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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A384319 Number of strict integer partitions of n with exactly two possible ways to choose disjoint strict partitions of each part.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 3, 1, 0, 4, 4, 4, 2, 0, 6, 7, 8, 8, 3, 2, 9, 9, 14, 13, 6, 7, 3, 15, 13, 20
Offset: 0

Views

Author

Gus Wiseman, May 28 2025

Keywords

Examples

			For y = (5,4,2) we have choices ((5),(4),(2)) and ((5),(3,1),(2)), so y is counted under a(11).
The a(3) = 1 through a(11) = 4 partitions:
  (3)  (4)  .  (4,2)  (4,3)  (6,2)  .  (5,3,2)  (5,4,2)
               (5,1)  (5,2)            (5,4,1)  (6,3,2)
                      (6,1)            (6,3,1)  (7,3,1)
                                       (7,2,1)  (8,2,1)

Crossrefs

The case of a unique choice is A179009, ranks A383707.
Choices of this type for each prime index are counted by A383706.
The non-strict version for at least one choice is A383708, ranks A382913.
The non-strict version for no choices is A383710, ranks A382912.
The non-strict version for more than one choice is A384317, ranks A384321.
The version for at least one choice is A384322, counted by A384318.
The non-strict version is A384323, ranks A384347.
These partitions are ranked by A384390.
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. A098859, A279375, A299200, A317142, A357982, A381454, A383533, A383711, A384320.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[pof[#]]==2&]],{n,0,30}]

A384347 Heinz numbers of integer partitions with exactly two possible ways to choose disjoint strict partitions of each part.

Original entry on oeis.org

5, 7, 21, 22, 25, 26, 33, 35, 39, 49, 102, 114, 130, 147, 154, 165, 170, 175, 190, 195, 231, 238, 242, 255, 275, 285
Offset: 1

Views

Author

Gus Wiseman, May 27 2025

Keywords

Comments

Positions of 2 in A383706.
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.

Examples

			The prime indices of 275 are {3,3,5}, with two ways to choose disjoint strict partitions of each part: ((3),(2,1),(5)) and ((2,1),(3),(5)). Hence 275 is in the sequence.
The terms together with their prime indices begin:
    5: {3}
    7: {4}
   21: {2,4}
   22: {1,5}
   25: {3,3}
   26: {1,6}
   33: {2,5}
   35: {3,4}
   39: {2,6}
   49: {4,4}
  102: {1,2,7}
  114: {1,2,8}
  130: {1,3,6}
  147: {2,4,4}
  154: {1,4,5}
  165: {2,3,5}

Crossrefs

The case of no choices is A382912, counted by A383710, odd case A383711.
These are positions of 2 in A383706.
The case of no proper choices is A383707, counted by A179009.
The case of some proper choice is A384321, strict A384322, count A384317, strict A384318.
These partitions are counted by A384323, strict A384319.
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.
A357982 counts strict partitions of prime indices, non-strict A299200.
Cf. A048767, A382525, A382771, A382857, A383533, A383708, A384320.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
Select[Range[100],Length[pof[prix[#]]]==2&]

A386581 Number of normal multisets of size n with no permutation having all distinct run lengths.

Original entry on oeis.org

0, 0, 1, 1, 5, 11, 20, 51, 108, 229, 448, 953, 1940, 3951, 7986, 15972
Offset: 0

Views

Author

Gus Wiseman, Aug 12 2025

Keywords

Comments

A multiset is normal iff it covers an initial interval of positive integers.

Examples

			The normal multiset m = {1,1,1,2,2,2} has permutation (1,2,2,2,1,1) with run lengths (1,3,2), so m is not counted under a(6).
The a(1) = 0 through a(6) = 20 multisets:
  .  (12)  (123)  (1122)  (11123)  (111123)
                  (1123)  (11223)  (111234)
                  (1223)  (11233)  (112233)
                  (1233)  (11234)  (112234)
                  (1234)  (12223)  (112334)
                          (12233)  (112344)
                          (12234)  (112345)
                          (12333)  (122223)
                          (12334)  (122234)
                          (12344)  (122334)
                          (12345)  (122344)
                                   (122345)
                                   (123333)
                                   (123334)
                                   (123344)
                                   (123345)
                                   (123444)
                                   (123445)
                                   (123455)
                                   (123456)

