A333487 -id:A333487 - cp's OEIS frontend

A386635 Triangle read by rows where T(n,k) is the number of separable type set partitions of {1..n} into k blocks.

1, 0, 1, 0, 0, 1, 0, 0, 3, 1, 0, 0, 3, 6, 1, 0, 0, 10, 25, 10, 1, 0, 0, 10, 75, 65, 15, 1, 0, 0, 35, 280, 350, 140, 21, 1, 0, 0, 35, 770, 1645, 1050, 266, 28, 1, 0, 0, 126, 2737, 7686, 6951, 2646, 462, 36, 1, 0, 0, 126, 7455, 32725, 42315, 22827, 5880, 750, 45, 1

Offset: 0

Gus Wiseman, Aug 10 2025

A set partition is of separable type iff the underlying set has a permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.
A set partition is also of separable type iff its greatest block size is at most one more than the sum of all its other blocks sizes.
This is different from separable partitions (A325534) and partitions of separable type (A336106).

			Row n = 4 counts the following set partitions:
  .  .  {{1,2},{3,4}}  {{1},{2},{3,4}}  {{1},{2},{3},{4}}
        {{1,3},{2,4}}  {{1},{2,3},{4}}
        {{1,4},{2,3}}  {{1},{2,4},{3}}
                       {{1,2},{3},{4}}
                       {{1,3},{2},{4}}
                       {{1,4},{2},{3}}
Triangle begins:
    1
    0    1
    0    0    1
    0    0    3    1
    0    0    3    6    1
    0    0   10   25   10    1
    0    0   10   75   65   15    1
    0    0   35  280  350  140   21    1

Column k = 2 appears to be A128015.
For separable partitions we have A386583, sums A325534, ranks A335433.
For inseparable partitions we have A386584, sums A325535, ranks A335448.
For separable type partitions we have A386585, sums A336106, ranks A335127.
For inseparable type partitions we have A386586, sums A386638 or A025065, ranks A335126.
Row sums are A386633.
The complement is counted by A386636, row sums A386634.
A000110 counts set partitions, row sums of A048993.
A000670 counts ordered set partitions.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A335434 counts separable factorizations, inseparable A333487.
A336103 counts normal separable multisets, inseparable A336102.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
A386587 counts disjoint families of strict partitions of each prime exponent.
Cf. A072233, A097805, A106351, A116861, A190945, A279790, A333382, A335125, A374252, A386575, A386578.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
stnseps[stn_]:=Select[Permutations[Union@@stn],And@@Table[Position[stn,#[[i]]][[1,1]]!=Position[stn,#[[i+1]]][[1,1]],{i,Length[#]-1}]&];
Table[Length[Select[sps[Range[n]],Length[#]==k&&stnseps[#]!={}&]],{n,0,5},{k,0,n}]

A386636 Triangle read by rows where T(n,k) is the number of inseparable type set partitions of {1..n} into k blocks.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4, 0, 0, 0, 1, 5, 0, 0, 0, 0, 1, 21, 15, 0, 0, 0, 0, 1, 28, 21, 0, 0, 0, 0, 0, 1, 92, 196, 56, 0, 0, 0, 0, 0, 1, 129, 288, 84, 0, 0, 0, 0, 0, 0, 1, 385, 1875, 1380, 210, 0, 0, 0, 0, 0, 0, 1, 561, 2860, 2145, 330, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Aug 10 2025

A set partition is of inseparable type iff the underlying set has no permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.
A set partition is also of inseparable type iff its greatest block size is at least 2 more than the sum of all its other block sizes.
This is different from inseparable partitions (A325535) and partitions of inseparable type (A386638 or A025065).

