

1, 1, 3, 1, 9, 15, 1, 21, 90, 113, 1, 45, 375, 1130, 1153, 1, 93, 1350, 7345, 17295, 15125, 1, 189, 4515, 39550, 161420, 317625, 245829, 1, 381, 14490, 192213, 1210650, 4023250, 6883212, 4815403, 1, 765, 45375, 878010, 8014503, 40020750, 113572998, 173354508, 111308699
Offset: 1

Views

Author

Gus Wiseman, Aug 25 2018

Keywords

Examples

			The T(3,2) = 9 pairs of set partitions:
  {{1},{2,3}}  {{1},{2,3}}
  {{1},{2,3}}   {{1,2,3}}
  {{1,2},{3}}  {{1,2},{3}}
  {{1,2},{3}}   {{1,2,3}}
  {{1,3},{2}}  {{1,3},{2}}
  {{1,3},{2}}   {{1,2,3}}
   {{1,2,3}}   {{1},{2,3}}
   {{1,2,3}}   {{1,2},{3}}
   {{1,2,3}}   {{1,3},{2}}
Triangle begins:
     1
     1     3
     1     9    15
     1    21    90   113
     1    45   375  1130  1153
     1    93  1350  7345 17295 15125

Links

Crossrefs

Row sums are A001247. Last column is A059849.
Cf. A000110, A000258, A008277, A048994, A060639, A181939, A318389, A318390, A318392, A318393.

Programs

  • Mathematica
    Table[StirlingS2[n,k]*Sum[StirlingS1[k,i]*BellB[i]^2,{i,k}],{n,10},{k,n}]
  • PARI
    row(n) = {my(b=Vec(serlaplace(exp(exp(x + O(x*x^n))-1)-1))); vector(n, k, stirling(n,k,2)*sum(i=1, k, stirling(k,i,1)*b[i]^2))}
{ for(n=1, 10, print(row(n))) } \\ Andrew Howroyd, Jan 19 2023

Formula

T(n,k) = S(n,k) * Sum_{i=1..k} s(k,i) * B(i)^2 where S = A008277, s = A048994, B = A000110.

A318389 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with meet {{1},...,{n}} and join of length k.

Original entry on oeis.org

1, 2, 1, 8, 6, 1, 56, 44, 12, 1, 552, 440, 140, 20, 1, 7202, 5632, 1920, 340, 30, 1, 118456, 89278, 31192, 6160, 700, 42, 1, 2369922, 1708016, 595448, 124432, 16240, 1288, 56, 1, 56230544, 38592786, 13214672, 2830632, 400512, 37296, 2184, 72, 1, 1552048082
Offset: 1

Views

Author

Gus Wiseman, Aug 25 2018

Keywords

Examples

			The T(3,2) = 6 pairs of set partitions:
  {{1},{2},{3}}  {{1},{2,3}}
  {{1},{2},{3}}  {{1,2},{3}}
  {{1},{2},{3}}  {{1,3},{2}}
   {{1},{2,3}}  {{1},{2},{3}}
   {{1,2},{3}}  {{1},{2},{3}}
   {{1,3},{2}}  {{1},{2},{3}}
Triangle begins:
     1
     2    1
     8    6    1
    56   44   12    1
   552  440  140   20    1
  7202 5632 1920  340   30    1

Crossrefs

Row sums are A059849. First column is A181939.
Cf. A000110, A000258, A001247, A008277, A048994, A060639, A318390, A318391, A318392, A318393.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
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]]]]]]]]];
spmeet[a_,b_]:=DeleteCases[Union@@Outer[Intersection,a,b,1],{}];spmeet[a_,b_,c__]:=spmeet[spmeet[a,b],c];
Table[Length[Select[Tuples[sps[Range[n]],2],And[Max@@Length/@spmeet@@#==1,Length[csm[Union@@#]]==k]&]],{n,5},{k,n}]

A318392 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with join of length k.