Crossrefs

The complement for partitions appears to be A239455, ranks A351294 or A381432.
For integer partitions we appear to have A351293, ranks A351295 or A381433.
For weakly decreasing multiplicities we appear to have A383710, ranks A382912.
The complement is counted by A386580, see A383708.
A032020 counts normal multisets with distinct multiplicities.
A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
A098859 counts partitions with distinct multiplicities, compositions A242882.
Cf. A000009, A025065, A047966, A072233, A116540, A130091, A320347, A326083, A382771, A382913, A383706.

Programs

  • Mathematica
    allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
nodrm[y_]:=Select[Permutations[y],UnsameQ@@Length/@Split[#]&];
Table[Length[Select[allnorm[n],nodrm[#]=={}&]],{n,0,7}]

A384323 Number of integer partitions of n with exactly two possible ways to choose disjoint strict partitions of each part.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 3, 3, 2, 0, 6, 6, 6, 6, 4, 10, 10, 14, 16, 15, 16, 17, 20, 25, 27, 28, 37, 43, 31, 42, 44
Offset: 0

Views

Author

Gus Wiseman, May 30 2025

Keywords

Examples

			For y = (4,3,3) we have two ways: ((4),(3),(2,1)) and ((4),(2,1),(3)), so y is counted under a(10).
The a(0) = 0 through a(15) = 10 partitions:
  .  .  .  3  4  .  33  43  44  .  433  533  543  544  554  5433
                    42  52  62     442  542  552  553  644  5442
                    51  61         532  551  633  652  662  5532
                                   541  632  732  661  833  5541
                                   631  731  741  733       6432
                                   721  821  831  832       6531
                                                            7431
                                                            7521
                                                            8421
                                                            9321

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 more than one choice we have A384317, ranked by A384321.
The strict version for at least one choice is A384318, ranked by A384322.
The strict version is A384319, ranked by A384390.
These partitions are ranked by A384347 = positions of 2 in A383706.
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.
Cf. A098859, A299200, A317142, A381454, A382525, A382912, A382913, A384005.

Programs

  • Mathematica
    pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Length[pof[#]]==2&]],{n,0,15}]

A386580 Number of normal multisets of size n having a permutation with all distinct run lengths.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 12, 13, 20, 27, 64, 71, 108, 145, 206, 412
Offset: 0

Views

Author

Gus Wiseman, Aug 07 2025

Keywords

Comments

A multiset is normal iff it covers an initial interval of positive integers.
Conjecture: Also the number of normal multisets of size n having a disjoint family of strict integer partitions, one of each multiplicity.

Examples

			The normal multiset m = {1,1,1,2,2,2} has permutation (1,2,2,2,1,1) with run lengths (1,3,2), so m is counted under a(6).
The a(n) multisets for n = 1..7:
  (1)  (11)  (111)  (1111)  (11111)  (111111)  (1111111)
             (112)  (1112)  (11112)  (111112)  (1111112)
             (122)  (1222)  (11122)  (111122)  (1111122)
                            (11222)  (111222)  (1111222)
                            (12222)  (111223)  (1111223)
                                     (111233)  (1111233)
                                     (112222)  (1112222)
                                     (112223)  (1122222)
                                     (112333)  (1122223)
                                     (122222)  (1123333)
                                     (122233)  (1222222)
                                     (122333)  (1222233)
                                               (1223333)

Crossrefs

For integer partitions we appear to have A239455, ranks A351294 or A381432.
For weakly decreasing multiplicities we appear to have A383708.
The complement is counted by A386581, see A383710 (ranks A382912).
A000041 counts integer partitions, strict A000009.
A032020 counts normal multisets with distinct multiplicities, increasing A000009.
A098859 counts partitions with distinct multiplicities, compositions A242882.
Cf. A025065, A047966, A048767, A072233, A116540, A130091, A320347, A326083, A382771, A382913, A383706.