			Row n = 6 counts the following set partitions:
  .  {123456}  {1}{23456}  {1}{2}{3456}  .  .  .
               {12}{3456}  {1}{2345}{6}
               {13}{2456}  {1}{2346}{5}
               {14}{2356}  {1}{2356}{4}
               {15}{2346}  {1}{2456}{3}
               {16}{2345}  {1234}{5}{6}
               {1234}{56}  {1235}{4}{6}
               {1235}{46}  {1236}{4}{5}
               {1236}{45}  {1245}{3}{6}
               {1245}{36}  {1246}{3}{5}
               {1246}{35}  {1256}{3}{4}
               {1256}{34}  {1345}{2}{6}
               {1345}{26}  {1346}{2}{5}
               {1346}{25}  {1356}{2}{4}
               {1356}{24}  {1456}{2}{3}
               {1456}{23}
               {12345}{6}
               {12346}{5}
               {12356}{4}
               {12456}{3}
               {13456}{2}
Triangle begins:
    0
    0    0
    0    1    0
    0    1    0    0
    0    1    4    0    0
    0    1    5    0    0    0
    0    1   21   15    0    0    0
    0    1   28   21    0    0    0    0
    0    1   92  196   56    0    0    0    0
    0    1  129  288   84    0    0    0    0    0
    0    1  385 1875 1380  210    0    0    0    0    0

For separable partitions we have A386583, sums A325534, ranks A335433.
For inseparable partitions we have A386584, sums A325535, ranks A335448.
For separable type partitions we have A386585, sums A336106, ranks A335127.
For inseparable type partitions we have A386586, sums A386638 or A025065, ranks A335126.
Row sums are A386634.
The complement is counted by A386635, row sums A386633.
A000110 counts set partitions, row sums of A048993.
A000670 counts ordered set partitions.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A279790 counts disjoint families on strongly normal multisets.
A335434 counts separable factorizations, inseparable A333487.
A336103 counts normal separable multisets, inseparable A336102.
A386587 counts disjoint families of strict partitions of each prime exponent.
Cf. A008284, A072233, A097805, A106351, A116861, A124762, A129506, A190945, A239455, A335125, A374252, A386575.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
stnseps[stn_]:=Select[Permutations[Union@@stn],And@@Table[Position[stn,#[[i]]][[1,1]]!=Position[stn,#[[i+1]]][[1,1]],{i,Length[#]-1}]&]
Table[Length[Select[sps[Range[n]],Length[#]==k&&stnseps[#]=={}&]],{n,0,5},{k,0,n}]

A348381 Number of inseparable factorizations of n that are not a twin (x*x).

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

Offset: 1

Gus Wiseman, Oct 30 2021

nonn

First differs from A347706 at a(216) = 3, A347706(216) = 4.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
A multiset is inseparable if it has no permutation that is an anti-run, meaning there are always adjacent equal parts. Alternatively, a multiset is inseparable if its maximal multiplicity is at most one plus the sum of its remaining multiplicities.

			The a(n) factorizations for n = 96, 192, 384, 576:
  2*2*2*12      3*4*4*4         4*4*4*6           4*4*4*9
  2*2*2*2*6     2*2*2*24        2*2*2*48          2*2*2*72
  2*2*2*2*2*3   2*2*2*2*12      2*2*2*2*24        2*2*2*2*36
                2*2*2*2*2*6     2*2*2*2*3*8       2*2*2*2*4*9
                2*2*2*2*3*4     2*2*2*2*4*6       2*2*2*2*6*6
                2*2*2*2*2*2*3   2*2*2*2*2*12      2*2*2*2*2*18
                                2*2*2*2*2*2*6     2*2*2*2*3*12
                                2*2*2*2*2*3*4     2*2*2*2*2*2*9
                                2*2*2*2*2*2*2*3   2*2*2*2*2*3*6
                                                  2*2*2*2*2*2*3*3

Positions of nonzero terms are A046099.
Partitions not of this type are counted by A325534 - A000035.
Partitions of this type are counted by A325535 - A000035.
Allowing twins gives A333487.
The case without an alternating permutation is A347706, with twins A348380.
The complement is counted by A348383, without twins A335434.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations of sets.
A008480 counts permutations of prime indices, strict A335489.
A025047 counts alternating or wiggly compositions.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A344654 counts non-twin partitions without an alternating permutation.
A348382 counts non-anti-run compositions that are not a twin.
A348611 counts anti-run ordered factorizations.
Cf. A038548, A336102, A344614, A344653, A344740, A347050, A347437, A347438, A347456, A348379, A348609.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],!MatchQ[#,{x_,x_}]&&Select[Permutations[#],!MatchQ[#,{_,x_,x_,_}]&]=={}&]],{n,100}]

a(n > 1) = A333487(n) - A010052(n).

a(2^n) = A325535(n) - 1 for odd n, otherwise A325535(n).

