A351293 -id:A351293 - cp's OEIS frontend

A386586 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k parts such that any multiset whose multiplicities are the parts of y is inseparable.

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 3, 2, 1, 0, 0, 0, 0, 0, 1, 3, 2, 1, 0, 0, 0, 0, 0, 0, 1, 4, 4, 2, 1, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Aug 05 2025

We say that such partitions are of inseparable type. This is different from inseparable partitions (see A386584). A multiset is separable iff it has a permutation without any adjacent equal parts.

			The partition y = (7,2,1) is the multiplicities of the multiset {1,1,1,1,1,1,1,2,2,3}, which is inseparable, so y is counted under T(10,3).
Row n = 10 counts the following partitions (A = 10):
  .  A  91  811  7111  61111  .  .  .  .  .
        82  721  6211
        73  631
        64  622
Triangle begins:
  0
  0 0
  0 1 0
  0 1 0 0
  0 1 1 0 0
  0 1 1 0 0 0
  0 1 2 1 0 0 0
  0 1 2 1 0 0 0 0
  0 1 3 2 1 0 0 0 0
  0 1 3 2 1 0 0 0 0 0
  0 1 4 4 2 1 0 0 0 0 0

This is the inseparable type case of A008284 or A072233.
Row sums shifted left once are A025065 (ranks A335126), separable version A336106 (ranks A335127).
For separable instead of inseparable type we have A386583.
For integer partitions instead of normal multisets we have A386584.
For separable type instead of inseparable type we have A386585.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A239455 counts Look-and-Say partitions, ranks A351294.
A325534 counts separable multisets, ranks A335433.
A325535 counts inseparable multisets, ranks A335448.
A336103 counts normal separable multisets, inseparable A336102.
A351293 counts non-Look-and-Say partitions, ranks A351295.
Cf. A000670, A005651, A106351, A238130, A279790, A386575, A386576, A386579, A386582, A386634.

insepQ[y_]:=Select[Permutations[y],Length[Split[#]]==Length[y]&]=={};
ptm[y_]:=Join@@Table[ConstantArray[k,y[[k]]],{k,Length[y]}];
Table[Length[Select[IntegerPartitions[n,{k}],insepQ[ptm[#]]&]],{n,0,5},{k,0,n}]

a(n) = A072233(n) - A386585(n).

A381440 Irregular triangle read by rows where row k is the Look-and-Say partition of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 28 2025

Row lengths are A066328.
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 Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.

			The prime indices of 24 are (2,1,1,1), with Look-and-Say partition (3,1,1), so row 24 is (3,1,1).
The prime indices of 36 are (2,2,1,1), with Look-and-Say partition (2,2,2), so row 36 is (2,2,2).
Triangle begins:
   1: (empty)
   2: 1
   3: 1 1
   4: 2
   5: 1 1 1
   6: 1 1 1
   7: 1 1 1 1
   8: 3
   9: 2 2
  10: 1 1 1 1
  11: 1 1 1 1 1
  12: 2 1 1
  13: 1 1 1 1 1 1
  14: 1 1 1 1 1
  15: 1 1 1 1 1
  16: 4
  17: 1 1 1 1 1 1 1
  18: 2 2 1
  19: 1 1 1 1 1 1 1 1

Heinz numbers are A048767 (union A351294, complement A351295, fixed A048768, A217605).
First part in each row is A051903, conjugate A066328.
Last part in each row is A051904, conjugate A381437 (counted by A381438).
Row sums are A056239.
Row lengths are A066328.
Partitions of this type are counted by A239455, complement A351293.
The conjugate is A381436, Heinz numbers A381431 (union A381432, complement A381433).
Rows appearing only once have Heinz numbers A381540, more than once A381541.
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, counted by A001222.
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, A003557, A047966, A071178, A091602, A116861, A130091, A181819, A212166, A238744, A380955.

