A381432 -id:A381432 - cp's OEIS frontend

A382771 Number of ways to permute the prime indices of n so that the run-lengths are all different.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 2, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 2, 1, 2, 0, 2, 1, 2, 0, 2, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 2, 1, 0, 2, 2, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Apr 07 2025

nonn

The first x with a(x) > 0 but A382857(x) > 1 is a(216) = 4, A382857(216) = 4.

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, sum A056239.

			The a(96) = 4 permutations are:
  (1,1,1,1,1,2)
  (1,1,1,2,1,1)
  (1,1,2,1,1,1)
  (2,1,1,1,1,1)
The a(216) = 4 permutations are:
  (1,1,2,2,2,1)
  (1,2,2,2,1,1)
  (2,1,1,1,2,2)
  (2,2,1,1,1,2)
The a(360) = 6 permutations are:
  (1,1,1,2,2,3)
  (1,1,1,3,2,2)
  (2,2,1,1,1,3)
  (2,2,3,1,1,1)
  (3,1,1,1,2,2)
  (3,2,2,1,1,1)

Positions of 1 are A000961.
Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433, equal A382879.
Sorted positions of first appearances are A382772, equal A382878.
For prescribed signature we have A382773, equal A382858.
The restriction to factorials is A382774, equal A335407.
For equal instead of distinct run-lengths we have A382857.
For run-sums instead of run-lengths we have A382876, equal A382877.
Positions of terms > 1 are A383113.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, complement A351293.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
Cf. A000720, A001221, A001222, A003242, A048767, A051903, A051904, A130091, A238130, A351013, A351202.

Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],UnsameQ@@Length/@Split[#]&]],{n,30}]

a(A181821(n)) = a(A304660(n)) = A382773(n).

a(n!) = A382774(n).

A383512 Heinz numbers of conjugate Wilf partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85

Offset: 1

Gus Wiseman, May 13 2025

nonn

First differs from A364347 in having 130 and lacking 110.
First differs from A381432 in lacking 65 and 133.
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.
An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
     1: {}           17: {7}            35: {3,4}
     2: {1}          19: {8}            37: {12}
     3: {2}          20: {1,1,3}        38: {1,8}
     4: {1,1}        22: {1,5}          39: {2,6}
     5: {3}          23: {9}            40: {1,1,1,3}
     7: {4}          25: {3,3}          41: {13}
     8: {1,1,1}      26: {1,6}          43: {14}
     9: {2,2}        27: {2,2,2}        44: {1,1,5}
    10: {1,3}        28: {1,1,4}        45: {2,2,3}
    11: {5}          29: {10}           46: {1,9}
    13: {6}          31: {11}           47: {15}
    14: {1,4}        32: {1,1,1,1,1}    49: {4,4}
    15: {2,3}        33: {2,5}          50: {1,3,3}
    16: {1,1,1,1}    34: {1,7}          51: {2,7}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Partitions of this type are counted by A098859.
The conjugate version is A130091, complement A130092.
Including differences of 0 gives A325367, counted by A325324.
The strict case is A325388, counted by A320348.
The complement is A383513, counted by A336866.
Also requiring distinct multiplicities gives A383532, counted by A383507.
These are the positions of strict rows in A383534, or squarefree numbers in A383535.
A000040 lists the primes, differences A001223.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A325349 counts partitions with distinct augmented differences, ranks A325366.
A383530 counts partitions that are not Wilf or conjugate Wilf, ranks A383531.
A383709 counts Wilf partitions with distinct augmented differences, ranks A383712.
Cf. A000720, A005117, A048768, A109298, A325325, A325351, A325355, A325368, A381431, A383506.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], UnsameQ@@DeleteCases[Differences[Prepend[prix[#],0]],0]&]

A383707 Heinz numbers of maximally refined strict integer partitions.

Original entry on oeis.org

1, 2, 3, 6, 10, 14, 15, 30, 42, 66, 70, 78, 105, 110, 182, 210, 330, 390

Offset: 1

Gus Wiseman, May 15 2025

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.
Also squarefree numbers such that every strict partition of a prime index contains a prime index.
Also squarefree numbers such that no prime index is a sum of distinct non prime indices.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   30: {1,2,3}
   42: {1,2,4}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
  105: {2,3,4}
  110: {1,3,5}
  182: {1,4,6}
  210: {1,2,3,4}
  330: {1,2,3,5}
  390: {1,2,3,6}

Partitions of this type are counted by A179009.
Appears to be positions of 1 in A383706.
For distinct prime indices see A384320.
The proper version appears to be A384390.
The conjugate version is A384723.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A048767, A130091, A299200, A351294, A381432, A351295, A357982, A381454, A382525, A384321, A384349, A384389.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Select[Range[30],SquareFreeQ[#]&&With[{y=prix[#]},Intersection[y,Total/@nonsets[y]]=={}]&]

A384321 Numbers whose distinct prime indices are not maximally refined.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 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, 102, 103, 106, 107, 109, 111, 113, 114, 115, 118, 119

Offset: 1

Gus Wiseman, Jun 01 2025

nonn

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.
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.