A348383 Number of factorizations of n that are either separable (have an anti-run permutation) or are a twin (x*x).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 9, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2

Offset: 1

Gus Wiseman, Oct 30 2021

nonn

First differs from A347050 at a(216) = 28, A347050(216) = 27.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts. Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of the remaining multiplicities plus one.

			The a(216) = 28 factorizations:
  (2*2*2*3*3*3)  (2*2*2*3*9)  (2*2*6*9)   (3*8*9)   (3*72)   (216)
                 (2*2*3*3*6)  (2*3*4*9)   (4*6*9)   (4*54)
                 (2*3*3*3*4)  (2*3*6*6)   (2*2*54)  (6*36)
                              (3*3*4*6)   (2*3*36)  (8*27)
                              (2*2*3*18)  (2*4*27)  (9*24)
                              (2*3*3*12)  (2*6*18)  (12*18)
                                          (2*9*12)  (2*108)
                                          (3*3*24)
                                          (3*4*18)
                                          (3*6*12)
The a(270) = 20 factorizations:
  (2*3*3*3*5)  (2*3*5*9)   (5*6*9)   (3*90)   (270)
               (3*3*5*6)   (2*3*45)  (5*54)
               (2*3*3*15)  (2*5*27)  (6*45)
                           (2*9*15)  (9*30)
                           (3*3*30)  (10*27)
                           (3*5*18)  (15*18)
                           (3*6*15)  (2*135)
                           (3*9*10)

Positions of 1's are 1 and A000040.
Not requiring separability gives A010052 for n > 1.
Positions of 2's are A323644.
Partitions of this type are counted by A325534(n) + A000035(n + 1).
Partitions of this type are ranked by A335433 \/ A001248.
Partitions not of this type are counted by A325535(n) - A000035(n + 1).
Partitions not of this type are ranked by A345193 = A335448 \ A001248.
Not allowing twins gives A335434, complement A333487,
The case with an alternating permutation is A347050, no twins A348379.
The case without an alternating permutation is A347706, no twins A348380.
The complement is counted by A348381.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations.
A003242 counts anti-run compositions, ranked by A333489.
A025047 counts alternating or wiggly compositions.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A119620, A138364, A336103, A344654, A344740, A347437, A347438, A347456, A347458, A348382.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sepQ[m_]:=Select[Permutations[m],!MatchQ[#,{_,x_,x_,_}]&]!={};
Table[Length[Select[facs[n],MatchQ[#,{x_,x_}]||sepQ[#]&]],{n,100}]

a(n > 1) = A335434(n) + A010052(n), where A010052(n) = 1 if n is a perfect square, otherwise 0.

A348611 Number of ordered factorizations of n with no adjacent equal factors.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 14, 1, 3, 3, 6, 1, 13, 1, 7, 3, 3, 3, 17, 1, 3, 3, 14, 1, 13, 1, 6, 6, 3, 1, 29, 1, 6, 3, 6, 1, 14, 3, 14, 3, 3, 1, 36, 1, 3, 6, 14, 3, 13, 1, 6, 3, 13, 1, 45, 1, 3, 6, 6, 3, 13, 1, 29, 4, 3

Offset: 1

Gus Wiseman, Nov 07 2021

nonn

First differs from A348610 at a(24) = 14, A348610(24) = 12.
An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
In analogy with Carlitz compositions, these may be called Carlitz ordered factorizations.

