A279790 -id:A279790 - cp's OEIS frontend

A358830 Number of twice-partitions of n into partitions with all different lengths.

1, 1, 2, 4, 9, 15, 31, 53, 105, 178, 330, 555, 1024, 1693, 2991, 5014, 8651, 14242, 24477, 39864, 67078, 109499, 181311, 292764, 483775, 774414, 1260016, 2016427, 3254327, 5162407, 8285796, 13074804, 20812682, 32733603, 51717463, 80904644, 127305773, 198134675, 309677802

Offset: 0

Gus Wiseman, Dec 03 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(1) = 1 through a(5) = 15 twice-partitions:
  (1)  (2)   (3)      (4)       (5)
       (11)  (21)     (22)      (32)
             (111)    (31)      (41)
             (11)(1)  (211)     (221)
                      (1111)    (311)
                      (11)(2)   (2111)
                      (2)(11)   (11111)
                      (21)(1)   (21)(2)
                      (111)(1)  (22)(1)
                                (3)(11)
                                (31)(1)
                                (111)(2)
                                (211)(1)
                                (111)(11)
                                (1111)(1)

The version for set partitions is A007837.
For sums instead of lengths we have A271619.
For constant instead of distinct lengths we have A306319.
The case of distinct sums also is A358832.
The version for multiset partitions of integer partitions is A358836.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A273873 counts strict trees.
Cf. A000009, A000219, A001970, A141199, A279375, A279785, A279790, A336342, A358334, A358831.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],UnsameQ@@Length/@#&]],{n,0,10}]

seq(n)={ local(Cache=Map());
  my(g=Vec(-1+1/prod(k=1, n, 1 - y*x^k + O(x*x^n))));
  my(F(m,r,b) = my(key=[m,r,b], z); if(!mapisdefined(Cache,key,&z),
  z = if(r<=0||m==0, r==0, self()(m-1, r, b) + sum(k=1, m, my(c=polcoef(g[m],k)); if(!bittest(b,k)&&c, c*self()(min(m,r-m), r-m, bitor(b, 1<Andrew Howroyd, Dec 31 2022

Terms a(26) and beyond from Andrew Howroyd, Dec 31 2022

A375136 Number of maximal strictly increasing runs in the weakly increasing prime factors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 04 2024

nonn

For n > 1, this is one more than the number of adjacent equal terms in the multiset of prime factors of n.

			The prime factors of 540 are {2,2,3,3,3,5}, with maximal strictly increasing runs ({2},{2,3},{3},{3,5}), so a(540) = 4.

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For compositions we have A124768, row-lengths of A374683, sum A374684.
For sum of prime indices we have A374706.
Row-lengths of A375128.
A112798 lists prime indices:
- distinct A001221
- length A001222
- leader A055396
- sum A056239
- reverse A296150
Cf. A034296, A046660, A066328, A141199, A141809, A279790, A320324, A333213, A358836, A374700.

Table[Length[Split[Flatten[ConstantArray@@@FactorInteger[n]],Less]],{n,100}]

For n > 1, a(n) = A046660(n) + 1 = A001222(n) - A001221(n) + 1.

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&]

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).

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).

A384320 Heinz numbers of integer partitions whose distinct parts are maximally refined.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 45, 48, 50, 54, 56, 60, 64, 66, 70, 72, 75, 78, 80, 81, 84, 90, 96, 98, 100, 105, 108, 110, 112, 120, 126, 128, 132, 135, 140, 144, 150, 156, 160, 162, 168, 180, 182, 192, 196

Offset: 1

Gus Wiseman, Jun 01 2025

nonn

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.
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 terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   27: {2,2,2}
   28: {1,1,4}
   30: {1,2,3}
   32: {1,1,1,1,1}

The squarefree case is A383707, counted by A179009.
The complement appears to be A384321, strict case A384322, counted by A384318.
Partitions of this type are counted by A384392.
A048767 is the Look-and-Say transform, fixed points A048768.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A383706, A357982 (non-disjoint), A299200 (non-strict).
Cf. A130091, A279375, A279790, A317142, A326080, A351294, A351295, A381454, A382525, 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[20],With[{y=Union[prix[#]]},UnsameQ@@y&&Intersection[y,Total/@nonsets[y]]=={}]&]

A384390 Heinz numbers of integer partitions with a unique proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

5, 7, 21, 22, 26, 33, 35, 39, 102, 114, 130, 154, 165, 170, 190, 195, 231, 238, 255, 285

Offset: 1

Gus Wiseman, Jun 02 2025

nonn

By "proper" we exclude the case of all singletons, which is disjoint in the strict case.