Original entry on oeis.org

1, 3, 1, 15, 9, 1, 119, 87, 18, 1, 1343, 1045, 285, 30, 1, 19905, 15663, 4890, 705, 45, 1, 369113, 286419, 95613, 16450, 1470, 63, 1, 8285261, 6248679, 2147922, 410053, 44870, 2730, 84, 1, 219627683, 159648795, 55211229, 11202534, 1394883, 105714, 4662, 108, 1
Offset: 1

Views

Author

Gus Wiseman, Aug 25 2018

Keywords

Examples

			The T(3,2) = 9 pairs of set partitions:
  {{1},{2},{3}}  {{1},{2,3}}
  {{1},{2},{3}}  {{1,2},{3}}
  {{1},{2},{3}}  {{1,3},{2}}
   {{1},{2,3}}  {{1},{2},{3}}
   {{1},{2,3}}   {{1},{2,3}}
   {{1,2},{3}}  {{1},{2},{3}}
   {{1,2},{3}}   {{1,2},{3}}
   {{1,3},{2}}  {{1},{2},{3}}
   {{1,3},{2}}   {{1,3},{2}}
Triangle begins:
      1
      3     1
     15     9     1
    119    87    18     1
   1343  1045   285    30     1
  19905 15663  4890   705    45     1

Crossrefs

Row sums are A001247. First column is A060639.
Cf. A000110, A000258, A008277, A048994, A059849, A181939, A318389, A318390, A318391, A318393.

Programs

  • Mathematica
    nn=5;Table[n!*SeriesCoefficient[Sum[BellB[n]^2*x^n/n!,{n,0,nn}]^t,{x,0,n},{t,0,k}],{n,nn},{k,n}]

Formula

E.g.f.: (Sum_{n>=0} B(n)^2 x^n/n!)^t where B = A000110.

A318394 Number of finite sets of set partitions of {1,...,n} such that any two have meet {{1},...,{n}}.

Original entry on oeis.org

2, 4, 18, 316, 37492
Offset: 1

Views

Author

Gus Wiseman, Aug 25 2018

Keywords

Examples

			The a(3) = 18 sets of set partitions:
        0
    {{1,2,3}}
   {{1,3},{2}}
   {{1,2},{3}}
   {{1},{2,3}}
  {{1},{2},{3}}
   {{1,2},{3}}   {{1,3},{2}}
   {{1},{2,3}}   {{1,3},{2}}
   {{1},{2,3}}   {{1,2},{3}}
  {{1},{2},{3}}   {{1,2,3}}
  {{1},{2},{3}}  {{1,3},{2}}
  {{1},{2},{3}}  {{1,2},{3}}
  {{1},{2},{3}}  {{1},{2,3}}
   {{1},{2,3}}   {{1,2},{3}}  {{1,3},{2}}
  {{1},{2},{3}}  {{1,2},{3}}  {{1,3},{2}}
  {{1},{2},{3}}  {{1},{2,3}}  {{1,3},{2}}
  {{1},{2},{3}}  {{1},{2,3}}  {{1,2},{3}}
  {{1},{2},{3}}  {{1},{2,3}}  {{1,2},{3}}  {{1,3},{2}}

Crossrefs

Cf. A000110, A000258, A008277, A059849, A060639, A096827, A181939.
Cf. A318389, A318390, A318391, A318392, A318531, A318532, A318531, A318532.

Programs

  • Mathematica
    stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
spmeet[a_,b_]:=DeleteCases[Union@@Outer[Intersection,a,b,1],{}];spmeet[a_,b_,c__]:=spmeet[spmeet[a,b],c];
Table[Length[stableSets[sps[Range[n]],Max@@Length/@spmeet[#1,#2]>1&]],{n,5}]

A318399 Number of triples of set partitions of {1,...,n} with meet {{1},...,{n}} and join {{1,...,n}}.

