A353932 -id:A353932 - cp's OEIS frontend

A354584 Irregular triangle read by rows where row k lists the run-sums of the multiset (weakly increasing sequence) of prime indices of n.

1, 2, 2, 3, 1, 2, 4, 3, 4, 1, 3, 5, 2, 2, 6, 1, 4, 2, 3, 4, 7, 1, 4, 8, 2, 3, 2, 4, 1, 5, 9, 3, 2, 6, 1, 6, 6, 2, 4, 10, 1, 2, 3, 11, 5, 2, 5, 1, 7, 3, 4, 2, 4, 12, 1, 8, 2, 6, 3, 3, 13, 1, 2, 4, 14, 2, 5, 4, 3, 1, 9, 15, 4, 2, 8, 1, 6, 2, 7, 2, 6, 16

Offset: 1

Gus Wiseman, Jun 17 2022

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

			Triangle begins:
  .
  1
  2
  2
  3
  1 2
  4
  3
  4
  1 3
  5
  2 2
  6
  1 4
  2 3
For example, the prime indices of 630 are {1,2,2,3,4}, so row 630 is (1,4,3,4).

Positions of first appearances are A308495 plus 1.
The version for compositions is A353932, ranked by A353847.
Classes:
- singleton rows: A000961
- constant rows: A353833, nonprime A353834, counted by A304442
- strict rows: A353838, counted by A353837, complement A353839
Statistics:
- row lengths: A001221
- row sums: A056239
- row products: A304117
- row ranks (as partitions): A353832
- row image sizes: A353835
- row maxima: A353862
- row minima: A353931
A001222 counts prime factors with multiplicity.
A112798 and A296150 list partitions by rank.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353840-A353846 pertain to partition run-sum trajectory.
A353861 counts distinct sums of partial runs of prime indices.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A000040, A002110, A027748, A071625, A073093, A181819, A238279/A333755, A353850, A353852, A353867.

Table[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k],{n,30}]

A353860 Number of collapsible integer compositions of n.

Original entry on oeis.org

0, 1, 2, 2, 5, 2, 12, 2, 26, 9, 36, 2, 206, 2, 132, 40, 677, 2, 1746, 2, 3398, 136, 2052, 2, 44388, 33, 8196, 730, 79166, 2, 263234, 2, 458330, 2056, 131076, 160, 8804349, 2, 524292, 8200, 13662156, 2, 36036674, 2, 48844526, 90282, 8388612, 2, 1971667502, 129

Offset: 0

Gus Wiseman, Jun 04 2022

nonn

If a collapse is a joining of some number of adjacent equal parts of an integer composition, we call a composition collapsible iff by some sequence of collapses it can be reduced to a single part. An example of such a sequence of collapses is (1,1,1,3,2,1,1,2) -> (3,3,2,1,1,2) -> (3,3,2,2,2) -> (6,2,2,2) -> (6,6) -> (12), which shows that (1,1,1,3,2,1,1,2) is a collapsible composition of 12.

			The a(0) = 0 through a(6) = 12 compositions:
  .  (1)  (2)   (3)    (4)     (5)      (6)
          (11)  (111)  (22)    (11111)  (33)
                       (112)            (222)
                       (211)            (1113)
                       (1111)           (1122)
                                        (2112)
                                        (2211)
                                        (3111)
                                        (11112)
                                        (11211)
                                        (21111)
                                        (111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for partitions is A275870, ranked by A300273.
A003242 counts anti-run compositions, ranked by A333489, complement A261983.
A011782 counts compositions.
A353847 represents the run-sums of a composition, partitions A353832.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions.
Cf. A237685, A238279, A304442, A318928, A333755, A353844, A353848, A353849, A353850, A353852.

repcams[q_List]:=repcams[q]=Union[{q},If[UnsameQ@@q,{},Union@@repcams/@ Union[Insert[Drop[q,#],Plus@@Take[q,#],First[#]]&/@ Select[Tuples[Range[Length[q]],2],And[Less@@#,SameQ@@Take[q,#]]&]]]];
Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],MemberQ[repcams[#],{n}]&]],{n,0,15}]

a(n) = if(n==0, 0, 1 - sumdiv(n, d, if(d>1, moebius(d)*a(n/d)^d ))) \\ Andrew Howroyd, Feb 04 2023

Sum_{d|n} mu(d)*a(n/d)^d = 1 for n > 0. - Andrew Howroyd, Feb 04 2023

Terms a(16) and beyond from Andrew Howroyd, Feb 04 2023

A382876 Number of ways to permute the prime indices of n so that the run-sums are all different.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 0, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 6, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 0, 1, 2, 0, 1, 2, 6, 1, 2, 2, 6, 1, 4, 1, 2, 2, 2, 2, 6, 1, 2, 1, 2, 1, 0, 2, 2, 2

Offset: 1

Gus Wiseman, Apr 12 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

A run in a sequence is a constant consecutive subsequence. The run-sums of a sequence are obtained by splitting it into maximal runs and taking their sums. See A353932 for run-sums of standard compositions.