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.

			The strict partition (7,2,1) with Heinz number 102 can only be properly refined as ((4,3),(2),(1)), so 102 is in the sequence. The other refinement ((7),(2),(1)) is not proper.
The terms together with their prime indices begin:
    5: {3}
    7: {4}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   33: {2,5}
   35: {3,4}
   39: {2,6}
  102: {1,2,7}
  114: {1,2,8}
  130: {1,3,6}
  154: {1,4,5}
  165: {2,3,5}
  170: {1,3,7}
  190: {1,3,8}
  195: {2,3,6}
  231: {2,4,5}
  238: {1,4,7}
  255: {2,3,7}
  285: {2,3,8}

The non-proper version is A383707, counted by A179009.
Partitions of this type are counted by A384319, non-strict A384323 (ranks A384347).
This is the unique case of A384321, counted by A384317.
This is the case of a unique proper choice in A384322.
The complement is A384349 \/ A384393.
These are positions of 1 in A384389.
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 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 strict partitions of each prime index, non-strict A299200.
Cf. A382912, counted by A383710, odd case A383711.
Cf. A382913, counted by A383708, odd case A383533.
Cf. A098859, A122111, A130091, A279375, A279790, A317142, A351201, A381454, A384005, A384320.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
Select[Range[100],Length[pofprop[prix[#]]]==1&]

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

A279791 Number of twice-partitions of type (Q,R,Q) and weight n.

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 5, 8, 8, 16, 12, 23, 18, 36, 33, 50, 38, 84, 54, 106, 100, 155, 104, 244, 142, 301, 270, 436, 256, 684, 340, 788, 670, 1044, 585, 1868, 760, 1878, 1600, 2647

Offset: 1

Gus Wiseman, Dec 18 2016

			The a(8)=8 twice-partitions of type (Q,R,Q) are:
((8)), ((71)), ((62)), ((53)),
((521)), ((4)(31)), ((31)(4)), ((431)).

Gus Wiseman, Sequences enumerating triangles of integer partitions

Cf. A063834, A275972, A279790.

nn=20;
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Total[Total[Factorial/@Length/@Select[sps[Sort[#]],SameQ@@Total/@#&]]&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,nn}]

A300300 Number of ways to choose a multiset of strict partitions, or odd partitions, of odd numbers, whose weights sum to n.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 9, 14, 20, 32, 48, 69, 105, 150, 225, 322, 472, 669, 977, 1379, 1980, 2802, 3977, 5602, 7892, 11083, 15494, 21688, 30147, 42007, 58143, 80665, 111199, 153640, 211080, 290408, 397817, 545171, 744645, 1016826, 1385124, 1885022, 2561111, 3474730

Offset: 0

Gus Wiseman, Mar 02 2018

nonn

			The a(6) = 9 multiset partitions using odd-weight strict partitions: (5)(1), (14)(1), (3)(3), (32)(1), (3)(21), (3)(1)(1)(1), (21)(21), (21)(1)(1)(1), (1)(1)(1)(1)(1)(1).
The a(6) = 9 multiset partitions using odd partitions: (5)(1), (3)(3), (311)(1), (3)(111), (3)(1)(1)(1), (11111)(1), (111)(111), (111)(1)(1)(1), (1)(1)(1)(1)(1)(1).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000009, A063834, A078408, A089259, A270995, A271619, A279374, A279375, A279785, A279790, A294617, A300301.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
      `if`(d::odd, d, 0), d=divisors(j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      `if`(d::odd, b(d)*d, 0), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Mar 02 2018

nn=50;
ser=Product[1/(1-x^n)^PartitionsQ[n],{n,1,nn,2}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

Euler transform of {Q(1), 0, Q(3), 0, Q(5), 0, ...} where Q = A000009.

cp's OEIS Frontend

A358830 Number of twice-partitions of n into partitions with all different lengths.

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A375136 Number of maximal strictly increasing runs in the weakly increasing prime factors of n.

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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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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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A384320 Heinz numbers of integer partitions whose distinct parts are maximally refined.

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A384390 Heinz numbers of integer partitions with a unique proper way to choose disjoint strict partitions of each part.

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

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