A386632 -id:A386632 - cp's OEIS frontend

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

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}]

A386587 Number of ways to choose a pairwise disjoint family 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, 2, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 3, 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, 4, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Aug 06 2025

nonn

First differs from A382525 at a(216) = 1, A382525(216) = 2.

			The prime exponents of 864 = 2^5 * 3^3 are (5,3), with disjoint families {{3},{5}}, {{3},{1,4}}, {{5},{1,2}}, so a(864) = 3.

Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433.
For ordered set partitions we have A382525.
Positions of first appearances are A382775.
The separable case is A386575.
The inseparable case is A386582, see A386632.
A000110 counts set partitions, ordered A000670.
A003242 and A335452 count separations, ranks A333489.
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.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A001221, A001222, A008480, A025065, A051903, A051904, A056239, A130091, A336106, A373957, A386580.

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}]]]];
Table[Length[disjointFamilies[prix[n]]],{n,100}]

A386638 Number of integer partitions of n of inseparable type.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 19, 19, 30, 30, 45, 45, 67, 67, 97, 97, 139, 139, 195, 195, 272, 272, 373, 373, 508, 508, 684, 684, 915, 915, 1212, 1212, 1597, 1597, 2087, 2087, 2714, 2714, 3506, 3506, 4508, 4508, 5763, 5763, 7338, 7338, 9296, 9296

Offset: 0

Gus Wiseman, Aug 14 2025

nonn

A multiset is inseparable iff it has no permutation without adjacent equal parts. It is of inseparable type iff any multiset with those multiplicities (type) is inseparable. For example, {1,1,2} is separable but {1,1,1,2} is not; hence (2,1) is of separable type but (3,1) is not.

Also the number of integer partitions of n whose greatest part is at least two more than the sum of all the other parts.

			The a(2) = 1 through a(10) = 12 partitions (A=10):
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)     (A)
            (31)  (41)  (42)   (52)   (53)    (63)    (64)
                        (51)   (61)   (62)    (72)    (73)
                        (411)  (511)  (71)    (81)    (82)
                                      (521)   (621)   (91)
                                      (611)   (711)   (622)
                                      (5111)  (6111)  (631)
                                                      (721)
                                                      (811)
                                                      (6211)
                                                      (7111)
                                                      (61111)

Reduplication of A000070 shifted right.
Same as A025065 shifted right twice.
The Heinz numbers of these partitions are A335126.
Row sums of A386586.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A239455 counts Look-and-Say partitions, inseparable case A386632.
A325534 counts separable multisets, ranks A335433, sums of A386583.
A325535 counts inseparable multisets, ranks A335448, sums of A386584.
A335434 counts separable factorizations, inseparable A333487.
A336103 counts normal separable multisets, inseparable A336102.
A336106 counts separable type partitions, ranks A335127, sums of A386585.
A386633 counts separable type set partitions, row sums of A386635.
A386634 counts inseparable type set partitions, row sums of A386636.
Cf. A000110, A008284, A047966, A072233, A106351, A124762, A217605, A239964, A279790, A335125, A351293, A386577.

Table[Length[Select[IntegerPartitions[n],2*Max@@#>1+n&]],{n,0,15}]

For n>1, a(n) = A025065(n-2).

a(n) = A000041(n) - A336106(n).

A386577 Irregular triangle read by rows where T(n,k) is the number of permutations of the multiset of prime factors of n with k adjacent equal terms.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 0, 1, 0, 0, 1, 0, 1, 2, 0, 1, 1, 2, 0, 1, 2, 0, 2, 0, 0, 0, 0, 1, 1, 1, 2, 0, 1, 1, 2, 0, 2, 0, 2, 0, 1, 0, 2, 2, 0, 0, 1, 2, 0, 0, 0, 1, 1, 2, 0, 1, 6, 0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 2, 0, 2, 0, 2, 2, 2, 0, 1, 2, 0, 2, 0, 0, 2, 2, 0, 1

Offset: 1

Gus Wiseman, Aug 01 2025

Are the rows all unimodal?

Counts permutations of prime factors by "inseparability". For "separability" we have A374252.