			The prime indices of 25 are {3,3}, which has refinements: ((3),(1,2)) and ((1,2),(3)), so 25 is in the sequence.
The prime indices of 102 are {1,2,7}, which has refinement ((1),(2),(3,4)), so 102 is in the sequence.
The terms together with their prime indices begin:
     5: {3}      39: {2,6}      73: {21}
     7: {4}      41: {13}       74: {1,12}
    11: {5}      43: {14}       77: {4,5}
    13: {6}      46: {1,9}      79: {22}
    17: {7}      47: {15}       82: {1,13}
    19: {8}      49: {4,4}      83: {23}
    21: {2,4}    51: {2,7}      85: {3,7}
    22: {1,5}    53: {16}       86: {1,14}
    23: {9}      55: {3,5}      87: {2,10}
    25: {3,3}    57: {2,8}      89: {24}
    26: {1,6}    58: {1,10}     91: {4,6}
    29: {10}     59: {17}       93: {2,11}
    31: {11}     61: {18}       94: {1,15}
    33: {2,5}    62: {1,11}     95: {3,8}
    34: {1,7}    65: {3,6}      97: {25}
    35: {3,4}    67: {19}      101: {26}
    37: {12}     69: {2,9}     102: {1,2,7}
    38: {1,8}    71: {20}      103: {27}

These appear to be positions of terms > 1 in A383706, non-disjoint A357982, non-strict A299200.
The strict complement is A383707, counted by A179009.
Partitions of this type appear to be counted by A384317.
The complement is A384320.
The strict (squarefree) case appears to be A384322, counted by A384318.
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. A098859, A122111, A130091, A317142, A326080, A381454, A382525, A384005, A384323, A384390.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Select[Range[30],With[{y=Union[prix[#]]},UnsameQ@@y&&Intersection[y,Total/@nonsets[y]]!={}]&]

A384886 Number of strict integer partitions of n with all equal lengths of maximal runs (decreasing by 1).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 4, 7, 7, 8, 11, 11, 14, 17, 19, 20, 27, 27, 35, 38, 45, 47, 60, 63, 75, 84, 97, 104, 127, 134, 155, 175, 196, 218, 251, 272, 307, 346, 384, 424, 480, 526, 586, 658, 719, 798, 890, 979, 1078, 1201, 1315, 1451, 1603, 1762, 1934, 2137

Offset: 0

Gus Wiseman, Jun 13 2025

			The strict partition y = (7,6,5,3,2,1) has maximal runs ((7,6,5),(3,2,1)), with lengths (3,3), so y is counted under a(24).
The a(1) = 1 through a(14) = 14 partitions (A-E = 10-14):
  1  2  3   4   5   6    7   8   9    A     B    C     D    E
        21  31  32  42   43  53  54   64    65   75    76   86
                41  51   52  62  63   73    74   84    85   95
                    321  61  71  72   82    83   93    94   A4
                                 81   91    92   A2    A3   B3
                                 432  631   A1   B1    B2   C2
                                 531  4321  641  543   C1   D1
                                            731  642   742  752
                                                 741   751  842
                                                 831   841  851
                                                 5421  931  941
                                                            A31
                                                            5432
                                                            6521

John Tyler Rascoe, Table of n, a(n) for n = 0..100

For subsets instead of strict partitions we have A243815, distinct lengths A384175.
For distinct instead of equal lengths we have A384178, for anti-runs A384880.
This is the strict case of A384904, distinct lengths A384884.
A000041 counts integer partitions, strict A000009.
A047993 counts partitions with max part = length (A106529).
A098859 counts Wilf partitions (complement A336866), compositions A242882.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
Cf. A000217, A008284, A044813, A047966, A089259, A325324, A325325, A329739, A382857, A383013, A383708, A384176.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&SameQ@@Length/@Split[#,#2==#1-1&]&]],{n,0,15}]

A_q(N) = {Vec(1+sum(k=1,floor(-1/2+sqrt(2+2*N)), sum(i=1,(N/(k*(k+1)/2))+1, q^(k*(k+1)*i^2/2)/prod(j=1,i, 1 - q^(j*k)))) + O('q^(N+1)))} \\ John Tyler Rascoe, Aug 21 2025