Table[Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>ConstantArray[k,PrimePi[p]]]]//Reverse,{n,30}]

A384178 Number of strict integer partitions of n with all distinct lengths of maximal runs (decreasing by 1).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 8, 10, 11, 13, 13, 16, 15, 19, 19, 23, 22, 26, 28, 31, 35, 39, 37, 47, 51, 52, 60, 65, 67, 78, 85, 86, 99, 108, 110, 127, 136, 138, 159, 170, 171, 196, 209, 213, 240, 257, 260, 292, 306, 313, 350, 371, 369, 417, 441

Offset: 0

Gus Wiseman, Jun 12 2025

nonn

			The strict partition y = (9,7,6,5,2,1) has maximal runs ((9),(7,6,5),(2,1)), with lengths (1,3,2), so y is counted under a(30).
The a(1) = 1 through a(14) = 8 strict partitions (A-E = 10-14):
  1  2  3   4  5   6    7    8    9    A     B     C     D     E
        21     32  321  43   431  54   532   65    543   76    653
                        421  521  432  541   542   651   643   743
                                  621  721   632   732   652   761
                                       4321  821   921   832   932
                                             5321  6321  A21   B21
                                                         5431  5432
                                                         7321  8321

For subsets instead of strict partitions we have A384175, complement A384176.
For anti-runs instead of runs we have A384880.
This is the strict version of A384884.
For equal instead of distinct lengths we have A384886.
A000041 counts integer partitions, strict A000009.
A047993 counts partitions with max part = length.
A098859 counts Wilf partitions (complement A336866), 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. A008284, A044813, A325324, A325325, A329739, A351202.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Length/@Split[#,#1==#2+1&]&]],{n,0,30}]

A384318 Number of strict integer partitions of n that are not maximally refined.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 4, 4, 5, 9, 10, 13, 15, 17, 26, 29, 36, 43, 49, 57, 74, 84, 101, 118, 136, 158, 181, 219, 248, 291

Offset: 0

Gus Wiseman, May 28 2025

This is the number of strict integer partitions of n containing at least one sum of distinct non-parts.

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

			For y = (5,4,2) we have 4 = 3+1 so y is counted under a(11).
On the other hand, no part of z = (6,4,1) is a subset-sum of the non-parts {2,3,5}, so z is not counted under a(11).
The a(3) = 1 through a(11) = 10 strict partitions:
  (3)  (4)  (5)  (6)    (7)    (8)    (9)    (10)     (11)
                 (4,2)  (4,3)  (5,3)  (5,4)  (6,4)    (6,5)
                 (5,1)  (5,2)  (6,2)  (6,3)  (7,3)    (7,4)
                        (6,1)  (7,1)  (7,2)  (8,2)    (8,3)
                                      (8,1)  (9,1)    (9,2)
                                             (5,3,2)  (10,1)
                                             (5,4,1)  (5,4,2)
                                             (6,3,1)  (6,3,2)
                                             (7,2,1)  (7,3,1)
                                                      (8,2,1)

The strict complement is A179009, ranks A383707.
The non-strict version for at least one choice is A383708, for none A383710.
The non-strict version is A384317, ranks A384321, complement A384392, ranks A384320.
These partitions are ranked by A384322.
For subsets instead of partitions we have A384350, complement A326080.
Cf. A357982, A383706 (disjoint), A384319, A384323 (non-strict).
Cf. A048767, A098859, A179822, A239455, A279375, A317142, A351293, A382525, A383533, A383711, A384391.

nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,Total/@nonsets[#]]!={}&]],{n,0,30}]

a(n) = A000009(n) - A179009(n).