			The prime indices of 12 are {1,1,2}, and we have:
- 1 permutation (1,2,1) with 0 adjacent equal parts
- 2 permutations (1,1,2), (2,1,1) with 1 adjacent equal part
- 0 permutations with 2 adjacent equal parts
so row 12 is (1,2,0).
Row 48 counts the following permutations:
  .  .  (1,1,1,2,1)  (1,1,1,1,2)  .
        (1,1,2,1,1)  (2,1,1,1,1)
        (1,2,1,1,1)
Row 144 counts the following permutations:
  .  (1,1,2,1,2,1)  (1,1,1,2,1,2)  (1,1,1,2,2,1)  (1,1,1,1,2,2)  .
     (1,2,1,1,2,1)  (1,1,2,1,1,2)  (1,1,2,2,1,1)  (2,2,1,1,1,1)
     (1,2,1,2,1,1)  (1,2,1,1,1,2)  (1,2,2,1,1,1)
                    (2,1,1,1,2,1)  (2,1,1,1,1,2)
                    (2,1,1,2,1,1)
                    (2,1,2,1,1,1)
Triangle begins:
   1:
   2: 1
   3: 1
   4: 0  1
   6: 1
   6: 2  0
   7: 1
   8: 0  0  1
   9: 0  1
  10: 2  0
  11: 1
  12: 1  2  0
  13: 1
  14: 2  0
  15: 2  0
  16: 0  0  0  1
  17: 1
  18: 1  2  0
  19: 1
  20: 1  2  0
  21: 2  0
  22: 2  0
  23: 1
  24: 0  2  2  0

Row lengths are A001222.
The minima of each row are A010051.
Sorted positions of first appearances appear to be A025487.
Column k = last is A069513.
Row sums are A168324 or A008480.
The number of trailing zeros in each row is A297155 = A001221-1.
Column k = 1 is A335452.
The number of leading zeros in each row is A374246.
For separability instead of inseparability we have A374252.
For a multiset with prescribed multiplicities we have A386578, separability A386579.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
A124762 gives inseparability of standard compositions, separability A333382.
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.
Cf. A001221, A051903, A051904, A106351, A238130, A336102, A373957, A382525, A386581, A386632.

Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],Function[q,Length[Select[Range[Length[q]-1],q[[#]]==q[[#+1]]&]]==k]]],{n,30},{k,0,PrimeOmega[n]-1}]

A386582 Number of distinct inseparable and pairwise disjoint sets of strict integer partitions, one of each exponent in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 31 2025

nonn

A set partition is inseparable iff the underlying set has no permutation whose adjacent elements all belong to different blocks. Note that inseparability only depends on the sizes of the blocks.

			The prime indices of 9216 are {1,1,1,1,1,1,1,1,1,1,2,2}, with a(9216) = 2 choices: {{2},{1,4,5}} and {{2},{1,3,6}}. The other 4 disjoint families {{2},{10}}, {{2},{4,6}}, {{2},{3,7}}, {{2},{1,9}} are separable.
The prime indices of 15552 are {1,1,1,1,1,1,2,2,2,2,2}, with a(15552) = 1 choice: {{5},{1,2,3}}. The other 5 disjoint families {{5},{6}}, {{5},{2,4}}, {{6},{2,3}}, {{6},{1,4}}, {{1,5},{2,3}} are separable.

For separable instead of inseparable we have A386575.
This is the inseparable case of A386587 (ordered version A382525).
Positions of positive terms are A386632.
Positions of first appearances are A386637.
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 (ranks A351294), complement A351293 (ranks A351295).
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, A111133, 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}]

a(2^n) = A111133(n).

cp's OEIS Frontend

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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A386587 Number of ways to choose a pairwise disjoint family of strict integer partitions, one of each exponent in the prime factorization of n.

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A386638 Number of integer partitions of n of inseparable type.

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A386577 Irregular triangle read by rows where T(n,k) is the number of permutations of the multiset of prime factors of n with k adjacent equal terms.

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A386582 Number of distinct inseparable and pairwise disjoint sets of strict integer partitions, one of each exponent in the prime factorization of n.

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