A383513 -id:A383513 - cp's OEIS frontend

A383512 Heinz numbers of conjugate Wilf partitions.

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]&]

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

A383530 Number of non Wilf and non conjugate Wilf integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 3, 2, 5, 12, 14, 19, 35, 38, 55, 83, 107, 137, 209, 252, 359, 462, 612, 757, 1032, 1266, 1649, 2050, 2617, 3210, 4111, 4980, 6262, 7659, 9479, 11484, 14224, 17132, 20962, 25259, 30693, 36744, 44517, 53043, 63850, 75955, 90943, 107721, 128485

Offset: 0

Gus Wiseman, May 14 2025

nonn

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 a(0) = 0 through a(9) = 12 partitions:
  .  .  .  (21)  .  .  (42)    (421)   (431)    (63)
                       (321)   (3211)  (521)    (432)
                       (2211)          (3221)   (531)
                                       (4211)   (621)
                                       (32111)  (3321)
                                                (4221)
                                                (4311)
                                                (5211)
                                                (32211)
                                                (42111)
                                                (222111)
                                                (321111)

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

Negating both sides gives A383507, ranks A383532.
These partitions are ranked by A383531.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A239455 counts Look-and-Say partitions, complement A351293.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A381431 is the section-sum transform, union A381432, complement A381433.
A383534 gives 0-prepended differences by rank, see A325351.
A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.
Cf. A033461, A047966, A111133, A320348, A325324, A325325, A325349, A325367, A325368, A325388, A383506.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]], {k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Length/@Split[#]&&!UnsameQ@@Length/@Split[conj[#]]&]], {n,0,30}]

These partitions have Heinz numbers A130092 /\ A383513.

A383531 Heinz numbers of integer partitions that do not have distinct multiplicities (Wilf) or distinct nonzero 0-appended differences (conjugate Wilf).

Original entry on oeis.org

6, 21, 30, 36, 42, 60, 65, 66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 132, 133, 138, 140, 150, 154, 156, 165, 168, 174, 180, 186, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246, 252, 258, 264, 270, 273, 276, 280, 282, 286, 294, 300, 306, 308

Offset: 1

Gus Wiseman, May 15 2025

nonn

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:
    6: {1,2}
   21: {2,4}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   60: {1,1,2,3}
   65: {3,6}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
  102: {1,2,7}
  105: {2,3,4}
  110: {1,3,5}
  114: {1,2,8}
  120: {1,1,1,2,3}

These partitions are counted by A383530.
Negating both sides gives A383532, counted by A383507.
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.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A122111 represents conjugation in terms of Heinz numbers.
A325324 counts integer partitions with distinct 0-appended differences, ranks A325367.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.
Cf. A001223, A047966, A181819, A238745, A320348, A325325, A325349, A325366, A325368, A325388, A381431, A383506.

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]}]];
Select[Range[100],!UnsameQ@@Length/@Split[prix[#]] && !UnsameQ@@Length/@Split[conj[prix[#]]]&]

Equals A130092 /\ A383513.

A384006 Heinz numbers of Look-and-Say partitions without distinct multiplicities (non Wilf).

Original entry on oeis.org

216, 1000, 1296, 2744, 3375, 7776, 9261, 10000, 10648, 17576, 32400, 35937, 38416, 38880, 39304, 42875, 46656, 50625, 54000, 54432, 54872, 59319, 63504, 81000, 85536, 90000, 97336, 100000

Offset: 1

Gus Wiseman, May 19 2025

nonn

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, complement A130092.
An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

