A381433 -id:A381433 - cp's OEIS frontend

A383094 Number of integer partitions of n having exactly one permutation with all equal run-lengths.

1, 1, 2, 2, 4, 4, 5, 6, 9, 7, 11, 10, 13, 12, 17, 14, 21, 16, 21, 18, 27, 22, 29, 22, 34, 25, 35, 28, 41, 28, 43, 30, 48, 38, 47, 38, 55, 36, 53, 46, 64, 40, 67, 42, 69, 54, 65, 46, 84, 51, 75, 62, 83, 52, 86, 62, 94, 70, 83, 58, 111, 60, 89, 80, 106, 74, 115, 66, 111

Offset: 0

Gus Wiseman, Apr 20 2025

nonn

			The partition (222211) has exactly one permutation with all equal run-lengths: (221122), so is counted under a(10).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (11111)  (411)     (511)      (422)
                                     (111111)  (22111)    (611)
                                               (1111111)  (2222)
                                                          (22211)
                                                          (221111)
                                                          (11111111)

The complement is ranked by A382879 \/ A383089.
For no choices we have A382915, ranks A382879.
For at least one choice we have A383013, for run-sums A383098, ranks A383110.
For more than one choice we have A383090, ranks A383089.
For at most one choice we have A383092, ranks A383091.
For run-sums instead of lengths we have A383095, ranks A383099.
Partitions of this type are ranked by A383112 = positions of 1 in A382857.
Cf. A047966, A072774, A098859, A100471, A100881, A100882, A100883, A304442.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A047993, A362608, A381871, A382076, A382771, A383096, A383097, A383111.

Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Length/@Split[#]&]]==1&]],{n,0,20}]

More terms from Bert Dobbelaere, Apr 26 2025

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[#]]]!={}&]

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

Gus Wiseman, May 27 2025

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.

			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}

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.

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&]

A386575 Number of distinct separable and pairwise disjoint sets of strict integer partitions, one of each exponent in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 30 2025

nonn

A set partition is separable iff the underlying set has a permutation whose adjacent elements all belong to different blocks. Note that separability only depends on the sizes of the blocks.

Conjecture: a(n) > 0 iff the multiset of prime factors of n has a permutation with all distinct run lengths.

			The prime indices of 6144 are {1,1,1,1,1,1,1,1,1,1,1,2}, and we have the following a(6144) = 5 choices: {{1},{11}}, {{1},{5,6}}, {{1},{4,7}}, {{1},{3,8}}, {{1},{2,9}}. The other 2 disjoint families {{1},{2,4,5}} and {{1},{2,3,6}} are not separable.
The prime indices of 7776 are {1,1,1,1,1,2,2,2,2,2}, with separable disjoint families {{5},{2,3}}, {{5},{1,4}}, {{1,4},{2,3}}, so a(7776) = 3.
The prime indices of 15552 are {1,1,1,1,1,1,2,2,2,2,2}, with a(15552) = 5 choices: {{5},{6}}, {{5},{2,4}}, {{6},{2,3}}, {{6},{1,4}}, {{1,5},{2,3}}. The other disjoint family {{5},{1,2,3}} is not separable.
The a(n) families for n = 2, 96, 384, 1536, 3456, 20736:
  {{1}}  {{1},{5}}    {{1},{7}}    {{1},{9}}    {{3},{7}}      {{4},{8}}
         {{1},{2,3}}  {{1},{2,5}}  {{1},{2,7}}  {{3},{1,6}}    {{4},{1,7}}
                      {{1},{3,4}}  {{1},{3,6}}  {{3},{2,5}}    {{4},{2,6}}
                                   {{1},{4,5}}  {{7},{1,2}}    {{4},{3,5}}
                                                {{1,2},{3,4}}  {{8},{1,3}}
                                                               {{1,3},{2,6}}

Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433.
For inseparable instead of separable we have A386582, see A386632.
This is the separable case of A386587 (ordered version A382525).
A000110 counts set partitions, ordered A000670.
A003242 and A335452 count separations, ranks A333489.
A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
A239455 counts Look-and-Say partitions, complement A351293.
A279790 counts disjoint families on strongly normal multisets.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A336106 counts partitions of separable type, ranks A335127, sums of A386585.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A001221, A001222, A008480, A051903, A051904, A056239, A130091, A373957, A386580, A386581.

disjointFamilies[y_]:=Union[Sort/@Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
seps[ptn_,fir_]:=If[Total[ptn]==1,{{fir}},Join@@Table[Prepend[#,fir]&/@seps[MapAt[#-1&,ptn,fir],nex],{nex,Select[DeleteCases[Range[Length[ptn]],fir],ptn[[#]]>0&]}]];
seps[ptn_]:=If[Total[ptn]==0,{{}},Join@@(seps[ptn,#]&/@Range[Length[ptn]])];
Table[Length[Select[disjointFamilies[prix[n]],seps[Length/@#]!={}&]],{n,100}]

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

Gus Wiseman, Aug 12 2025

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

			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)

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.