A351592 Number of Look-and-Say partitions (A239455) of n without distinct multiplicities, i.e., those that are not Wilf partitions (A098859).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 5, 2, 8, 9, 8, 6, 21, 14, 20, 26, 31, 24, 53, 35, 60, 68, 78, 76, 140, 115, 163, 183, 232, 218, 343, 301, 433, 432, 565, 542, 774, 728, 958, 977, 1251, 1220, 1612, 1561, 2053, 2090, 2618, 2609, 3326, 3378

Offset: 0

Gus Wiseman, Feb 16 2022

nonn

A partition is Look-and-Say iff it has a permutation with all distinct run-lengths. For example, the partition y = (2,2,2,1,1,1) has the permutation (2,2,1,1,1,2), with run-lengths (2,3,1), which are distinct, so y is counted under A239455(9).
A partition is Wilf iff it has distinct multiplicities of parts. For example, (2,2,2,1,1,1) has multiplicities (3,3), so is not counted under A098859(9).
The Heinz numbers of these partitions are given by A351294 \ A130091.
Is a(17) = 0 the last zero of the sequence?

			The a(9) = 1 through a(18) = 5 partitions are (empty columns not shown):
  n=9:      n=12:       n=15:         n=16:       n=18:
  --------------------------------------------------------------
  (222111)  (333111)    (333222)      (33331111)  (444222)
            (22221111)  (444111)                  (555111)
                        (2222211111)              (3322221111)
                                                  (32222211111)
                                                  (222222111111)

Wilf partitions are counted by A098859, ranked by A130091.
Look-and-Say partitions are counted by A239455, ranked by A351294.
Non-Wilf partitions are counted by A336866, ranked by A130092.
Non-Look-and-Say partitions are counted by A351293, ranked by A351295.
A000569 = number of graphical partitions, complement A339617.
A032020 = number of binary expansions with all distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A225485/A325280 = frequency depth, ranked by A182850/A323014.
A329738 = compositions with all equal run-lengths.
A329739 = compositions with all distinct run-lengths
A351013 = compositions with all distinct runs.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
A351292 = patterns with all distinct run-lengths, for all runs A351200.
Cf. A000041, A008284, A018783, A047966, A181819, A182857, A238130, A305563, A319149, A351203, A351204, A351290.

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Length/@Split[#]&&Select[Permutations[#], UnsameQ@@Length/@Split[#]&]!={}&]],{n,0,15}]

a(n) = A239455(n) - A098859(n). Here we assume A239455(0) = 1.

More terms from Jinyuan Wang, Feb 14 2025

A381437 Last part of the section-sum partition of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 4, 1, 2, 4, 5, 1, 6, 5, 5, 1, 7, 2, 8, 1, 6, 6, 9, 1, 3, 7, 2, 1, 10, 6, 11, 1, 7, 8, 7, 3, 12, 9, 8, 1, 13, 7, 14, 1, 2, 10, 15, 1, 4, 3, 9, 1, 16, 2, 8, 1, 10, 11, 17, 1, 18, 12, 2, 1, 9, 8, 19, 1, 11, 8, 20, 1, 21, 13, 3, 1, 9, 9, 22, 1, 2

Offset: 1

Gus Wiseman, Feb 28 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.
The section-sum partition (A381436) 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 972 are {1,1,2,2,2,2,2}, with section-sum partition (3,3,2,2,2), so a(972) = 2.

Positions of first appearances are A008578.
The length of this partition is A051903.
The conjugate version is A051904.
For first instead of last part we get A066328.
These partitions are counted by A239455, complement A351293.
Positions of 1 are A360013, complement A381439.
This is the least prime index of A381431 (see A381432, A381433, A381434, A381435).
This is the last part of row n of A381436 (see A381440, A048767, A351294, A351295).
Counting partitions by this statistic gives A381438.
A000040 lists the primes, differences A001223.
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.
Cf. A000720, A001221, A001222, A003557, A047966, A116861, A130091, A181819, 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[If[n==1,0,Last[egs[prix[n]]]],{n,100}]

a(n) = A055396(A381431(n)).

