A386579 -id:A386579 - cp's OEIS frontend

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.

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).

A386586 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k parts such that any multiset whose multiplicities are the parts of y is inseparable.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 3, 2, 1, 0, 0, 0, 0, 0, 1, 3, 2, 1, 0, 0, 0, 0, 0, 0, 1, 4, 4, 2, 1, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Aug 05 2025

We say that such partitions are of inseparable type. This is different from inseparable partitions (see A386584). A multiset is separable iff it has a permutation without any adjacent equal parts.

			The partition y = (7,2,1) is the multiplicities of the multiset {1,1,1,1,1,1,1,2,2,3}, which is inseparable, so y is counted under T(10,3).
Row n = 10 counts the following partitions (A = 10):
  .  A  91  811  7111  61111  .  .  .  .  .
        82  721  6211
        73  631
        64  622
Triangle begins:
  0
  0 0
  0 1 0
  0 1 0 0
  0 1 1 0 0
  0 1 1 0 0 0
  0 1 2 1 0 0 0
  0 1 2 1 0 0 0 0
  0 1 3 2 1 0 0 0 0
  0 1 3 2 1 0 0 0 0 0
  0 1 4 4 2 1 0 0 0 0 0

This is the inseparable type case of A008284 or A072233.
Row sums shifted left once are A025065 (ranks A335126), separable version A336106 (ranks A335127).
For separable instead of inseparable type we have A386583.
For integer partitions instead of normal multisets we have A386584.
For separable type instead of inseparable type we have A386585.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A239455 counts Look-and-Say partitions, ranks A351294.
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.
Cf. A000670, A005651, A106351, A238130, A279790, A386575, A386576, A386579, A386582, A386634.

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

a(n) = A072233(n) - A386585(n).

A386634 Number of inseparable type set partitions of {1..n}.

Original entry on oeis.org

0, 0, 1, 1, 5, 6, 37, 50, 345, 502, 3851, 5897, 49854, 79249, 730745, 1195147, 11915997, 19929390, 213332101, 363275555, 4150104224, 7172334477, 87003759195, 152231458128, 1952292972199, 3451893361661, 46625594567852, 83183249675125, 1179506183956655, 2120758970878892

Offset: 0

Gus Wiseman, Aug 09 2025

nonn

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 its other block sizes.
This is different from inseparable partitions (A325535) and partitions of inseparable type (A386638 or A025065).

			The a(2) = 1 through a(5) = 6 set partitions:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}    {{1,2,3,4,5}}
                      {{1},{2,3,4}}  {{1},{2,3,4,5}}
                      {{1,2,3},{4}}  {{1,2,3,4},{5}}
                      {{1,2,4},{3}}  {{1,2,3,5},{4}}
                      {{1,3,4},{2}}  {{1,2,4,5},{3}}
                                     {{1,3,4,5},{2}}

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

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.
The complement is counted by A386633, sums of A386635.
Row sums of A386636.
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. A001055, A124762, A239455, A335125, A386575, A386579.

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(29) from Alois P. Heinz, Aug 10 2025

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}]

A386578 Irregular triangle read by rows where T(n,k) is the number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent equal parts.

Original entry on oeis.org

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

Offset: 2

Gus Wiseman, Aug 04 2025

Row 1 is empty, so offset is 2.
Same as A386579 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:
  .  112121  111212  111221  111122  .
     121121  112112  112211  221111
     121211  121112  122111
             211121  211112
             211211
             212111
Triangle begins
   .
   1
   0  1
   2  0
   0  0  1
   1  2  0
   0  0  0  1
   6  0  0
   2  2  2  0
   0  2  2  0
   0  0  0  0  1
   6  6  0  0
   0  0  0  0  0  1
   0  0  3  2  0
   1  4  3  2  0
  24  0  0  0
   0  0  0  0  0  0  1
  12 12  6  0  0
   0  0  0  0  0  0  0  1
   2 12  6  0  0
   0  3  6  4  2  0

Column k = last is A010051.
Row lengths are A056239.
Initial zeros are counted by A252736 = A001222 - 1.
Row sums are A318762.
Column k = 0 is A335125.
For prime indices we have A386577.
Reversing all rows gives 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.
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, A003963, A051903, A051904, A106351, A112798, A238130, A336102, A373957, A374246, A382525, A386581.

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

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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.

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A386586 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k parts such that any multiset whose multiplicities are the parts of y is inseparable.

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A386634 Number of inseparable type set partitions of {1..n}.

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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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A386578 Irregular triangle read by rows where T(n,k) is the number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent equal parts.

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