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.

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A384395 Number of integer partitions of n with more than one proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 1, 4, 5, 8, 8, 12, 17, 22, 29, 31, 40, 50, 65, 77, 101, 112, 135, 162, 201
Offset: 0

Views

Author

Gus Wiseman, May 30 2025

Keywords

Comments

By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.

Examples

			For the partition (8,5,2) we have four choices:
  ((8),(4,1),(2))
  ((7,1),(5),(2))
  ((5,3),(4,1),(2))
  ((4,3,1),(5),(2))
Hence (8,5,2) is counted under a(15).
The a(5) = 1 through a(12) = 12 partitions:
  (5)  (6)    (7)  (8)    (9)    (10)     (11)     (12)
       (3,3)       (4,4)  (5,4)  (5,5)    (6,5)    (6,6)
                   (5,3)  (6,3)  (6,4)    (7,4)    (7,5)
                   (7,1)  (7,2)  (7,3)    (8,3)    (8,4)
                          (8,1)  (8,2)    (9,2)    (9,3)
                                 (9,1)    (10,1)   (10,2)
                                 (4,3,3)  (5,3,3)  (11,1)
                                 (4,4,2)  (5,5,1)  (5,5,2)
                                                   (6,3,3)
                                                   (6,4,2)
                                                   (6,5,1)
                                                   (9,2,1)

Crossrefs

For just one choice we have A179009, ranked by A383707.
Twice-partitions of this type are counted by A279790.
For at least one choice we have A383708, odd case A383533.
For no choices we have A383710, odd case A383711.
For at least one proper choice we have A384317, ranked by A384321.
The strict version for at least one proper choice is A384318, ranked by A384322.
The strict version for just one proper choice is A384319, ranked by A384390.
For just one proper choice we have A384323, ranks A384347 = positions of 2 in A383706.
For no proper choices we have A384348, ranked by A384349.
These partitions are ranked by A384393.
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
A357982 counts choices of strict partitions of each prime index, non-strict A299200.
Cf. A098859, A279375, A317142, A381454, A382525, A382912, A382913, A384005.

Programs

  • Mathematica
    pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Length[pofprop[#]]>1&]],{n,0,15}]

A384179 Number of ways to choose strict integer partitions of each conjugate prime index of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 4, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 4, 1, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 1, 4, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 4, 2, 1, 2, 1, 3, 4, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, May 23 2025

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.

Examples

			The prime indices of 180 are {1,1,2,2,3}, conjugate {5,3,1}, and we have choices:
  {{5},{3},{1}}
  {{5},{2,1},{1}}
  {{4,1},{3},{1}}
  {{4,1},{2,1},{1}}
  {{3,2},{3},{1}}
  {{3,2},{2,1},{1}}
so a(180) = 6.

Crossrefs

Positions of 1 are A037143, complement A033942.
For multiplicities instead of indices we have A050361.
Adding up over all integer partitions gives A270995, disjoint A279790, strict A279375.
The conjugate version is A357982, disjoint A383706.
The disjoint case is A384005.
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non Look-and-Say or non section-sum partitions, ranks A351295 or A381433.
Cf. A122111, A130091, A179009, A382913, A383707, A383710, A384010, A384011.

Programs

  • Mathematica
    fop[y_]:=Join@@@Tuples[strptns/@y];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[fop[conj[prix[n]]]],{n,100}]

A384723 Heinz numbers of conjugates of maximally refined strict integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 24, 30, 60, 90, 120, 150, 180, 210, 240, 420, 540, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2310, 2520, 3360, 4620, 6300, 6930, 7560, 9240
Offset: 1

Views

Author

Gus Wiseman, Jun 09 2025

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Given a partition, the following are equivalent:
1) The distinct parts are maximally refined.
2) Every strict partition of a part contains a part. In other words, if y is the set of parts and z is any strict partition of any element of y, then z must contain at least one element from y.
3) No part is a sum of distinct non-parts.

Examples

			The terms together with their prime indices begin:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
    12: {1,1,2}
    18: {1,2,2}
    24: {1,1,1,2}
    30: {1,2,3}
    60: {1,1,2,3}
    90: {1,2,2,3}
   120: {1,1,1,2,3}
   150: {1,2,3,3}
   180: {1,1,2,2,3}
   210: {1,2,3,4}
   240: {1,1,1,1,2,3}
   420: {1,1,2,3,4}
   540: {1,1,2,2,2,3}
   630: {1,2,2,3,4}
   840: {1,1,1,2,3,4}

Crossrefs

Partitions of this type are counted by A179009.
The conjugate version is A383707, proper A384390.
Appears to be the positions of 1 in A384005 (conjugate A383706).
For at least one instead of exactly one choice we appear to have A384010.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
A357982 counts strict partitions of prime indices, non-strict A299200.
Cf. A003963, A048767, A130091, A382525, A384317, A384318, A384320, A384347, A384349, A384394.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Select[Range[100],With[{y=conj[prix[#]]},UnsameQ@@y&&Intersection[y,Total/@nonsets[y]]=={}]&]
Previous Showing 11-13 of 13 results.

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