A382915 Number of integer partitions of n having no permutation with all equal run-lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 4, 4, 9, 11, 18, 21, 34, 41, 55, 69, 98, 120, 160, 189, 249, 309, 396, 472, 605, 734, 913, 1099, 1371, 1632, 2021, 2406, 2937, 3514, 4251, 5039, 6101, 7221, 8646, 10205, 12209, 14347, 17086, 20041, 23713, 27807, 32803, 38262, 45043, 52477, 61471, 71496

Offset: 0

Gus Wiseman, Apr 12 2025

nonn

			The partition y = (2,2,1,1,1) has permutations and run-lengths:
  (2,2,1,1,1) (2,3)
  (2,1,2,1,1) (1,1,1,2)
  (2,1,1,2,1) (1,2,1,1)
  (2,1,1,1,2) (1,3,1)
  (1,2,2,1,1) (1,2,2)
  (1,2,1,2,1) (1,1,1,1,1)
  (1,2,1,1,2) (1,1,2,1)
  (1,1,2,2,1) (2,2,1)
  (1,1,2,1,2) (2,1,1,1)
  (1,1,1,2,2) (3,2)
Since (1,2,1,2,1) has all equal run-lengths (1,1,1,1,1), y is not counted under a(7).
The a(5) = 1 through a(10) = 11 partitions:
  (2111)  (3111)   (2221)    (5111)     (3222)      (3331)
          (21111)  (4111)    (41111)    (6111)      (4222)
                   (31111)   (311111)   (22221)     (7111)
                   (211111)  (2111111)  (51111)     (61111)
                                        (321111)    (421111)
                                        (411111)    (511111)
                                        (2211111)   (3211111)
                                        (3111111)   (4111111)
                                        (21111111)  (22111111)
                                                    (31111111)
                                                    (211111111)

The complement for distinct run-lengths is A239455, ranked by A351294.
For distinct instead of equal run-lengths we have A351293, ranked by A351295.
These partitions are ranked by A382879, by signature A382914.
The complement is counted by A383013.
A000041 counts integer partitions, strict A000009.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A329738 counts compositions with equal run-lengths, ranks A353744.
A382857 counts permutations of prime indices with equal run-lengths.
Cf. A003242, A047966, A238279, A329739, A351201, A351290, A351596, A382773.

Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],SameQ@@Length/@Split[#]&]=={}&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

A384390 Heinz numbers of integer partitions with a unique proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

5, 7, 21, 22, 26, 33, 35, 39, 102, 114, 130, 154, 165, 170, 190, 195, 231, 238, 255, 285

Offset: 1

Gus Wiseman, Jun 02 2025

nonn

By "proper" we exclude the case of all singletons, which is disjoint in the strict case.

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.

			The strict partition (7,2,1) with Heinz number 102 can only be properly refined as ((4,3),(2),(1)), so 102 is in the sequence. The other refinement ((7),(2),(1)) is not proper.
The terms together with their prime indices begin:
    5: {3}
    7: {4}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   33: {2,5}
   35: {3,4}
   39: {2,6}
  102: {1,2,7}
  114: {1,2,8}
  130: {1,3,6}
  154: {1,4,5}
  165: {2,3,5}
  170: {1,3,7}
  190: {1,3,8}
  195: {2,3,6}
  231: {2,4,5}
  238: {1,4,7}
  255: {2,3,7}
  285: {2,3,8}

The non-proper version is A383707, counted by A179009.
Partitions of this type are counted by A384319, non-strict A384323 (ranks A384347).
This is the unique case of A384321, counted by A384317.
This is the case of a unique proper choice in A384322.
The complement is A384349 \/ A384393.
These are positions of 1 in A384389.
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.
A357982 counts strict partitions of each prime index, non-strict A299200.
Cf. A382912, counted by A383710, odd case A383711.
Cf. A382913, counted by A383708, odd case A383533.
Cf. A098859, A122111, A130091, A279375, A279790, A317142, A351201, A381454, A384005, A384320.

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&]

A383506 Number of non Wilf section-sum partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 3, 4, 4, 7, 9, 12, 18, 25, 32, 42, 55, 64, 87, 101, 128, 147, 192, 218, 273, 314, 394, 450, 552, 631, 772, 886, 1066, 1221, 1458, 1677, 1980, 2269, 2672, 3029

Offset: 0

Gus Wiseman, May 18 2025

An integer partition is Wilf iff its multiplicities are all different, ranked by A130091.

