A353932 -id:A353932 - cp's OEIS frontend

A383096 Number of integer partitions of n having no permutation with all equal run-sums.

0, 0, 0, 1, 1, 5, 4, 13, 15, 25, 35, 54, 58, 99, 128, 168, 217, 295, 358, 488, 603, 784, 995, 1253, 1517, 1953, 2429, 2997, 3688, 4563, 5532, 6840, 8311, 10135, 12303, 14875, 17842, 21635, 26008, 31177, 37247, 44581, 53062, 63259, 75130, 89096, 105551, 124752, 147015, 173520

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The a(3) = 1 through a(8) = 15 partitions:
  (21)  (31)  (32)    (42)   (43)      (53)
              (41)    (51)   (52)      (62)
              (221)   (321)  (61)      (71)
              (311)   (411)  (322)     (332)
              (2111)         (331)     (431)
                             (421)     (521)
                             (511)     (611)
                             (2221)    (3221)
                             (3211)    (3311)
                             (4111)    (4211)
                             (22111)   (5111)
                             (31111)   (22211)
                             (211111)  (32111)
                                       (311111)
                                       (2111111)

For distinct instead of equal run-sums we appear to have A381717, q.v.
For run-lengths instead of sums we have A382915, ranks A382879, by signature A382914.
For more than one permutation we have A383097, ranks A383015.
The complement is counted by A383098, ranks A383110
These partitions are ranked by A383100, positions of 0 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.
A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
A383095 counts partitions having a unique permutation with equal run-sums, ranks A383099.
Cf. A006171, A329738, A353832, A353839, A353932, A354584, A382076, A382857, A383094.

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

More terms from Bert Dobbelaere, Apr 26 2025

A353929 Number of distinct sums of runs (of 0's or 1's) in the binary expansion of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 3, 2, 2, 3

Offset: 0

Gus Wiseman, Jun 26 2022

Assuming the binary digits are not all 1, this is one more than the number of different lengths of runs of 1's in the binary expansion of n.

			The binary expansion of 183 is (1,0,1,1,0,1,1,1), with runs (1), (0), (1,1), (0), (1,1,1), with sums 1, 0, 2, 0, 3, of which four are distinct, so a(183) = 4.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

For lengths of all runs we have A165413, firsts A165933.
Numbers whose binary expansion has distinct runs are A175413.
For runs instead of run-sums we have A297770, firsts A350952.
For prime indices we have A353835, weak A353861, firsts A006939.
For standard compositions we have A353849, firsts A246534.
Positions of first appearances are A353930.
A005811 counts runs in binary expansion.
A044813 lists numbers with distinct run-lengths in binary expansion.
A318928 gives runs-resistance of binary expansion.
A351014 counts distinct runs in standard compositions.
Cf. A215203, A353743, A353832, A353847, A353932, A354579.

Table[Length[Union[Total/@Split[IntegerDigits[n,2]]]],{n,0,100}]

from itertools import groupby
def A353929(n): return len(set(sum(map(int,y[1])) for y in groupby(bin(n)[2:]))) # Chai Wah Wu, Jun 26 2022

A353854 Length of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 3, 2, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 3, 1, 1, 3, 2, 1, 1, 1, 2, 2, 2, 2

Offset: 0

Gus Wiseman, Jun 01 2022

nonn

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). The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353847) until the rank of an anti-run is reached. For example, the trajectory 11 -> 10 -> 8 corresponds to the trajectory (2,1,1) -> (2,2) -> (4), with length a(11) = 3.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The trajectory of 94685 and the a(94685) = 5 corresponding compositions:
  94685: (2,1,1,4,1,1,2,1,1,2,1)
  86357: (2,2,4,2,2,2,2,1)
  69889: (4,4,8,1)
  65793: (8,8,1)
  65537: (16,1)

Positions of first appearances are A072639.
Positions of 1's are A333489, counted by A003242 (complement A261983).
The version for partitions is A353841.
The last part of the same trajectory is A353855.
This is the rank statistic counted by A353859.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A333627 represents the run-lengths of standard compositions.
A353832 represents the run-sum transformation of a partition.
A353840-A353846 pertain to the partition run-sum trajectory.
A353847 represents the run-sum transformation of a composition.
A353853-A353859 pertain to the composition run-sum trajectory.
A353932 lists run-sums of standard compositions, represented by A353847.
Cf. A237685, A238279, A304442, A325277, A333755, A353848, A353849, A353850, A353852, A353860.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[FixedPointList[Total/@Split[#]&,stc[n]]]-1,{n,0,100}]

A353858 Number of integer compositions of n with run-sum trajectory ending in a singleton.