Original entry on oeis.org

1, 6, 84, 2226, 93246, 5616492, 459173406, 48933260388, 6595445513412, 1098326915060730, 221772386369110242, 53460963703982862534, 15185890964240671486740, 5026315912246843181692776, 1919721040169845172603949966, 838872819016448052585038291124
Offset: 1

Views

Author

Gus Wiseman, Aug 25 2018

Keywords

Examples

			The a(2) = 6 triples:
  {{1},{2}} {{1},{2}}  {{1,2}}
  {{1},{2}}  {{1,2}}  {{1},{2}}
  {{1},{2}}  {{1,2}}   {{1,2}}
   {{1,2}}  {{1},{2}} {{1},{2}}
   {{1,2}}  {{1},{2}}  {{1,2}}
   {{1,2}}   {{1,2}}  {{1},{2}}

Crossrefs

Cf. A000110, A000258, A001247, A008277, A048994, A059849, A060639, A181939, A318389, A318390, A318391, A318392, A318393, A318398.

Programs

  • Mathematica
    nn=20;
Table[n!*SeriesCoefficient[Log[1+Sum[x^k/k!*Sum[StirlingS1[k,i]*BellB[i]^3,{i,0,k}],{k,nn}]],{x,0,n}],{n,nn}]

Formula

Logarithmic transform of A318398.

A318531 Number of finite sets of set partitions of {1,...,n} such that any two have join {{1,...,n}}.

Original entry on oeis.org

2, 4, 18, 450, 436270
Offset: 1

Views

Author

Gus Wiseman, Aug 28 2018

Keywords

Examples

			The a(3) = 18 sets of set partitions:
        0
    {{1,2,3}}
   {{1,3},{2}}
   {{1,2},{3}}
   {{1},{2,3}}
  {{1},{2},{3}}
   {{1,3},{2}}   {{1,2,3}}
   {{1,2},{3}}   {{1,2,3}}
   {{1,2},{3}}  {{1,3},{2}}
   {{1},{2,3}}   {{1,2,3}}
   {{1},{2,3}}  {{1,3},{2}}
   {{1},{2,3}}  {{1,2},{3}}
  {{1},{2},{3}}  {{1,2,3}}
   {{1,2},{3}}  {{1,3},{2}}   {{1,2,3}}
   {{1},{2,3}}  {{1,3},{2}}   {{1,2,3}}
   {{1},{2,3}}  {{1,2},{3}}   {{1,2,3}}
   {{1},{2,3}}  {{1,2},{3}}  {{1,3},{2}}
   {{1},{2,3}}  {{1,2},{3}}  {{1,3},{2}}  {{1,2,3}}

Crossrefs

Cf. A000110, A000258, A008277, A059849, A060639, A096827, A181939.
Cf. A318389, A318390, A318391, A318392, A318394, A318532.

Programs

  • Mathematica
    stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
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]]]]]]]]];
Table[Length[stableSets[sps[Range[n]],Length[csm[Union[#1,#2]]]>1&]],{n,4}]

A318532 Number of finite sets of set partitions of {1,...,n} such that any two have meet {{1},...,{n}} and join {{1,...,n}}.

Original entry on oeis.org

2, 4, 11, 51, 635, 15591
Offset: 1

Views

Author

Gus Wiseman, Aug 28 2018

Keywords

Examples

			The a(3) = 11 sets of set partitions:
        0
    {{1,2,3}}
   {{1,3},{2}}
   {{1,2},{3}}
   {{1},{2,3}}
  {{1},{2},{3}}
   {{1,2},{3}}   {{1,3},{2}}
   {{1},{2,3}}   {{1,3},{2}}
   {{1},{2,3}}   {{1,2},{3}}
  {{1},{2},{3}}   {{1,2,3}}
   {{1},{2,3}}   {{1,2},{3}}  {{1,3},{2}}

Crossrefs

Cf. A000110, A000258, A008277, A059849, A060639, A096827, A181939.
Cf. A318389, A318390, A318391, A318392, A318394, A318531.
Showing 1-8 of 8 results.

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