A351295 -id:A351295 - cp's OEIS frontend

A384321 Numbers whose distinct prime indices are not maximally refined.

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]]!={}]&]

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}]

A383100 Numbers whose prime indices have no permutation with all equal run-sums.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 45, 46, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108

Offset: 1

Gus Wiseman, Apr 20 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, sum A056239.

			The prime indices of 18 are {1,2,2}, with permutations (1,2,2), (2,1,2), (2,2,1), with run sums (1,4), (2,1,2), (4,1) respectively, so 18 is in the sequence.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   46: {1,9}
   50: {1,3,3}

For distinct instead of equal run-sums we appear to have A381636, counted by A381717.
For run-lengths instead of sums we have A382879, counted by complement of A383013.
These are the positions of 0 in A382877.
For more than one choice we have A383015.
The complement is A383110, counted by A383098.
Partitions of this type are counted by A383096.
For a unique choice we have A383099, counted by A383095.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A353851 counts compositions with equal run-sums, ranks A353848.
Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.
Cf. A351294, A351295, A353832, A353837, A353838, A354584, A381871, A382857, A382876, A383094, A383097.

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]==0&]

A386583 Triangle read by rows where T(n,k) is the number of length k integer partitions of n having a permutation without any adjacent equal parts (separable).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 2, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 1, 3, 4, 1, 1, 0, 0, 0, 1, 3, 5, 3, 2, 0, 0, 0, 0, 1, 4, 6, 4, 3, 1, 0, 0, 0, 0, 1, 4, 8, 6, 5, 1, 1, 0, 0, 0, 0, 1, 5, 10, 8, 8, 3, 2, 0, 0, 0, 0, 0, 1, 5, 11, 12, 11, 5, 3, 1, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Aug 03 2025

A multiset is separable iff it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
Separable partitions (A325534) are different from partitions of separable type (A386585).
Are the rows all unimodal?
Some rows are not unimodal: T(200, k=26..30) = 149371873744, 153304102463, 152360653274, 152412869411, 147228477998. - Alois P. Heinz, Aug 04 2025

			Row n = 9 counts the following partitions:
  (9)  (5,4)  (4,3,2)  (3,3,2,1)  (3,2,2,1,1)  (2,2,2,1,1,1)
       (6,3)  (4,4,1)  (4,2,2,1)  (3,3,1,1,1)
       (7,2)  (5,2,2)  (4,3,1,1)  (4,2,1,1,1)
       (8,1)  (5,3,1)  (5,2,1,1)
              (6,2,1)
              (7,1,1)
Triangle begins:
  1
  0  1
  0  1  0
  0  1  1  0
  0  1  1  1  0
  0  1  2  2  0  0
  0  1  2  2  1  0  0
  0  1  3  4  1  1  0  0
  0  1  3  5  3  2  0  0  0
  0  1  4  6  4  3  1  0  0  0
  0  1  4  8  6  5  1  1  0  0  0
  0  1  5 10  8  8  3  2  0  0  0  0
  0  1  5 11 12 11  5  3  1  0  0  0  0
  0  1  6 14 14 15  8  6  1  1  0  0  0  0
  0  1  6 16 19 20 11  9  3  2  0  0  0  0  0
  0  1  7 18 23 27 17 14  5  3  1  0  0  0  0  0
  0  1  7 21 29 34 23 20  9  6  1  1  0  0  0  0  0
  0  1  8 24 34 43 32 28 13 10  3  2  0  0  0  0  0  0
  0  1  8 26 42 53 42 38 20 15  5  3  1  0  0  0  0  0  0
  0  1  9 30 48 66 55 52 28 23  9  6  1  1  0  0  0  0  0  0
  0  1  9 33 58 80 70 68 41 33 14 10  3  2  0  0  0  0  0  0  0
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Separable case of A008284.
Row sums are A325534, ranked by A335433.
For inseparable instead separable we have A386584, sums A325535, ranks A335448.
For separable type instead of separable we have A386585, sums A336106, ranks A335127.
For inseparable type instead of separable we have A386586, sums A025065, ranks A335126.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A124762 gives inseparability of standard compositions, separability A333382.
A239455 counts Look-and-Say partitions, ranks A351294.
A336103 counts normal separable multisets, inseparable A336102.
A351293 counts non-Look-and-Say partitions, ranks A351295.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A106351, A111133, A238130, A335434, A386575, A386576, A386580, A386581, A386577.

sepQ[y_]:=Select[Permutations[y],Length[Split[#]]==Length[y]&]!={};
Table[Length[Select[IntegerPartitions[n,{k}],sepQ]],{n,0,15},{k,0,n}]

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).