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}]

A381434 Numbers appearing only once in A381431 (section-sum partition of prime indices).

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 10, 14, 15, 16, 20, 22, 27, 28, 32, 33, 35, 40, 44, 45, 50, 55, 56, 64, 75, 77, 80, 81, 88, 98, 99, 100, 112, 128, 130, 135, 160, 170, 175, 176, 182, 190, 195, 196, 200

Offset: 1

Gus Wiseman, Feb 27 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.
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 terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   20: {1,1,3}
   22: {1,5}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}

In A381431:
- fixed points are A000961, A000005
- conjugate is A048767, fixed points A048768, A217605
- all numbers present are A381432, conjugate A351294
- numbers missing are A381433, conjugate A351295
- numbers appearing only once are A381434 (this), conjugate A381540
- numbers appearing more than once are A381435, conjugate A381541
A000040 lists the primes, differences A001223.
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 section-sum partitions, complement A351293.
A381436 lists section-sum partition of prime indices, conjugate A381440.
Set multipartitions: A050320, A089259, A116540, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A051903, A066328, A116861, A130091, A212166, A239964, A317081, A381437.

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]}]];
Select[Range[100],Count[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]==1&]

The complement is A381433 U A381435.

A381438 Triangle read by rows where T(n>0,k>0) is the number of integer partitions of n whose section-sum partition ends with k.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 2, 1, 0, 2, 3, 1, 0, 0, 3, 4, 1, 2, 0, 0, 4, 7, 2, 1, 0, 0, 0, 5, 9, 4, 1, 2, 0, 0, 0, 6, 13, 4, 4, 1, 0, 0, 0, 0, 8, 18, 6, 3, 2, 3, 0, 0, 0, 0, 10, 26, 9, 5, 2, 2, 0, 0, 0, 0, 0, 12, 32, 12, 8, 4, 2, 4, 0, 0, 0, 0, 0, 15

Offset: 1

Gus Wiseman, Mar 01 2025

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.

			Triangle begins:
   1
   1  1
   1  0  2
   2  1  0  2
   3  1  0  0  3
   4  1  2  0  0  4
   7  2  1  0  0  0  5
   9  4  1  2  0  0  0  6
  13  4  4  1  0  0  0  0  8
  18  6  3  2  3  0  0  0  0 10
  26  9  5  2  2  0  0  0  0  0 12
  32 12  8  4  2  4  0  0  0  0  0 15
  47 16 11  4  3  2  0  0  0  0  0  0 18
  60 23 12  8  3  2  5  0  0  0  0  0  0 22
  79 27 20  7  9  4  3  0  0  0  0  0  0  0 27
 Row n = 9 counts the following partitions:
  (711)        (522)    (333)     (441)  .  .  .  .  (9)
  (6111)       (4221)   (3321)                       (81)
  (5211)       (3222)   (32211)                      (72)
  (51111)      (22221)  (222111)                     (63)
  (4311)                                             (621)
  (42111)                                            (54)
  (411111)                                           (531)
  (33111)                                            (432)
  (321111)
  (3111111)
  (2211111)
  (21111111)
  (111111111)

Last column (k=n) is A000009.
Row sums are A000041.
Row sums without the last column (k=n) are A047967.
For first instead of last part we have A116861, rank A066328.
First column (k=1) is A241131 shifted right and starting with 1 instead of 0.
Using Heinz numbers, this statistic is given by A381437.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions, complement A351293.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Section-sum partition: A381431, A381432, A381433, A381434, A381435, A381436.
Look-and-Say partition: A048767, A351294, A351295, A381440.
Cf. A047966, A051903, A051904, A091602, A181819, A212166.

egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[Length[Select[IntegerPartitions[n],k==Last[egs[#]]&]],{n,15},{k,n}]

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}]

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A383094 Number of integer partitions of n having exactly one permutation with all equal run-lengths.

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

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A386575 Number of distinct separable and pairwise disjoint sets of strict integer partitions, one of each exponent in the prime factorization of n.

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

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A381434 Numbers appearing only once in A381431 (section-sum partition of prime indices).

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A381438 Triangle read by rows where T(n>0,k>0) is the number of integer partitions of n whose section-sum partition ends with k.

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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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