			The a(n) ordered factorizations without adjacent equal factors for n = 1, 6, 12, 16, 24, 30, 32, 36 are:
  ()   6     12      16      24      30      32      36
       2*3   2*6     2*8     3*8     5*6     4*8     4*9
       3*2   3*4     8*2     4*6     6*5     8*4     9*4
             4*3     2*4*2   6*4     10*3    16*2    12*3
             6*2             8*3     15*2    2*16    18*2
             2*3*2           12*2    2*15    2*8*2   2*18
                             2*12    3*10    4*2*4   3*12
                             2*3*4   2*3*5           2*3*6
                             2*4*3   2*5*3           2*6*3
                             2*6*2   3*2*5           2*9*2
                             3*2*4   3*5*2           3*2*6
                             3*4*2   5*2*3           3*4*3
                             4*2*3   5*3*2           3*6*2
                             4*3*2                   6*2*3
                                                     6*3*2
                                                     2*3*2*3
                                                     3*2*3*2
Thus, of total A074206(12) = 8 ordered factorizations of 12, only factorizations 2*2*3 and 3*2*2 (see A348616) are not included in this count, therefore a(12) = 6. - _Antti Karttunen_, Nov 12 2021

The additive version (compositions) is A003242, complement A261983.
The additive alternating version is A025047, ranked by A345167.
Factorizations without a permutation of this type are counted by A333487.
As compositions these are ranked by A333489, complement A348612.
Factorizations with a permutation of this type are counted by A335434.
The non-alternating additive version is A345195, ranked by A345169.
The alternating case is A348610, which is dominated at positions A122181.
The complement is counted by A348616.
A001055 counts factorizations, strict A045778, ordered A074206.
A325534 counts separable partitions, ranked by A335433.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A348613 counts non-alternating ordered factorizations.
Cf. A001250, A138364, A336103, A347050, A347438, A347463, A347466, A347706, A348379, A348383.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
antirunQ[y_]:=Length[y]==Length[Split[y]]
Table[Length[Select[ordfacs[n],antirunQ]],{n,100}]

A348611(n, e=0) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d!=e), s += A348611(n/d, d))); (s)); \\ Antti Karttunen, Nov 12 2021

a(n) = A074206(n) - A348616(n).

A350139 Number of non-weakly alternating ordered factorizations of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 12, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 20, 0, 0, 0, 0, 0, 2, 0, 10, 0, 0, 0, 12, 0

Offset: 1

Gus Wiseman, Dec 24 2021

nonn

The first odd term is a(180) = 69, which has, for example, the non-weakly alternating ordered factorization 2*3*5*3*2.
An ordered factorization of n is a finite sequence of positive integers > 1 with product n. Ordered factorizations are counted by A074206.
We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

			The a(n) ordered factorizations for n = 24, 36, 48, 60:
  (2*3*4)  (2*3*6)    (2*3*8)    (2*5*6)
  (4*3*2)  (6*3*2)    (2*4*6)    (3*4*5)
           (2*3*3*2)  (6*4*2)    (5*4*3)
           (3*2*2*3)  (8*3*2)    (6*5*2)
                      (2*2*3*4)  (10*3*2)
                      (2*3*4*2)  (2*3*10)
                      (2*4*3*2)  (2*2*3*5)
                      (3*2*2*4)  (2*3*5*2)
                      (4*2*2*3)  (2*5*3*2)
                      (4*3*2*2)  (3*2*2*5)
                                 (5*2*2*3)
                                 (5*3*2*2)

Positions of nonzero terms are A122181.
The strong version for compositions is A345192, ranked by A345168.
The strong case is A348613, complement A348610.
The version for compositions is A349053, complement A349052.
As compositions with ones allowed these are ranked by A349057.
The complement is counted by A349059.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations, complement A348615.
A025047 counts weakly alternating compositions, ranked by A345167.
A335434 counts separable factorizations, complement A333487.
A345164 counts alternating perms of prime factors, with twins A344606.
A345170 counts partitions with an alternating permutation.
A348379 counts factorizations w/ alternating perm, complement A348380.
A348611 counts anti-run ordered factorizations, complement A348616.
A349060 counts weakly alternating partitions, complement A349061.
Cf. A003242, A138364, A339846, A339890, A344604, A345194, A347050, A347438, A347463, A347706, A349054.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@facs[n],!whkQ[#]&&!whkQ[-#]&]],{n,100}]

a(2^n) = A349053(n).