G.f.: 1 + Sum_{i,k>0} q^(k*(k+1)*i^2/2)/Product_{j=1..i} (1 - q^(j*k)). - John Tyler Rascoe, Aug 21 2025

A384884 Number of integer partitions of n with all distinct lengths of maximal gapless runs (decreasing by 0 or 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 35, 46, 60, 79, 104, 131, 170, 215, 271, 342, 431, 535, 670, 830, 1019, 1258, 1547, 1881, 2298, 2787, 3359, 4061, 4890, 5849, 7010, 8361, 9942, 11825, 14021, 16558, 19561, 23057, 27084, 31821, 37312, 43627, 50999, 59500, 69267

Offset: 0

Gus Wiseman, Jun 13 2025

nonn

			The partition y = (6,6,4,3,3,2) has maximal gapless runs ((6,6),(4,3,3,2)), with lengths (2,4), so y is counted under a(24).
The a(1) = 1 through a(8) = 18 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (211)   (221)    (222)     (322)      (332)
                    (1111)  (311)    (321)     (331)      (422)
                            (2111)   (411)     (421)      (431)
                            (11111)  (2211)    (511)      (521)
                                     (3111)    (2221)     (611)
                                     (21111)   (3211)     (2222)
                                     (111111)  (4111)     (3221)
                                               (22111)    (4211)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

For subsets instead of strict partitions we have A384175.
The strict case is A384178, for anti-runs A384880.
For anti-runs we have A384885.
For equal instead of distinct lengths we have A384887.
A000041 counts integer partitions, strict A000009.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A355394 counts partitions without a neighborless part, singleton case A355393.
A356236 counts partitions with a neighborless part, singleton case A356235.
A356606 counts strict partitions without a neighborless part, complement A356607.
Cf. A008284, A044813, A047993, A242882, A287170, A325324, A325325, A356226, A356230, A356233, A356234, A384176, A384177, A384886.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#,#2>=#1-1&]&]],{n,0,15}]

A383533 Number of integer partitions of n with no ones such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 5, 8, 8, 11, 13, 17, 22, 25, 30, 37, 44, 53, 69, 77, 93, 111, 130, 153, 181, 220, 249, 295

Offset: 0

Gus Wiseman, May 07 2025

The Heinz numbers of these partitions are the odd terms of A382913.

Also the number of integer partitions y of n with no ones such that the normal multiset (in which i appears y_i times) is a Look-and-Say partition.

			For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is counted under a(6).
The a(2) = 1 through a(10) = 8 partitions:
  (2)  (3)  (4)  (5)    (6)    (7)    (8)    (9)      (10)
                 (3,2)  (3,3)  (4,3)  (4,4)  (5,4)    (5,5)
                        (4,2)  (5,2)  (5,3)  (6,3)    (6,4)
                                      (6,2)  (7,2)    (7,3)
                                             (4,3,2)  (8,2)
                                                      (4,3,3)
                                                      (4,4,2)
                                                      (5,3,2)

The number of such families is A383706.
Allowing ones gives A383708 (ranks A382913), complement A383710 (ranks A382912).
The complement is counted by A383711.
A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
A098859 counts partitions with distinct multiplicities, compositions A242882.
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. A044813, A047966, A089259, A116540, A130091, A317141, A318361, A381441, A381454, A383013.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n], FreeQ[#,1]&&!pof[#]=={}&]],{n,0,15}]

A383711 Number of integer partitions of n with no ones such that it is not possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 3, 3, 4, 6, 10, 11, 17, 19, 30, 36, 51, 61, 84, 96, 133, 160, 209, 253, 325, 393, 488, 598, 744

Offset: 0

Gus Wiseman, May 07 2025

The Heinz numbers of these partitions are the odd terms of A382912.

Also the number of integer partitions of n with no ones whose normal multiset (in which i appears y_i times) is not a Look-and-Say partition.

			For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is not counted under a(6).
The a(4) = 1 through a(12) = 10 partitions:
  (22)  .  (222)  (322)  (332)   (333)   (622)    (443)    (444)
                         (422)   (522)   (3322)   (722)    (822)
                         (2222)  (3222)  (4222)   (3332)   (3333)
                                         (22222)  (4322)   (4332)
                                                  (5222)   (4422)
                                                  (32222)  (5322)
                                                           (6222)
                                                           (33222)
                                                           (42222)
                                                           (222222)

The complement without ones is counted by A383533.
The number of these families is A383706.
Allowing ones gives A383710 (ranks A382912), complement A383708 (ranks A382913).
A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
A098859 counts partitions with distinct multiplicities, compositions A242882.
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. A044813, A047966, A089259, A116540, A130091, A317141, A318361, A381441, A381454, A383013.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&pof[#]=={}&]],{n,0,15}]

A381436 Irregular triangle read by rows where row k is the section-sum partition of the prime indices of n.