A384322 Heinz numbers of strict integer partitions with more than one possible way to choose disjoint strict partitions of each part, i.e., strict partitions that can be properly refined.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 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, 122

Offset: 1

Gus Wiseman, Jun 01 2025

nonn

			The strict partition (7,2,1) with Heinz number 102 can be properly refined into (4,3,2,1), so 102 is in the sequence.
The terms together with their prime indices begin:
     5: {3}      46: {1,9}      85: {3,7}
     7: {4}      47: {15}       86: {1,14}
    11: {5}      51: {2,7}      87: {2,10}
    13: {6}      53: {16}       89: {24}
    17: {7}      55: {3,5}      91: {4,6}
    19: {8}      57: {2,8}      93: {2,11}
    21: {2,4}    58: {1,10}     94: {1,15}
    22: {1,5}    59: {17}       95: {3,8}
    23: {9}      61: {18}       97: {25}
    26: {1,6}    62: {1,11}    101: {26}
    29: {10}     65: {3,6}     102: {1,2,7}
    31: {11}     67: {19}      103: {27}
    33: {2,5}    69: {2,9}     106: {1,16}
    34: {1,7}    71: {20}      107: {28}
    35: {3,4}    73: {21}      109: {29}
    37: {12}     74: {1,12}    111: {2,12}
    38: {1,8}    77: {4,5}     113: {30}
    39: {2,6}    79: {22}      114: {1,2,8}
    41: {13}     82: {1,13}    115: {3,9}
    43: {14}     83: {23}      118: {1,17}

The non-strict version for no choices appears to be A382912, count A383710, odd A383711.
The non-strict version for > 0 choice appears to be A382913, count A383708, odd A383533.
These are the squarefree positions of terms > 1 in A383706, see A357982, A299200.
The case of a unique choice is A383707, counted by A179009.
Partitions of this type are counted by A384318.
This is the strict/squarefree case of A384321, counted by A384317.
The case of a unique proper choice is A384390, counted by A384319, non-strict A384323.
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, A381454, A382525, A384005, A384320, A384347.

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],UnsameQ@@prix[#]&&Length[pof[prix[#]]]>1&]

A384880 Number of strict integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 6, 6, 9, 10, 12, 15, 18, 21, 25, 30, 34, 41, 46, 55, 63, 75, 85, 99, 114, 133, 152, 178, 201, 236, 269, 308, 352, 404, 460, 525, 594, 674, 763, 865, 974, 1098, 1236, 1385, 1558, 1745, 1952, 2181, 2435, 2712, 3026, 3363, 3740, 4151, 4612

Offset: 0

Gus Wiseman, Jun 13 2025

nonn

			The strict partition y = (10,7,6,4,2,1) has maximal anti-runs ((10,7),(6,4,2),(1)), with lengths (2,3,1), so y is counted under a(30).
The a(1) = 1 through a(14) = 18 partitions (A-E = 10-14):
  1  2  3  4   5   6   7    8    9    A    B    C    D     E
           31  41  42  52   53   63   64   74   75   85    86
                   51  61   62   72   73   83   84   94    95
                       421  71   81   82   92   93   A3    A4
                            431  531  91   A1   A2   B2    B3
                            521  621  532  542  B1   C1    C2
                                      541  632  642  643   D1
                                      631  641  651  652   653
                                      721  731  732  742   743
                                           821  741  751   752
                                                831  832   761
                                                921  841   842
                                                     931   851
                                                     A21   932
                                                     6421  941
                                                           A31
                                                           B21
                                                           7421

For subsets instead of strict partitions we have A384177.
For runs instead of anti-runs we have A384178.
This is the strict case of A384885.
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. A008289, A089259, A325324, A325325, A329739, A357982, A384175, A384176, A384884, A384886.

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

A386585 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k = 0..n parts such that any multiset whose multiplicities are the parts of y is separable.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 2, 2, 1, 1, 0, 0, 1, 3, 3, 2, 1, 1, 0, 0, 1, 3, 4, 3, 2, 1, 1, 0, 0, 1, 5, 5, 5, 3, 2, 1, 1, 0, 0, 1, 4, 7, 6, 5, 3, 2, 1, 1

Offset: 0

Gus Wiseman, Aug 02 2025

We say that such partitions are of separable type.

A multiset is separable iff it has a permutation without any adjacent equal parts.

			Row n = 8 counts the following partitions:
  .  .  44  431  4211  41111  311111  2111111  11111111
            422  3311  32111  221111
            332  3221  22211
                 2222
with the following separable multisets:
  . . 11112222 11112223 11112234 11112345 11123456 11234567 12345678
               11112233 11122234 11122345 11223456
               11122233 11122334 11223345
                        11223344
Triangle begins:
  1
  0  1
  0  0  1
  0  0  1  1
  0  0  1  1  1
  0  0  1  2  1  1
  0  0  1  2  2  1  1
  0  0  1  3  3  2  1  1
  0  0  1  3  4  3  2  1  1
  0  0  1  5  5  5  3  2  1  1
  0  0  1  4  7  6  5  3  2  1  1

This is the separable type case of A072233 or A008284.
Row sums are A336106, ranks A335127.
For separable instead of separable type we have A386583, inseparable A386584.
For inseparable instead of separable we have A386586, sums A025065, ranks A335126.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A239455 counts Look-and-Say partitions, ranks A351294.
A279790 counts disjoint families on strongly normal multisets.
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.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A005651, A106351, A111133, A238130, A335434, A386575, A386576, A386579, A386580, A386581, A386582.

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

a(n) = A072233(n) - A386586(n).

cp's OEIS Frontend

A384321 Numbers whose distinct prime indices are not maximally refined.

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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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A383100 Numbers whose prime indices have no permutation with all equal run-sums.

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A386583 Triangle read by rows where T(n,k) is the number of length k integer partitions of n having a permutation without any adjacent equal parts (separable).

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

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A384322 Heinz numbers of strict integer partitions with more than one possible way to choose disjoint strict partitions of each part, i.e., strict partitions that can be properly refined.

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A384880 Number of strict integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).

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