An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

			The a(4) = 1 through a(12) = 12 partitions (A=10, B=11):
  (31)  (32)  (51)  (43)  (53)    (54)  (64)    (65)    (75)
        (41)        (52)  (62)    (63)  (73)    (74)    (84)
                    (61)  (71)    (72)  (82)    (83)    (93)
                          (3311)  (81)  (91)    (92)    (A2)
                                        (631)   (A1)    (B1)
                                        (3322)  (632)   (732)
                                        (4411)  (641)   (831)
                                                (731)   (5511)
                                                (6311)  (6411)
                                                        (7311)
                                                        (63111)
                                                        (333111)

Ranking sequences are shown in parentheses below.
For Look-and-Say instead of section-sum we have A351592 (A384006).
The Look-and-Say case is A383511 (A383518).
These partitions are ranked by (A383514).
For Wilf instead of non Wilf we have A383519 (A383520).
A000041 counts integer partitions, strict A000009.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
Cf. A047966, A048767, A122111, A320348, A325324, A325325, A353837, A381431, A383530, A383709.

disjointDiffs[y_]:=Select[Tuples[IntegerPartitions /@ Differences[Prepend[Sort[y],0]]], UnsameQ@@Join@@#&];
Table[Length[Select[IntegerPartitions[n], disjointDiffs[#]!={} && !UnsameQ@@Length/@Split[#]&]],{n,0,15}]

A383709 Number of integer partitions of n with distinct multiplicities (Wilf) and distinct 0-appended differences.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 4, 4, 4, 5, 6, 5, 7, 8, 6, 8, 9, 9, 10, 9, 10, 12, 12, 11, 12, 14, 13, 14, 15, 14, 16, 16, 16, 18, 17, 17, 19, 20, 19, 19, 21, 21, 22, 22, 21, 24, 24, 23, 25, 25, 25, 26, 27, 27, 27, 28, 28, 30, 30, 28, 31, 32, 31, 32, 32, 33, 34, 34, 34

Offset: 0

Gus Wiseman, May 15 2025

nonn

Integer partitions with distinct multiplicities are called Wilf partitions.

			The a(1) = 1 through a(8) = 4 partitions:
  (1)  (2)    (3)  (4)    (5)      (6)      (7)      (8)
       (1,1)       (2,2)  (3,1,1)  (3,3)    (3,2,2)  (4,4)
                                   (4,1,1)  (3,3,1)  (3,3,2)
                                            (5,1,1)  (6,1,1)

For just distinct multiplicities we have A098859, ranks A130091, conjugate A383512.
For just distinct 0-appended differences we have A325324, ranks A325367.
For positive differences we have A383507, ranks A383532.
These partitions are ranked by A383712.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A239455 counts Look-and-Say partitions, complement A351293.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383530 counts partitions that are not Wilf or conjugate-Wilf, ranks A383531.
A383534 gives 0-prepended differences by rank, see A325351.
Cf. A047966, A320348, A325325, A325349, A325368, A325388, A381431, A383506.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#]&&UnsameQ@@Differences[Append[#,0]]&]],{n,0,30}]

Ranked by A130091 /\ A325367

cp's OEIS Frontend

A386586 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k parts such that any multiset whose multiplicities are the parts of y is inseparable.

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A381440 Irregular triangle read by rows where row k is the Look-and-Say partition of the prime indices of n.

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A384178 Number of strict integer partitions of n with all distinct lengths of maximal runs (decreasing by 1).

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A384318 Number of strict integer partitions of n that are not maximally refined.

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A351592 Number of Look-and-Say partitions (A239455) of n without distinct multiplicities, i.e., those that are not Wilf partitions (A098859).

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A381437 Last part of the section-sum partition of the prime indices of n.

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A382915 Number of integer partitions of n having no permutation with all equal run-lengths.

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A384390 Heinz numbers of integer partitions with a unique proper way to choose disjoint strict partitions of each part.

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A383506 Number of non Wilf section-sum partitions of n.

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