Programs

  • Mathematica
    allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
nodrm[y_]:=Select[Permutations[y],UnsameQ@@Length/@Split[#]&];
Table[Length[Select[allnorm[n],nodrm[#]!={}&]],{n,0,5}]

A384349 Heinz numbers of integer partitions with no proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 105, 108, 110, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135
Offset: 1

Views

Author

Gus Wiseman, Jun 03 2025

Keywords

Comments

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

Examples

			The prime indices of 102 are {1,2,7}, which has proper disjoint choice ((1),(2),(3,4)), so 102 is not in the sequence.
The terms together with their prime indices begin:
     1: {}           27: {2,2,2}        63: {2,2,4}
     2: {1}          28: {1,1,4}        64: {1,1,1,1,1,1}
     3: {2}          30: {1,2,3}        66: {1,2,5}
     4: {1,1}        32: {1,1,1,1,1}    68: {1,1,7}
     6: {1,2}        36: {1,1,2,2}      70: {1,3,4}
     8: {1,1,1}      40: {1,1,1,3}      72: {1,1,1,2,2}
     9: {2,2}        42: {1,2,4}        75: {2,3,3}
    10: {1,3}        44: {1,1,5}        76: {1,1,8}
    12: {1,1,2}      45: {2,2,3}        78: {1,2,6}
    14: {1,4}        48: {1,1,1,1,2}    80: {1,1,1,1,3}
    15: {2,3}        50: {1,3,3}        81: {2,2,2,2}
    16: {1,1,1,1}    52: {1,1,6}        84: {1,1,2,4}
    18: {1,2,2}      54: {1,2,2,2}      88: {1,1,1,5}
    20: {1,1,3}      56: {1,1,1,4}      90: {1,2,2,3}
    24: {1,1,1,2}    60: {1,1,2,3}      92: {1,1,9}

Crossrefs

The non-proper version appears to be A382912, counted by A383710.
The non-proper complement appears to be A382913, counted by A383708.
The complement is A384321, counted by A384317.
These partitions are counted by A384348.
These are the positions of 0 in A384389.
The case of a unique proper choice is A384390, counted by A384319.
A048767 is the Look-and-Say transform, fixed points A048768.
A056239 adds up prime indices, row sums of A112798.
A179009 counts maximally refined strict partitions, ranks A383707.
A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
Cf. A122111, A130091, A317142, A326080, A351294, A357982, A381454, A382525, A383706, A384320, A384322.

Programs

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

A384389 Number of proper ways to choose disjoint strict integer partitions of each prime index of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 3, 0, 0, 0, 4, 0, 5, 0, 1, 1, 7, 0, 2, 1, 0, 0, 9, 0, 11, 0, 1, 2, 1, 0, 14, 2, 1, 0, 17, 0, 21, 0, 0, 4, 26, 0, 2, 0, 2, 0, 31, 0, 2, 0, 3, 4, 37, 0, 45, 6, 0, 0, 3, 0, 53, 0, 4, 0, 63, 0, 75, 7, 0, 0, 2, 0, 88, 0, 0, 9
Offset: 1

Views

Author

Gus Wiseman, Jun 01 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.
By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.

Examples

			The prime indices of 65 are {3,6}, and we have proper choices: ((3),(5,1)), ((3),(4,2)), ((2,1),(6)). Hence a(65) = 3.
The prime indices of 175 are {3,3,4}, and we have choices: ((3),(2,1),(4)), ((2,1),(3),(4)), both already proper. Hence a(175) = 2.

Crossrefs

Without disjointness we have A357982 - 1, non-strict version A299200 - 1.
This is the proper case of A383706, conjugate version A384005.
Positions of positive terms are A384321.
Positions of 0 are A384349.
Positions of 1 are A384390.
Positions of terms > 1 are A384393.
The conjugate version is A384394.
Positions of first appearances are A384396.
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 partitions, ranks A351294
A351293 counts non-Look-and-Say partitions, ranks A351295.
Cf. A382912, A383710, A383711, A382913, A383708, A383533.
Cf. A179009, A279375, A279790, A381454, A382525, A384322, A384347.

Programs

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

Formula

a(prime(n)) = A000009(n) - 1.

