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 1-4 of 4 results.

A321738 Number of ways to partition the Young diagram of the integer partition with Heinz number n into vertical sections.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 7, 4, 1, 10, 1, 5, 13, 15, 1, 27, 1, 17, 21, 6, 1, 37, 34, 7, 87, 26, 1, 60, 1, 52, 31, 8, 73, 114, 1, 9, 43, 77, 1, 115, 1, 37, 235, 10, 1, 151, 209, 175, 57, 50, 1, 409, 136, 141, 73, 11, 1, 295, 1, 12, 543, 203, 229, 198, 1, 65, 91
Offset: 1

Views

Author

Gus Wiseman, Nov 19 2018

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A vertical section is a partial Young diagram with at most one square in each row. For example, a partition (shown as a coloring by positive integers) into vertical sections of the Young diagram of (322) is:
1 2 3
1 2
2 3

Examples

			The a(12) = 10 partitions of the Young diagram of (211) into vertical sections:
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  3     3     2     3     2     1     1     3     2     1
  4     3     3     2     2     3     2     1     1     1

Crossrefs

Cf. A000110, A000700, A000701, A006052, A056239, A122111, A320328, A321719-A321731, A321737, A321854.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];
Table[With[{y=Reverse[primeMS[n]]},Length[spsu[ptnverts[y],ptnpos[y]]]],{n,30}]

A321854 Irregular triangle where T(H(u),H(v)) is the number of ways to partition the Young diagram of u into vertical sections whose sizes are the parts of v, where H is Heinz number.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 1, 1, 3, 1, 0, 2, 0, 4, 1, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 6, 0, 6, 1, 1, 3, 4, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Gus Wiseman, Nov 19 2018

Keywords

Comments

Row n has length A000041(A056239(n)).
A vertical section is a partial Young diagram with at most one square in each row.

Examples

			Triangle begins:
  1
  1
  0  1
  1  1
  0  0  1
  0  2  1
  0  0  0  0  1
  1  3  1
  0  2  0  4  1
  0  0  0  3  1
  0  0  0  0  0  0  1
  0  2  2  5  1
  0  0  0  0  0  0  0  0  0  0  1
  0  0  0  0  0  4  1
  0  0  0  6  0  6  1
  1  3  4  6  1
  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
  0  0  4 10  4  8  1
The 12th row counts the following partitions of the Young diagram of (211) into vertical sections (shown as colorings by positive integers):
  T(12,7) = 0:
.
  T(12,9) = 2:    1 2   1 2
                  1     2
                  2     1
.
  T(12,10) = 2:   1 2   1 2
                  2     1
                  2     1
.
  T(12,12) = 5:   1 2   1 2   1 2   1 2   1 2
                  3     2     3     1     3
                  3     3     2     3     1
.
  T(12,16) = 1:   1 2
                  3
                  4

Crossrefs

Cf. A000085, A000110, A007016, A056239, A122111, A153452, A215366, A296188, A300121, A318396, A321719-A321731, A321737, A321738, A321742-A321765.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];
Table[With[{y=Reverse[primeMS[n]]},Table[Length[Select[spsu[ptnverts[y],ptnpos[y]],Sort[Length/@#]==primeMS[k]&]],{k,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}]],{n,18}]

A321737 Number of ways to partition the Young diagram of an integer partition of n into vertical sections.

Original entry on oeis.org

1, 1, 3, 9, 37, 152, 780, 3965, 23460, 141471, 944217, 6445643, 48075092, 364921557, 2974423953, 24847873439, 219611194148, 1987556951714, 18930298888792, 184244039718755, 1874490999743203, 19510832177784098, 210941659716920257, 2331530519337226199, 26692555830628617358
Offset: 0

Views

Author

Gus Wiseman, Nov 19 2018

Keywords

Comments

A vertical section is a partial Young diagram with at most one square in each row. For example, a partition (shown as a coloring by positive integers) into vertical sections of the Young diagram of (322) is:
1 2 3
1 2
2 3

Examples

			The a(4) = 37 partitions into vertical sections of integer partitions of 4:
  1 2 3 4
.
  1 2 3   1 2 3   1 2 3   1 2 3
  4       3       2       1
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2
  3 4   2 3   3 2   1 3   1 2   3 1   2 1
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  3     3     2     3     2     1     1     3     2     1
  4     3     3     2     2     3     2     1     1     1
.
  1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
  2   2   2   2   2   1   1   2   2   2   2   1   1   2   1
  3   3   2   3   2   2   2   1   1   3   2   1   2   1   1
  4   3   3   2   2   3   2   3   2   1   1   2   1   1   1

Crossrefs

Cf. A000110, A000258, A008277, A046682, A122111, A318396, A321728, A321729, A321730, A321731, A321738, A321854.

Programs

  • Mathematica
    spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];
Table[Sum[Length[spsu[ptnverts[y],ptnpos[y]]],{y,IntegerPartitions[n]}],{n,6}]

Extensions

a(11)-a(24) from Ludovic Schwob, Aug 28 2023

A321730 Number of ways to partition the Young diagram of an integer partition of n into vertical sections of the same sizes as the parts of the original partition.

Original entry on oeis.org

1, 1, 1, 3, 8, 23, 79, 303, 1294, 5934, 29385, 156232, 884893
Offset: 0

Views

Author

Gus Wiseman, Nov 18 2018

Keywords

Comments

A vertical section is a partial Young diagram with at most one square in each row. For example, a suitable partition (shown as a coloring by positive integers) of the Young diagram of (322) is:
1 2 3
1 2
2 3

Examples

			The a(5) = 23 partitions of Young diagrams of integer partitions of 5 into vertical sections of the same sizes as the parts of the original partition, shown as colorings by positive integers:
  1 2 3   1 2 3   1 2 3
  1       2       3
  1       2       3
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  1 2   1 3   1 3   2 1   3 1   3 1   2 3   3 2   2 3   3 2
  3     2     3     3     2     3     1     1     3     3
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  1     3     3     2     3     3     3     3     3
  3     1     4     3     2     4     3     4     4
  4     4     1     4     4     2     4     3     4
.
  1
  2
  3
  4
  5

Crossrefs

Cf. A000110, A000258, A000700, A000701, A006052, A007016, A008277, A321728, A321729, A321731, A321737, A321738.

Programs

  • Mathematica
    spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
Table[Sum[Length[Select[spsu[ptnverts[y],ptnpos[y]],Function[p,Sort[Length/@p]==Sort[y]]]],{y,IntegerPartitions[n]}],{n,5}]
Showing 1-4 of 4 results.

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