A279790 -id:A279790 - cp's OEIS frontend

A300335 Number of ordered set partitions of {1,...,n} with weakly increasing block-sums.

1, 1, 2, 6, 18, 65, 258, 1156, 5558, 29029, 161942, 967921, 6110687, 40807420, 286177944, 2107745450, 16202590638, 130043111849, 1085011337141, 9408577992091, 84501248359552, 786018565954838, 7550153439748394

Offset: 0

Gus Wiseman, Mar 03 2018

			The a(3) = 6 ordered set partitions: (123), (1)(23), (2)(13), (12)(3), (3)(12), (1)(2)(3).

A000110(n) <= a(n) <= A000670(n).

Cf. A005651, A063834, A275780, A279375, A279790, A279791, A281113.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Times@@Factorial/@Length/@GatherBy[sptn,Total],{sptn,sps[Range[n]]}],{n,8}]

a(12)-a(15) from Alois P. Heinz, Mar 03 2018

a(16)-a(22) from Christian Sievers, Aug 30 2024

A384005 Number of ways to choose disjoint strict integer partitions, one of each conjugate prime index of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 22 2025

nonn

			The prime indices of 96 are {1,1,1,1,1,2}, conjugate (6,1), and we have choices (6,1) and (4,2,1), so a(96) = 2.
The prime indices of 108 are {1,1,2,2,2}, conjugate (5,3), and we have choices (5,3), (5,2,1), (4,3,1), so a(108) = 3.

Adding up over all integer partitions gives A279790, strict A279375.
For multiplicities instead of indices we have conjugate of A382525.
The conjugate version is A383706.
Positive positions are A384010, conjugate A382913, counted by A383708, odd case A383533.
Positions of 0 are A384011.
Without disjointness we have A384179, conjugate A357982, non-strict version A299200.
A000041 counts integer partitions, strict A000009.
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.
Cf. `A098859, A122111, A130091, A179009, A317141, A382771, A383707, A383710.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[pof[conj[prix[n]]]],{n,100}]

a(n) = A383706(A122111(n)).

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

Original entry on oeis.org

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

A294788 Number of twice-factorizations of type (Q,P,Q) and product n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 1, 3, 3, 3, 1, 5, 1, 5, 3, 3, 1, 12, 1, 3, 3, 5, 1, 12, 1, 5, 3, 3, 3, 13, 1, 3, 3, 12, 1, 12, 1, 5, 5, 3, 1, 19, 1, 5, 3, 5, 1, 12, 3, 12, 3, 3, 1, 26, 1, 3, 5, 11, 3, 12, 1, 5, 3, 12, 1, 26, 1, 3, 5, 5, 3, 12, 1, 19, 3, 3

Offset: 1

Gus Wiseman, Nov 08 2017

nonn

a(n) is the number of ways to choose a product-preserving permutation of a set partition of a factorization of n into distinct factors greater than one.

			The a(36) = 13 twice-factorizations are: (2)*(3)*(6), (2)*(3*6), (6)*(2*3), (2*3)*(6), (2*6)*(3), (2*3*6), (2)*(18), (2*18), (3)*(12), (3*12), (4)*(9), (4*9), (36).

Cf. A001055, A063834, A279790, A281113, A294786, A294787.

nn=100;
sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Total[Sum[Times@@Factorial/@Length/@Split[Sort[Times@@@f]],{f,sps[Sort[#]]}]&/@sfs[n]],{n,nn}]

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}]

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}]

A386575 Number of distinct separable and pairwise disjoint sets 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, 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}]

A375399 Numbers k such that the minima of maximal anti-runs in the weakly increasing sequence of prime factors of k (with multiplicity) are not distinct.

Original entry on oeis.org

4, 8, 9, 12, 16, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 52, 54, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 162, 164, 168, 169, 171

Offset: 1

Gus Wiseman, Aug 16 2024

nonn

An anti-run is a sequence with no adjacent equal terms.
The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.
Note the prime factors can alternatively be taken in weakly decreasing order.