Original entry on oeis.org

0, 1, 2, 2, 5, 2, 8, 2, 20, 5, 8, 2, 78, 2, 8, 8, 223, 2, 179, 2, 142, 8, 8, 2, 4808

Offset: 0

Gus Wiseman, Jun 17 2022

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). The run-sum trajectory is obtained by repeatedly taking the run-sums (cf. A353847) until an anti-run composition (A003242) is reached. For example, the composition (2,2,1,1,2) is counted under a(8) because it has the following run-sum trajectory: (2,2,1,1,2) -> (4,2,2) -> (4,4) -> (8).

			The a(0) = 0 through a(8) = 20 compositions:
  .  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
          (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                       (112)            (222)                (224)
                       (211)            (1113)               (422)
                       (1111)           (2112)               (1124)
                                        (3111)               (2114)
                                        (11211)              (2222)
                                        (111111)             (4112)
                                                             (4211)
                                                             (11114)
                                                             (21122)
                                                             (22112)
                                                             (41111)
                                                             (111122)
                                                             (112112)
                                                             (211211)
                                                             (221111)
                                                             (1111211)
                                                             (1121111)
                                                             (11111111)

The version for partitions is A353845, ranked by A353844.
The trajectory itself is A353853, last part A353855.
The lengths of trajectories of standard compositions are A353854.
This is column k = 1 of A353856, for partitions A353843.
These compositions are ranked by A353857.
A011782 counts compositions.
A066099 lists compositions in standard order.
A238279 and A333755 count compositions by number of runs.
A275870 counts collapsible partitions, ranked by A300273.
A333489 ranks anti-runs, counted by A003242 (complement A261983).
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sums of a composition, partitions A353832.
A353851 counts compositions with equal run-sums, ranked by A353848.
A353859 counts compositions by length of run-sum trajectory.
A353860 counts collapsible compositions.
A353932 lists run-sums of standard compositions.
Cf. A072639, A237685, A304442, A304465, A318928, A353833, A353841, A353849, A353850, A353852.

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

A353930 Smallest number whose binary expansion has n distinct run-sums.

Original entry on oeis.org

1, 2, 11, 183, 5871, 375775, 48099263, 12313411455, 6304466665215, 6455773865180671, 13221424875890015231, 54154956291645502388223, 443637401941159955564326911, 7268555193403964711965932118015, 238176016577461115681699663643131903, 15609103422420491677315869156516292427775

Offset: 1

Gus Wiseman, Jun 07 2022

nonn

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).

			The terms, binary expansions, and standard compositions begin:
       1:                    1  (1)
       2:                   10  (2)
      11:                 1011  (2,1,1)
     183:             10110111  (2,1,2,1,1,1)
    5871:        1011011101111  (2,1,2,1,1,2,1,1,1,1)
  375775:  1011011101111011111  (2,1,2,1,1,2,1,1,1,2,1,1,1,1,1)

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Essentially the same as A215203.
For prime indices instead of binary expansion we have A006939.
For lengths instead of sums of runs we have A165933 = firsts in A165413.
Numbers whose binary expansion has all distinct runs are A175413.
For standard compositions we have A246534, firsts of A353849.
For runs instead of run-sums we have A350952, firsts of A297770.
These are the positions of first appearances in A353929.
A005811 counts runs in binary expansion.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A351014 counts distinct runs in standard compositions.
A353835 counts partitions with all distinct run-sums, weak A353861.
A353864 counts rucksack partitions.
Cf. A044813, A073093, A181819, A304442, A353743, A353840, A353841, A353842, A353847, A353848, A353850, A353853, A353932, A354582.

qe=Table[Length[Union[Total/@Split[IntegerDigits[n,2]]]],{n,1,10000}];
Table[Position[qe,i][[1,1]],{i,Max@@qe}]

a(n) = {my(t=1); if(n==2, t<<=1, for(k=3, n, t = (t<Andrew Howroyd, Jan 01 2023

Offset corrected and terms a(7) and beyond from Andrew Howroyd, Jan 01 2023

A354912 Numbers k such that the k-th composition in standard order is the sequence of run-sums of some other composition.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 48, 49, 50, 52, 54, 64, 65, 66, 68, 69, 70, 72, 73, 74, 76, 77, 80, 81, 82, 84, 85, 86, 88, 89, 90, 96, 97, 98, 100, 101, 102, 104, 105, 106, 108

Offset: 0

Gus Wiseman, Jun 22 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

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).