			The terms together with their prime indices begin:
     216: {1,1,1,2,2,2}
    1000: {1,1,1,3,3,3}
    1296: {1,1,1,1,2,2,2,2}
    2744: {1,1,1,4,4,4}
    3375: {2,2,2,3,3,3}
    7776: {1,1,1,1,1,2,2,2,2,2}
    9261: {2,2,2,4,4,4}
   10000: {1,1,1,1,3,3,3,3}
   10648: {1,1,1,5,5,5}
   17576: {1,1,1,6,6,6}
   32400: {1,1,1,1,2,2,2,2,3,3}
   35937: {2,2,2,5,5,5}
   38416: {1,1,1,1,4,4,4,4}
   38880: {1,1,1,1,1,2,2,2,2,2,3}
   39304: {1,1,1,7,7,7}
   42875: {3,3,3,4,4,4}
   46656: {1,1,1,1,1,1,2,2,2,2,2,2}
   50625: {2,2,2,2,3,3,3,3}
   54000: {1,1,1,1,2,2,2,3,3,3}
   54432: {1,1,1,1,1,2,2,2,2,2,4}
   54872: {1,1,1,8,8,8}
   59319: {2,2,2,6,6,6}
   63504: {1,1,1,1,2,2,2,2,4,4}
   81000: {1,1,1,2,2,2,2,3,3,3}
   85536: {1,1,1,1,1,2,2,2,2,2,5}
   90000: {1,1,1,1,2,2,3,3,3,3}
   97336: {1,1,1,9,9,9}
  100000: {1,1,1,1,1,3,3,3,3,3}

Ranking sequences are shown in parentheses below.
These partitions are counted by A351592.
For section-sum instead of Look-and-Say we have (A383514), counted by A383506.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A383511 counts partitions that are Look-and-Say and section-sum but not Wilf (A383518).
Cf. A000720, A001223, A048767, A051903, A212166, A325368, A383520, A383531.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],disjointFamilies[prix[#]]!={}&&!UnsameQ@@Last/@FactorInteger[#]&]

A383509 Number of Look-and-Say partitions of n that are not section-sum partitions.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 3, 4, 4, 7, 9, 11, 18, 25, 30, 41, 55, 63, 87, 98, 125, 147, 192, 213, 271, 313, 389, 444, 551, 621, 767, 874, 1055, 1209, 1444, 1646, 1965, 2244, 2644, 2991

Offset: 0

Gus Wiseman, May 18 2025

A partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

A partition is section-sum iff its conjugate is Look-and-Say, meaning 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(11) = 9 partitions:
  211  221   21111  2221    422      22221     442        222221
       2111         22111   22211    222111    4222       322211
                    211111  221111   2211111   222211     332111
                            2111111  21111111  322111     422111
                                               2221111    2222111
                                               22111111   3221111
                                               211111111  22211111
                                                          221111111
                                                          2111111111
Conjugates of the a(4) = 1 through a(11) = 9 partitions:
  (3,1)  (3,2)  (5,1)  (4,3)  (5,3)      (5,4)  (6,4)      (6,5)
         (4,1)         (5,2)  (6,2)      (6,3)  (7,3)      (7,4)
                       (6,1)  (7,1)      (7,2)  (8,2)      (8,3)
                              (3,3,1,1)  (8,1)  (9,1)      (9,2)
                                                (6,3,1)    (10,1)
                                                (3,3,2,2)  (6,3,2)
                                                (4,4,1,1)  (6,4,1)
                                                           (7,3,1)
                                                           (6,3,1,1)

Ranking sequences are shown in parentheses below.
These partitions are ranked by (A383516).
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).
A351592 counts non Wilf Look-and-Say partitions (A384006).
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
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).
A383519 counts section-sum Wilf partitions (A383520).
Cf. A047966, A048767, A320348, A325324, A325325, A353837, A381431, A383511, A383530, A383709.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions /@ Length/@Split[y]],UnsameQ@@Join@@#&];
conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]], {k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n], disjointFamilies[#]!={}&&disjointFamilies[conj[#]]=={}&]], {n,0,30}]

A383515 Heinz numbers of integer partitions that are both Look-and-Say and section-sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 49, 50, 52, 53, 56, 59, 61, 64, 67, 68, 71, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 97, 98, 99, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121, 124, 125

Offset: 1

Gus Wiseman, May 18 2025

nonn

First differs from A383532 in having 325.
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 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.
An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  28: {1,1,4}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}

Ranking sequences are shown in parentheses below.
These partitions are counted by A383508.
A048767 is the Look-and-Say transform.
A048768 gives Look-and-Say fixed points, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
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).
A381431 is the section-sum transform.
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).
A383511 counts partitions that are Look-and-Say and section-sum but not Wilf (A383518).
Cf. A000720, A001223, A047966, A051903, A109298, A212166, A238745, A325368, A351592, A384006.