			For n = 12, none of the permutations (1,1,2), (1,2,1), (2,1,1) has distinct run-sums, so a(12) = 0.
The prime indices of 36 are {1,1,2,2}, and we have permutations: (1,1,2,2), (2,2,1,1), so a(36) = 2.
For n = 90 we have:
  (1,2,2,3)
  (1,3,2,2)
  (2,2,1,3)
  (2,2,3,1)
  (3,1,2,2)
  (3,2,2,1)
So a(90) = 6. The 6 missing permutations are: (1,2,3,2), (2,1,2,3), (2,1,3,2), (2,3,1,2), (2,3,2,1), (3,2,1,2).

Positions of 1 are A000961.
Compositions of this type are counted by A353850, ranked by A353852.
Positions of 0 appear to be A381636, for equal run-sums A383100.
For run-lengths instead of sums we have A382771, equal A382857 (zeros A382879).
For equal instead of distinct run-sums we have A382877.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A304442 counts compositions with equal run-sums, complement A382076.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A353837 counts partitions with distinct run-sums, ranks A353838.
A353847 gives composition run-sum transformation, for partitions A353832.
A353932 lists run-sums of standard compositions.
Cf. A000720, A001221, A001222, A098859, A130091, A329738, A351013, A351202, A353848, A353851, A354580, A354584.

Table[Length[Select[Permutations[PrimePi /@ Join@@ConstantArray@@@FactorInteger[n]], UnsameQ@@Total/@Split[#]&]],{n,100}]

A353863 Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 11, 16, 20, 24, 30, 43, 47, 62, 79, 94, 113, 143, 170, 211, 256, 307, 372, 449, 531, 648, 779, 926, 1100, 1323, 1562, 1864, 2190, 2595, 3053, 3611, 4242, 4977, 5834, 6825, 7973, 9344, 10844, 12641, 14699, 17072, 19822

Offset: 0

Gus Wiseman, Jun 04 2022

nonn

A weak run-sum of a sequence is the sum of any consecutive constant subsequence. For example, the weak run-sums of (3,2,2,1) are {1,2,3,4}.

This is a kind of completeness property, cf. A126796.

			The a(1) = 1 through a(8) = 7 partitions:
  (1)  (11)  (21)   (211)   (311)    (321)     (3211)     (3221)
             (111)  (1111)  (2111)   (3111)    (4111)     (32111)
                            (11111)  (21111)   (22111)    (41111)
                                     (111111)  (31111)    (221111)
                                               (211111)   (311111)
                                               (1111111)  (2111111)
                                                          (11111111)

For parts instead of weak run-sums we have A000009.
For multiplicities instead of weak run-sums we have A317081.
If weak run-sums are distinct we have A353865, the completion of A353864.
A003242 counts anti-run compositions, ranked by A333489, complement A261983.
A005811 counts runs in binary expansion.
A165413 counts distinct run-lengths in binary expansion, sums A353929.
A300273 ranks collapsible partitions, counted by A275870, comps A353860.
A353832 represents taking run-sums of a partition, compositions A353847.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices.
A353837 counts partitions with distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353861 counts distinct weak run-sums of prime indices.
A353932 lists run-sums of standard compositions.
Rulers: A103295, A103300, A169942, A325768.
Complete: A002033, A325780, A126796, A276024, A325781, A188431, A353866.
Cf. A047967, A073093, A181819, A237685, A353844, A353867, A353930.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
wkrs[y_]:=Union[Total/@Select[msubs[y],SameQ@@#&]];
Table[Length[Select[IntegerPartitions[n],normQ[Rest[wkrs[#]]]&]],{n,0,15}]

\\ isok(p) tests the partition.
isok(p)={my(b=0, s=0, t=0); for(i=1, #p, if(p[i]<>t, t=p[i]; s=0); s += t; b = bitor(b, 1<<(s-1))); bitand(b,b+1)==0}
a(n) = {my(r=0); forpart(p=n, r+=isok(p)); r} \\ Andrew Howroyd, Jan 15 2024

a(31) onwards from Andrew Howroyd, Jan 15 2024

A382877 Number of ways to permute the prime indices of n so that the run-sums are all equal.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 14 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The a(144) = 4 permutations of {1,1,1,1,2,2} are:
  (1,1,1,1,2,2)
  (1,1,2,1,1,2)
  (2,1,1,2,1,1)
  (2,2,1,1,1,1)
The a(1728) = 4 permutations are:
  (1,1,1,1,1,1,2,2,2)
  (1,1,2,1,1,2,1,1,2)
  (2,1,1,2,1,1,2,1,1)
  (2,2,2,1,1,1,1,1,1)