			The terms and their corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   6: (1,2)
   8: (4)
   9: (3,1)
  10: (2,2)
  12: (1,3)
  13: (1,2,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  20: (2,3)
  21: (2,2,1)
  22: (2,1,2)
For example, the 21st composition in standard order (2,2,1) equals the run-sums of (1,1,2,1), so 21 is in the sequence. On the other hand, no composition has run-sums equal to the 29th composition (1,1,2,1), so 29 is not in the sequence.

The standard compositions used here are A066099, run-sums A353847/A353932.
These are the positions of nonzero terms in A354578.
The complement is A354904, counted by A354909.
These compositions are counted by A354910.
A003242 counts anti-run compositions, ranked by A333489.
A124767 counts runs in standard compositions.
A238279 and A333755 count compositions by number of runs.
A351014 counts distinct runs of standard compositions, firsts A351015.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
Cf. A000120, A029837, A124771, A239312, A333381, A334299, A351597, A353832, A353849, A353860, A354905.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MemberQ[Total/@Split[#]&/@ Join@@Permutations/@IntegerPartitions[Total[stc[#]]],stc[#]]&]

A354904 Numbers k such that the k-th composition in standard order is not the sequence of run-sums of any other composition.

Original entry on oeis.org

3, 7, 11, 14, 15, 19, 23, 27, 28, 29, 30, 31, 35, 39, 43, 46, 47, 51, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 75, 78, 79, 83, 87, 91, 92, 93, 94, 95, 99, 103, 107, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127

Offset: 1

Gus Wiseman, Jun 21 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
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).
The first term k such that the k-th composition in standard order does not have ones sandwiching the same prime number an even number of times is k = 3221, corresponding to the composition (1,3,3,2,2,1).

			The terms and their corresponding compositions begin:
   3: (1,1)
   7: (1,1,1)
  11: (2,1,1)
  14: (1,1,2)
  15: (1,1,1,1)
  19: (3,1,1)
  23: (2,1,1,1)
  27: (1,2,1,1)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  31: (1,1,1,1,1)

The standard compositions used here are A066099, run-sums A353847/A353932.
These are the positions of zeros in A354578, firsts A354905.
These compositions are counted by A354909.
The complement is A354912, counted by A354910.
A003242 counts anti-run compositions, ranked by A333489.
A124767 counts runs in standard compositions.
A238279 and A333755 count compositions by number of runs.
A351014 counts distinct runs of standard compositions, firsts A351015.
A353852 ranks compositions with distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
Cf. A000120, A000918, A005811, A029837, A333381, A334299, A353832, A353849, A353860, A354907.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],FreeQ[Total/@Split[#]&/@ Join@@Permutations/@IntegerPartitions[Total[stc[#]]],stc[#]]&]

A354905 First position of n in A354578, where A354578(k) is the number of integer compositions whose run-sums constitute the k-th composition in standard order (graded reverse-lexicographic, A066099).

Original entry on oeis.org

3, 0, 2, 8, 32, 68, 130, 290, 274, 580, 520, 1298, 2080, 1096, 2082, 4168, 2178, 4164, 4386, 35137, 8328, 8786, 10274, 8772, 16712, 20562, 8712, 16658, 33320, 41554, 33288, 82210, 34856, 66628, 33312, 66642, 34850, 69704, 140306, 133448, 69714, 74308, 133154

Offset: 0

Gus Wiseman, Jun 21 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

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).