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).

A348616 Number of ordered factorizations of n with adjacent equal factors.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 6, 1, 0, 1, 2, 0, 0, 0, 9, 0, 0, 0, 9, 0, 0, 0, 6, 0, 0, 0, 2, 2, 0, 0, 19, 1, 2, 0, 2, 0, 6, 0, 6, 0, 0, 0, 8, 0, 0, 2, 18, 0, 0, 0, 2, 0, 0, 0, 31, 0, 0, 2, 2, 0, 0, 0, 19, 4, 0, 0, 8, 0, 0

Offset: 1

Gus Wiseman, Nov 08 2021

nonn

First differs from A348613 at a(24) = 6, A348613(24) = 8.

An ordered factorization of n is a finite sequence of positive integers > 1 with product n.

			The a(n) ordered factorizations with at least one pair of adjacent equal factors for n = 12, 24, 36, 60:
   2*2*3    2*2*6      6*6        15*2*2
   3*2*2    6*2*2      2*2*9      2*2*15
            2*2*2*3    3*3*4      2*2*3*5
            2*2*3*2    4*3*3      2*2*5*3
            2*3*2*2    9*2*2      3*2*2*5
            3*2*2*2    2*2*3*3    3*5*2*2
                       2*3*3*2    5*2*2*3
                       3*2*2*3    5*3*2*2
                       3*3*2*2
See also examples in A348611.

Positions of 0's are A005117.
Positions of 4's appear to be A030514.
Positions of 2's appear to be A054753.
Positions of 1's appear to be A168363.
The additive version (compositions) is A261983, complement A003242.
Factorizations with a permutation of this type are counted by A333487.
Factorizations without a permutation of this type are counted by A335434.
The complement is counted by A348611.
As compositions these are ranked by A348612, complement A333489.
Dominated by A348613 (non-alternating ordered factorizations).
A001055 counts factorizations, strict A045778, ordered A074206.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A001250, A025047, A122181, A138364, A347050, A347463, A347706, A348379, A348380, A348382, A348610.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
antirunQ[y_]:=Length[y]==Length[Split[y]]
Table[Length[Select[ordfacs[n],!antirunQ[#]&]],{n,100}]

a(n) = A074206(n) - A348611(n).

A363265 Number of integer factorizations of n with a unique mode.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 7, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 6, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Jun 27 2023

nonn

An integer factorization of n is a multiset of positive integers > 1 with product n.
A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
Conjecture: 9 is missing from this sequence.

			The a(n) factorizations for n = 2, 4, 16, 24, 48, 72:
  (2)  (4)    (16)       (24)       (48)         (72)
       (2*2)  (4*4)      (2*2*6)    (3*4*4)      (2*6*6)
              (2*2*4)    (2*2*2*3)  (2*2*12)     (3*3*8)
              (2*2*2*2)             (2*2*2*6)    (2*2*18)
                                    (2*2*3*4)    (2*2*2*9)
                                    (2*2*2*2*3)  (2*2*3*6)
                                                 (2*3*3*4)
                                                 (2*2*2*3*3)

The complement for partitions is A362607, ranks A362605.
The version for partitions is A362608, ranks A356862.
A001055 counts factorizations, strict A045778, ordered A074206.
A089723 counts constant factorizations.
A316439 counts factorizations by length, A008284 partitions.
A339846 counts even-length factorizations, A339890 odd-length.
Cf. A240219, A326622, A333487, A335434, A347438, A362610, A362611, A362612, A362614, A363723.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[facs[n],Length[modes[#]]==1&]],{n,100}]

cp's OEIS Frontend

A386635 Triangle read by rows where T(n,k) is the number of separable type set partitions of {1..n} into k blocks.

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A348383 Number of factorizations of n that are either separable (have an anti-run permutation) or are a twin (x*x).

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A348611 Number of ordered factorizations of n with no adjacent equal factors.

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A350139 Number of non-weakly alternating ordered factorizations of n.

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A386638 Number of integer partitions of n of inseparable type.

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A348616 Number of ordered factorizations of n with adjacent equal factors.

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