Compositions of this type are counted by A353851, ranked by A353848.
For run-lengths instead of sums we have A382857 (zeros A382879), distinct A382771.
For distinct instead of equal run-sums we have A382876, counted by A353850.
Positions of terms > 1 are A383015.
Positions of 1 are A383099.
Positions of 0 are A383100 (complement A383110), counted by A383098.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A304442 counts compositions with equal run-sums, complement A382076.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A353837 counts partitions with distinct run-sums, ranks A353838.
A353847 gives composition run-sum transformation, for partitions A353832.
A353932 lists run-sums of standard compositions.
Cf. A000720, A000961, A001221, A001222, A130091, A329738, A353833, A353852, A354584, A381871.

Table[Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[n]], SameQ@@Total/@Split[#]&]],{n,100}]

A383015 Numbers whose prime indices have more than one permutation with all equal run-sums.

Original entry on oeis.org

12, 40, 63, 112, 144, 325, 351, 352, 675, 832, 931, 1008, 1539, 1600, 1728, 2176, 2875, 3509, 3969, 4864, 6253, 7047, 7056, 8775, 9072, 11776, 12427, 12544, 12691, 16128, 19133, 20736, 20800, 22464, 23125, 26973, 29403, 29696, 32269, 43200, 49392, 57967, 59711

Offset: 1

Gus Wiseman, Apr 14 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

All terms appear to have even sum of prime indices.

			The terms together with their prime indices begin:
     12: {1,1,2}
     40: {1,1,1,3}
     63: {2,2,4}
    112: {1,1,1,1,4}
    144: {1,1,1,1,2,2}
    325: {3,3,6}
    351: {2,2,2,6}
    352: {1,1,1,1,1,5}
    675: {2,2,2,3,3}
    832: {1,1,1,1,1,1,6}
    931: {4,4,8}
   1008: {1,1,1,1,2,2,4}
   1539: {2,2,2,2,8}
   1600: {1,1,1,1,1,1,3,3}
   1728: {1,1,1,1,1,1,2,2,2}

Compositions of this type are counted by A353851, ranked by A353848.
Positions of terms > 1 in A382877, zeros A383100 (complement A383014).
For run-lengths instead of sums we have A383089, counted by A383090.
The complement for run-lengths instead of sums is A383091, counted by A383092
Partitions of this type are counted by A383097.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A304442 counts compositions with equal run-sums, complement A382076.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A353837 counts partitions with distinct run-sums, ranks A353838.
A353847 gives composition run-sum transformation, for partitions A353832.
A353932 lists run-sums of standard compositions.
Cf. A000720, A000961, A001221, A001222, A329738, A353833, A354584, A381636, A381871, A382857, A382876, A382879.

Select[Range[100],Length[Select[Permutations[PrimePi/@Join@@ConstantArray@@@FactorInteger[#]],SameQ@@Total/@Split[#]&]]>1&]

A383097 Number of integer partitions of n having more than one permutation with all equal run-sums.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 9, 0, 7, 0, 12, 0, 1, 0, 38, 0, 1, 1, 18, 0, 38, 0, 32, 0, 1, 0, 90, 0, 1, 0, 71, 0, 78, 0, 33, 10, 1, 0, 228, 0, 31, 0, 42, 0, 156, 0, 123, 0, 1, 0, 447, 0, 1, 16, 146, 0, 222, 0, 63, 0, 102, 0, 811, 0, 1, 29, 75, 0, 334, 0

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The a(27) = 1 partition is: (9,3,3,3,1,1,1,1,1,1,1,1,1).
The a(4) = 1 through a(16) = 9 partitions (empty columns not shown):
  (211)  (3111)  (422)     (511111)  (633)        (71111111)  (844)
                 (41111)             (6222)                   (82222)
                 (221111)            (33222)                  (442222)
                                     (4221111)                (44221111)
                                     (6111111)                (422221111)
                                     (33111111)               (811111111)
                                     (222111111)              (4411111111)
                                                              (42211111111)
                                                              (222211111111)

These partitions are ranked by A383015, positions of terms > 1 in A382877.
For run-lengths instead of sums we have A383090, ranks A383089, unique A383094.
The complement is A383095 + A383096, ranks A383099 \/ A383100.
For any positive number of permutations we have A383098, ranks A383110.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
Cf. A006171, A329738, A353832, A353839, A353850, A353932, A354584, A381717, A382076.

Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Total/@Split[#]&]]>1&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

A383099 Numbers whose prime indices have exactly one permutation with all equal run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 48, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193

Offset: 1

Gus Wiseman, Apr 20 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   37: {12}
   41: {13}

For distinct instead of equal run-sums we have A000961, counted by A000005.
These are the positions of 1 in A382877.
For more than one choice we have A383015.
Partitions of this type are counted by A383095.
For no choices we have A383100, counted by A383096.
For at least one choice we have A383110, counted by A383098, see A383013.
For run-lengths instead of sums we have A383112 = positions of 1 in A382857.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A353851 counts compositions with equal run-sums, ranks A353848.
Cf. A047966, A072774, A130091, A242031, A304686, A304678, A334965.
Cf. A000720, A001221, A001222, A351294, A353832, A353838, A353932, A354584, A382876, A383091.

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], SameQ@@Total/@Split[#]&]]==1&]

The complement is A383015 \/ A383100, for run-lengths A382879 \/ A383089.

A383095 Number of integer partitions of n having exactly one permutation with all equal run-sums.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 6, 2, 4, 5, 6, 2, 12, 2, 6, 8, 5, 2, 20, 2, 12, 8, 6, 2, 20, 5, 6, 12, 12, 2, 34, 2, 6, 8, 6, 8, 45, 2, 6, 8, 20, 2, 34, 2, 12, 28, 6, 2, 30, 5, 20, 8, 12, 2, 52, 8, 20, 8, 6, 2, 78, 2, 6, 28, 7, 8, 34, 2, 12, 8, 34, 2, 80, 2, 6, 28, 12, 8, 34, 2, 30, 25

Offset: 0

Gus Wiseman, Apr 16 2025

nonn

			The partition (2,2,1,1) has permutation (2,1,1,2) so is counted under a(6).
The a(1) = 1 through a(10) = 6 partitions (A=10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              1111         222              2222      33111      22222
                           2211             11111111  3111111    2221111
                           21111                      111111111  22111111
                           111111                                1111111111

For distinct instead of equal run-sums we have A000005.
For run-lengths instead of sums we have A383094.
The complement is counted by A383096 + A383097, ranks A383100 \/ A383015.
These partitions are ranked by A383099 = positions of 1 in A382877.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A383098 counts partitions with a permutation having all equal run-sums, ranks A383110.
Cf. A329738, A353839, A353832, A353932, A354584, A381636, A381871, A382076, A382876.

Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Total/@Split[#]&]]==1&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

A383098 Number of integer partitions of n having at least one permutation with all equal run-sums.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 2, 7, 5, 7, 2, 19, 2, 7, 8, 14, 2, 27, 2, 24, 8, 7, 2, 58, 5, 7, 13, 30, 2, 72, 2, 38, 8, 7, 8, 135, 2, 7, 8, 91, 2, 112, 2, 45, 38, 7, 2, 258, 5, 51, 8, 54, 2, 208, 8, 143, 8, 7, 2, 525, 2, 7, 44, 153, 8, 256, 2, 75, 8, 136, 2, 891, 2, 7, 57, 87, 8

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The partition (4,4,4,2,2,1,1,1,1) has permutations (4,2,2,4,1,1,1,1,4) and (4,1,1,1,1,4,2,2,4) so is counted under a(20).
The a(1) = 1 through a(10) = 7 partitions (A=10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              211          222              422       33111      22222
              1111         2211             2222      3111111    511111
                           3111             41111     111111111  2221111
                           21111            221111               22111111
                           111111           11111111             1111111111

For distinct instead of equal run-sums we appear to have A382427.
For run-lengths instead of sums we have A383013, ranked by complement of A382879.
The case of a unique choice is A383095, ranks A383099 = positions of 1 in A382877.
The complement is counted by A383096, ranks A383100 = positions of 0 in A382877.
These partitions are ranked by A383110.
The case of more than one choice is A383097, ranks A383015.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
Cf. A006171, A329738, A353832, A353839, A353850, A353932, A354584, A382076, A382857, A382876, A383094, A383112.

Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],SameQ@@Total/@Split[#]&]!={}&]],{n,0,15}]

a(n) = A383097(n) + A383095(n), ranks A383015 \/ A383099.

More terms from Bert Dobbelaere, Apr 26 2025

cp's OEIS Frontend

A354584 Irregular triangle read by rows where row k lists the run-sums of the multiset (weakly increasing sequence) of prime indices of n.

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A353860 Number of collapsible integer compositions of n.

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A382876 Number of ways to permute the prime indices of n so that the run-sums are all different.

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A353863 Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.

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A382877 Number of ways to permute the prime indices of n so that the run-sums are all equal.

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A383015 Numbers whose prime indices have more than one permutation with all equal run-sums.

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A383097 Number of integer partitions of n having more than one permutation with all equal run-sums.

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A383099 Numbers whose prime indices have exactly one permutation with all equal run-sums.

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