Original entry on oeis.org

1, 2, 1, 1, 3, 3, 4, 1, 1, 1, 2, 2, 4, 5, 3, 1, 6, 5, 5, 1, 1, 1, 1, 7, 3, 2, 8, 4, 1, 6, 6, 9, 3, 1, 1, 3, 3, 7, 2, 2, 2, 5, 1, 10, 6, 11, 1, 1, 1, 1, 1, 7, 8, 7, 3, 3, 12, 9, 8, 4, 1, 1, 13, 7, 14, 6, 1, 5, 2, 10, 15, 3, 1, 1, 1, 4, 4, 4, 3, 9, 7, 1, 16, 3, 2, 2

Offset: 1

Gus Wiseman, Feb 28 2025

Row-lengths are A051903.
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.
The section-sum partition of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The prime indices of 24 are (2,1,1,1), with sections ((2,1),(1),(1)), so row 24 is (3,1,1).
Triangle begins:
   1: (empty)
   2: 1
   3: 2
   4: 1 1
   5: 3
   6: 3
   7: 4
   8: 1 1 1
   9: 2 2
  10: 4
  11: 5
  12: 3 1
  13: 6
  14: 5
  15: 5
  16: 1 1 1 1

Row-lengths are A051903.
Row sums are A056239.
First part in each row is A066328.
Taking length instead of sum gives A238744, Heinz numbers A238745, conjugate A181819.
Partitions of this type are counted by A239455, complement A351293.
Heinz numbers are A381431 (union A381432, complement A381433, fixed A000961, A000005).
Rows appearing only once have Heinz numbers A381434, more than once A381435.
Last part in each row is A381437, counted by A381438.
The conjugate is A381440, Heinz numbers A048767 (union A351294, complement A351295).
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A001222, A003557, A005117, A047966, A116861, A130091, A380955.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[egs[prix[n]],{n,100}]

A384317 Number of integer partitions of n with more than one possible way to choose disjoint strict partitions of each part.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 4, 4, 5, 5, 12, 12, 16, 19, 22, 35, 38, 48, 58, 68, 79, 110, 121, 149, 175, 207, 242, 281, 352, 397, 473

Offset: 0

Gus Wiseman, May 28 2025

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.

			There are two possibilities for (4,3), namely ((4),(3)) and ((4),(2,1)), so (4,3) is counted under a(7).
The a(3) = 1 through a(11) = 12 partitions:
  (3)  (4)  (5)  (6)    (7)    (8)    (9)    (10)     (11)
                 (3,3)  (4,3)  (4,4)  (5,4)  (5,5)    (6,5)
                 (4,2)  (5,2)  (5,3)  (6,3)  (6,4)    (7,4)
                 (5,1)  (6,1)  (6,2)  (7,2)  (7,3)    (8,3)
                               (7,1)  (8,1)  (8,2)    (9,2)
                                             (9,1)    (10,1)
                                             (4,3,3)  (5,3,3)
                                             (4,4,2)  (5,4,2)
                                             (5,3,2)  (5,5,1)
                                             (5,4,1)  (6,3,2)
                                             (6,3,1)  (7,3,1)
                                             (7,2,1)  (8,2,1)

The case of a unique choice is A179009, ranks A383707.
The case of at least one choice is A383708, ranks A382913.
The case of no choices is A383710, ranks A382912.
The strict case is A384318, ranks A384322.
These partitions are ranked by A384321, positions of terms > 1 in A383706.
The case of a unique proper choice is A384323, ranks A384347, strict A384319.
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 choices of strict partitions of prime indices, non-strict A299200.
Cf. A098859, A381454, A382525, A383533, A383711, A384320.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Length[pof[#]]>1&]],{n,0,30}]

a(n) = A383708(n) - A179009(n).

cp's OEIS Frontend

A382771 Number of ways to permute the prime indices of n so that the run-lengths are all different.

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A383512 Heinz numbers of conjugate Wilf partitions.

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A383707 Heinz numbers of maximally refined strict integer partitions.

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A384321 Numbers whose distinct prime indices are not maximally refined.

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A384886 Number of strict integer partitions of n with all equal lengths of maximal runs (decreasing by 1).

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A384884 Number of integer partitions of n with all distinct lengths of maximal gapless runs (decreasing by 0 or 1).

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A383533 Number of integer partitions of n with no ones such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

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A383711 Number of integer partitions of n with no ones such that it is not possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

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A381436 Irregular triangle read by rows where row k is the section-sum partition of the prime indices of n.

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