			The prime factors of 300 are {2,2,3,5,5}, with maximal anti-runs ((2),(2,3,5),(5)), with minima (2,2,5), so 300 is in the sequence.
The prime factors of 450 are {2,3,3,5,5}, with maximal anti-runs ((2,3),(3,5),(5)), with minima (2,3,5), so 450 is not in the sequence.
The terms together with their prime indices begin:
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    12: {1,1,2}
    16: {1,1,1,1}
    20: {1,1,3}
    24: {1,1,1,2}
    25: {3,3}
    27: {2,2,2}
    28: {1,1,4}
    32: {1,1,1,1,1}
    36: {1,1,2,2}
    40: {1,1,1,3}
    44: {1,1,5}
    45: {2,2,3}
    48: {1,1,1,1,2}

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

The complement for compositions is A374638, counted by A374518.
A version for compositions is A374639, counted by A374678.
Positions of non-strict rows in A375128, sums A374706, ranks A375400.
For identical instead of strict we have A375397, counted by A375405.
The complement is A375398, counted by A375134.
The complement for maxima instead of minima is A375402, counted by A375133.
For maxima instead of minima we have A375403, counted by A375401.
Partitions (or reversed partitions) of this type are counted by A375404.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
Both have length A001222, distinct A001221.
Cf. A034296, A046660, A065200, A065201, A115029, A279790, A374632, A374761, A375136, A375396.

Select[Range[100],!UnsameQ@@Min /@ Split[Flatten[ConstantArray@@@FactorInteger[#]],UnsameQ]&]

A375401 Number of integer partitions of n whose maximal anti-runs do not all have different maxima.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 7, 12, 16, 25, 33, 48, 63, 88, 116, 157, 204, 272, 349, 456, 581, 749, 946, 1205, 1511, 1904, 2371, 2960, 3661, 4538, 5577, 6862, 8389, 10257, 12472, 15164, 18348, 22192, 26731, 32177, 38593, 46254, 55256, 65952, 78500, 93340, 110706

Offset: 0

Gus Wiseman, Aug 17 2024

nonn

An anti-run is a sequence with no adjacent equal terms. The maxima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the greatest term of each.

			The partition y = (3,2,2,1) has maximal ant-runs ((3,2),(2,1)), with maxima (3,2), so y is not counted under a(8).
The a(2) = 1 through a(8) = 12 partitions:
  (11)  (111)  (22)    (221)    (33)      (331)      (44)
               (1111)  (2111)   (222)     (2221)     (332)
                       (11111)  (2211)    (4111)     (2222)
                                (3111)    (22111)    (3311)
                                (21111)   (31111)    (5111)
                                (111111)  (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

For identical instead of distinct we have A239955, ranks A073492.
The complement is counted by A375133, ranks A375402.
The complement for minima instead of maxima is A375134, ranks A375398.
These partitions have Heinz numbers A375403.
For minima instead of maxima we have A375404, ranks A375399.
The reverse for identical instead of distinct is A375405, ranks A375397.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A055887 counts sequences of partitions with total sum n.
A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
Cf. A034296, A115029, A141199, A279790, A358830, A358836, A374632, A374761, A375136, A375396, A375400.

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Max/@Split[#,UnsameQ]&]],{n,0,30}]

A375404 Number of integer partitions of n whose minima of maximal anti-runs are not all different.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 7, 9, 14, 19, 30, 38, 56, 73, 102, 133, 179, 231, 307, 392, 511, 647, 831, 1046, 1328, 1658, 2084, 2586, 3219, 3970, 4909, 6016, 7386, 9005, 10988, 13330, 16175, 19531, 23580, 28350, 34067, 40788, 48809, 58215, 69383, 82461, 97917, 115976

Offset: 0

Gus Wiseman, Aug 17 2024

nonn

An anti-run is a sequence with no adjacent equal terms. The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.

Also the number of reversed integer partitions of n such that the minima of maximal anti-runs are not all different.

			The a(0) = 0 through a(8) = 14 reversed partitions:
  .  .  (11)  (111)  (22)    (113)    (33)      (115)      (44)
                     (112)   (1112)   (114)     (223)      (116)
                     (1111)  (11111)  (222)     (1114)     (224)
                                      (1113)    (1123)     (1115)
                                      (1122)    (1222)     (1124)
                                      (11112)   (11113)    (1133)
                                      (111111)  (11122)    (2222)
                                                (111112)   (11114)
                                                (1111111)  (11123)
                                                           (11222)
                                                           (111113)
                                                           (111122)
                                                           (1111112)
                                                           (11111111)

The complement for maxima instead of minima is A375133, ranks A375402.
The complement is counted by A375134, ranks A375398.
These partitions are ranked by A375399.
For maxima instead of minima we have A375401, ranks A375403.
For identical instead of distinct we have A375405, ranks A375397.
A000041 counts integer partitions, strict A000009.
A003242 counts anti-run compositions, ranks A333489.
A055887 counts sequences of partitions with total sum n.
A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
Cf. A034296, A046660, A073492, A115029, A141199, A239955, A279790, A358830, A358836, A375136.

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Min/@Split[#,UnsameQ]&]],{n,0,30}]

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