A384393 Heinz numbers of integer partitions with more than one proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

11, 13, 17, 19, 23, 25, 29, 31, 34, 37, 38, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134
Offset: 1

Views

Author

Gus Wiseman, Jun 02 2025

Keywords

Comments

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

Examples

			The prime indices of 275 are {3,3,5}, with a total of 2 proper choices: ((3),(2,1),(5)) and ((2,1),(3),(5)), so 275 is in the sequence.
The terms together with their prime indices begin:
    11: {5}      51: {2,7}      82: {1,13}
    13: {6}      53: {16}       83: {23}
    17: {7}      55: {3,5}      85: {3,7}
    19: {8}      57: {2,8}      86: {1,14}
    23: {9}      58: {1,10}     87: {2,10}
    25: {3,3}    59: {17}       89: {24}
    29: {10}     61: {18}       91: {4,6}
    31: {11}     62: {1,11}     93: {2,11}
    34: {1,7}    65: {3,6}      94: {1,15}
    37: {12}     67: {19}       95: {3,8}
    38: {1,8}    69: {2,9}      97: {25}
    41: {13}     71: {20}      101: {26}
    43: {14}     73: {21}      103: {27}
    46: {1,9}    74: {1,12}    106: {1,16}
    47: {15}     77: {4,5}     107: {28}
    49: {4,4}    79: {22}      109: {29}

Crossrefs

Without "proper" we get A384321 (strict A384322), counted by A384317 (strict A384318).
The case of no choices is A384349, counted by A384348.
These are positions of terms > 1 in A384389.
The case of a unique proper choice is A384390, counted by A384319.
Partitions of this type are counted by A384395.
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 partitions, ranks A351294 or A381432.
A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
Cf. A382912, A383710, A382913, A383708, A383533.
Cf. A179009, A357982, A381454, A382525, A383706, A383707, A384320, A384323.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
Select[Range[100],Length[pofprop[prix[#]]]>1&]

A384010 Heinz numbers of integer partitions such that it is possible to choose a family of disjoint strict partitions, one of each conjugate part.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 27, 30, 32, 36, 48, 54, 60, 64, 72, 81, 90, 96, 108, 120, 128, 144, 150, 162, 180, 192
Offset: 1

Views

Author

Gus Wiseman, May 23 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.

Examples

			The prime indices of 96 are {1,1,1,1,1,2}, conjugate (6,1), disjoint family (4,2,1), so 96 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

Crossrefs

For multiplicities instead of indices we have A382525.
These partitions are counted by A383708, without ones A383533, complement A383711.
These are the positions of positive terms in A384005.
The complement is A384011, conjugate A383710.
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.
A122111 represent 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.
Cf. A098859, A130091, A279375, A279790, A299200, A357982, A382912, A383706, A383707.

Programs

  • Mathematica
    pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
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}]]]];
Select[Range[100],pof[conj[prix[#]]]!={}&]

A384350 Number of subsets of {1..n} containing at least one element that is a sum of distinct non-elements.

Original entry on oeis.org

0, 0, 0, 1, 4, 13, 33, 81, 183, 402, 856, 1801, 3721, 7646, 15567, 31575
Offset: 0

Views

Author

Gus Wiseman, Jun 05 2025

Keywords

Comments

Conjecture: Also the number of subsets of {1..n} such that it is possible in more than one way to choose a disjoint family of strict integer partitions, one of each element.

Examples

			For the set s = {1,5} we have 5 = 2+3, so s is counted under a(5).
The a(0) = 0 through a(5) = 13 subsets:
  .  .  .  {3}  {3}    {3}
                {4}    {4}
                {2,4}  {5}
                {3,4}  {1,5}
                       {2,4}
                       {2,5}
                       {3,4}
                       {3,5}
                       {4,5}
                       {1,4,5}
                       {2,3,5}
                       {2,4,5}
                       {3,4,5}

Crossrefs

The complement is counted by A326080, allowing repeats A326083.
For strict partitions of n instead of subsets of {1..n} we have A384318, ranks A384322.
First differences are A384391.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A179009 counts maximally refined strict partitions, ranks A383707.
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.
A383706 counts ways to choose disjoint strict partitions of prime indices, non-disjoint A357982, non-strict A299200.
Cf. A279375, A279790, A317141, A317142, A383708, A383710, A384317, A384319, A384320, A384321.

Programs

  • Mathematica
    nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@nonsets[#]]!={}&]],{n,0,10}]
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