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

A383508 Number of integer partitions of n that are both Look-and-Say and section-sum partitions.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 22, 27, 30, 35, 42, 50, 58, 68, 82, 92, 112, 126, 149, 174, 199, 225, 263, 299, 337, 388, 435, 488, 545, 635, 681, 775, 841, 948, 1051, 1181, 1271, 1446, 1553, 1765, 1896, 2141, 2285, 2608, 2799

Offset: 0

Gus Wiseman, May 17 2025

nonn

An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

An integer partition is section-sum iff its conjugate is Look-and-Say, meaning 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(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (1111)  (11111)  (222)     (331)      (332)
                                     (411)     (511)      (611)
                                     (3111)    (4111)     (2222)
                                     (111111)  (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)

Ranking sequences are shown in parentheses below.
The non Wilf case is A383511 (A383518).
These partitions are ranked by (A383515).
A000041 counts integer partitions, strict A000009.
A047993 counts partitions with max part = length (A106529).
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
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).
A351592 counts non Wilf Look-and-Say partitions (A384006).
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. A033461, A047966, A048767, A320348, A325324, A325325, A381431, A383519, A383530, A383709.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions /@ Length/@Split[y]],UnsameQ@@Join@@#&];
conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n], disjointFamilies[#]!={}&&disjointFamilies[conj[#]]!={}&]], {n,0,30}]

A383510 Number of integer partitions of n that are neither Look-and-Say nor section-sum.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 3, 2, 5, 10, 14, 19, 33, 38, 55, 81, 107, 137, 201, 248, 349, 450, 596, 745, 1000, 1242, 1611, 2007, 2567, 3164, 4025, 4920, 6166, 7545, 9347, 11360, 14004, 16932, 20686, 24949, 30305, 36366, 43939, 52521, 63098, 75221

Offset: 0

Gus Wiseman, May 18 2025

nonn

An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

An integer partition is section-sum iff its conjugate is Look-and-Say, meaning 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(3) = 1 through a(10) = 14 partitions:
  (21)  .  .  (42)    (421)   (431)    (432)     (532)
              (321)   (3211)  (521)    (531)     (541)
              (2211)          (3221)   (621)     (721)
                              (4211)   (3321)    (4321)
                              (32111)  (4221)    (5221)
                                       (4311)    (5311)
                                       (5211)    (6211)
                                       (32211)   (32221)
                                       (42111)   (33211)
                                       (321111)  (42211)
                                                 (43111)
                                                 (52111)
                                                 (421111)
                                                 (3211111)

Ranking sequences are shown in parentheses below.
These partitions are ranked by (A383517).
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 (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A351592 counts non Wilf Look-and-Say partitions (A384006).
A381431 is the section-sum transform.
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).
A383519 counts section-sum Wilf partitions (A383520).
A383530 counts partitions that are neither Wilf nor conjugate Wilf (A383531).
Cf. A033461, A047966, A098859, A122111, A320348, A325324, A325325, A383511, A383709.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions /@ Length/@Split[y]],UnsameQ@@Join@@#&];
conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]], {k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n], disjointFamilies[#]=={} && disjointFamilies[conj[#]]=={}&]], {n,0,15}]

cp's OEIS Frontend

A383512 Heinz numbers of conjugate Wilf partitions.

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

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

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A383530 Number of non Wilf and non conjugate Wilf integer partitions of n.

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A383531 Heinz numbers of integer partitions that do not have distinct multiplicities (Wilf) or distinct nonzero 0-appended differences (conjugate Wilf).

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A384006 Heinz numbers of Look-and-Say partitions without distinct multiplicities (non Wilf).

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A383509 Number of Look-and-Say partitions of n that are not section-sum partitions.

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A383515 Heinz numbers of integer partitions that are both Look-and-Say and section-sum.

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A383508 Number of integer partitions of n that are both Look-and-Say and section-sum partitions.

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