A386578 -id:A386578 - cp's OEIS frontend

A386633 Number of separable type set partitions of {1..n}.

1, 1, 1, 4, 10, 46, 166, 827, 3795, 20645, 112124, 672673, 4163743, 27565188, 190168577, 1381763398, 10468226150, 82844940414, 681863474058, 5832378929502, 51720008131148, 474862643822274, 4506628734688128, 44151853623626218, 445956917001833090, 4638586880336637692

Offset: 0

Gus Wiseman, Aug 09 2025

nonn

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 block sizes.
This is different from separable partitions (A325534) and partitions of separable type (A336106).

			The a(1) = 1 through a(4) = 10 set partitions:
  {{1}}  {{1},{2}}  {{1},{2,3}}    {{1,2},{3,4}}
                    {{1,2},{3}}    {{1,3},{2,4}}
                    {{1,3},{2}}    {{1,4},{2,3}}
                    {{1},{2},{3}}  {{1},{2},{3,4}}
                                   {{1},{2,3},{4}}
                                   {{1,2},{3},{4}}
                                   {{1},{2,4},{3}}
                                   {{1,3},{2},{4}}
                                   {{1,4},{2},{3}}
                                   {{1},{2},{3},{4}}

Alois P. Heinz, Table of n, a(n) for n = 0..250

For separable partitions see A386583, sums A325534, ranks A335433.
For inseparable partitions see A386584, sums A325535, ranks A335448.
For separable type partitions see A386585, sums A336106, ranks A335127.
For inseparable type partitions see A386586, sums A386638 or A025065, ranks A335126.
The complement is counted by A386634, sums of A386636.
Row sums of A386635.
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.
Cf. A001055, A072233, A106351, A190945, A239455, A239964, A335125, A386575, A386578, A386580, A386587.

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]],stnseps[#]!={}&]],{n,0,5}]

a(12)-a(25) from Alois P. Heinz, Aug 10 2025

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

Original entry on oeis.org

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}]

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}]

A386579 Number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent unequal parts.

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 0, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 6, 0, 2, 2, 2, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 6, 6, 1, 0, 0, 0, 0, 0, 0, 2, 3, 0, 0, 0, 2, 3, 4, 1, 0, 0, 0, 24, 1, 0, 0, 0, 0, 0, 0, 0, 0, 6, 12, 12, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 12, 2, 0, 2, 4, 6, 3, 0

Offset: 2

Gus Wiseman, Aug 04 2025

Row 1 is empty, so offset is 2.
Same as A386578 with rows reversed.
This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			Row n = 21 counts the following permutations:
  .  111122  111221  111212  112121  .
     221111  112211  112112  121121
             122111  121112  121211
             211112  211121
                     211211
                     212111
Triangle begins:
  .
  1
  1  0
  0  2
  1  0  0
  0  2  1
  1  0  0  0
  0  0  6
  0  2  2  2
  0  2  2  0
  1  0  0  0  0
  0  0  6  6
  1  0  0  0  0  0
  0  2  3  0  0
  0  2  3  4  1
  0  0  0 24
  1  0  0  0  0  0  0
  0  0  6 12 12
  1  0  0  0  0  0  0  0
  0  0  6 12  2
  0  2  4  6  3  0

Column k = 0 is A010051.
Row lengths are A056239.
Row sums are A318762.
Column k = last is A335125.
For prime indices we have A374252, reverse A386577.
Reversing all rows gives A386578.
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.
A305936 is a multiset whose multiplicities are the prime indices of n.
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, A001222, A003963, A051903, A051904, A106351, A112798, A238130, A336102, A373957, A386581.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
ugt[c_,x_]:=Select[Permutations[c],Function[q,Length[Select[Range[Length[q]-1],q[[#]]!=q[[#+1]]&]]==x]];
Table[Table[Length[ugt[nrmptn[n],k]],{k,0,Length[nrmptn[n]]-1}],{n,30}]

cp's OEIS Frontend

A386633 Number of separable type set partitions of {1..n}.

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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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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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A386579 Number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent unequal parts.

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