			The terms and their corresponding compositions begin:
     3: (1,1)
     0: ()
     2: (2)
     8: (4)
    32: (6)
    68: (4,3)
   130: (6,2)
   290: (3,4,2)
   274: (4,3,2)
   580: (3,4,3)
   520: (6,4)
  1298: (2,4,3,2)
The inverse run-sum compositions for n = 2, 8, 32, 68, 130, 290:
  (2)   (4)     (6)       (43)     (62)       (342)
  (11)  (22)    (33)      (223)    (332)      (3411)
        (1111)  (222)     (4111)   (611)      (11142)
                (111111)  (11113)  (3311)     (32211)
                          (22111)  (22211)    (111411)
                                   (1111112)  (311112)
                                              (1112211)

The standard compositions used here are A066099, run-sums A353847/A353932.
This is the position of the first appearance of n in A354578.
A011782 counts compositions.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A354904 lists positions of zeros in A354578, complement A354912.
A354908 ranks collapsible compositions, counted by A353860.
Cf. A000005, A029837, A124767, A238279/A333755, A274174, A333381, A353832, A353849, A353863, A354584.

nn=1000;
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
antirunQ[y_]:=Length[Split[y]]==Length[y];
q=Table[Length[Select[Tuples[Divisors/@stc[n]],antirunQ]],{n,0,nn}];
w=Last[Select[Table[Take[q+1,i],{i,nn}],Union[#]==Range[Max@@#]&]-1];
Table[Position[w,k][[1,1]]-1,{k,0,Max@@w}]

A354907 Number of distinct sums of contiguous constant subsequences (partial runs) of the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 23 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

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).

			Composition number 981 in standard order is (1,1,1,2,2,2,1), with partial runs (1), (2), (1,1), (2,2), (1,1,1), (2,2,2), with distinct sums {1,2,3,4,6}, so a(981) = 5.

Positions of 1's are A000051.
Positions of first appearances are A000079.
The standard compositions used here are A066099, run-sums A353847/A353932.
If we allow any subsequence we get A334968.
The case of full runs is A353849, firsts A246534.
A version for nonempty partitions is A353861, full A353835.
Counting all distinct runs (instead of their distinct sums) gives A354582.
A124767 counts runs in standard compositions.
A238279 and A333755 count compositions by number of runs.
A330036 counts distinct partial runs of prime indices, full A005811.
A351014 counts distinct runs of standard compositions, firsts A351015.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
A354584 lists run-sums of prime indices, rows ranked by A353832.
Cf. A029837, A124771, A274174, A333381, A334299, A353864.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pre[y_]:=NestWhileList[Most,y,Length[#]>1&];
Table[Length[Union[Total/@Join@@pre/@Split[stc[n]]]],{n,0,100}]

A357878 Number of integer partitions of n whose run-sums are not weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 11, 19, 25, 40, 55, 79, 104, 150, 196, 270, 350, 467, 600, 786, 997, 1293, 1632, 2077, 2597, 3283, 4067, 5088, 6268, 7769, 9517, 11704, 14238, 17405, 21092, 25598, 30861, 37278, 44729, 53742, 64226, 76811, 91448, 108929, 129174

Offset: 0

Gus Wiseman, Oct 18 2022

nonn

The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. 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).

			The a(0) = 0 through a(9) = 8 partitions:
  .  .  .  .  .  (2111)  (21111)  (322)     (3221)     (3222)
                                  (31111)   (32111)    (32211)
                                  (211111)  (311111)   (42111)
                                            (2111111)  (321111)
                                                       (411111)
                                                       (2211111)
                                                       (3111111)
                                                       (21111111)

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The complement is counted by A304405, ranked by A357875.
Number of rows in A354584 summing to n that are weakly increasing.
The opposite (not weakly increasing) version is A357865, ranked by A357850.
These partitions are ranked by A357876.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, distinct A353837.
Cf. A047966, A098859, A239312, A275870, A304406, A304428, A304430, A353832, A353864, A353932.

Table[Length[Select[IntegerPartitions[n],!LessEqual@@Total/@Split[Reverse[#]]&]],{n,0,30}]

cp's OEIS Frontend

A383096 Number of integer partitions of n having no permutation with all equal run-sums.

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A353929 Number of distinct sums of runs (of 0's or 1's) in the binary expansion of n.

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A353854 Length of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.

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A353858 Number of integer compositions of n with run-sum trajectory ending in a singleton.

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A353930 Smallest number whose binary expansion has n distinct run-sums.

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A354912 Numbers k such that the k-th composition in standard order is the sequence of run-sums of some other composition.

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A354904 Numbers k such that the k-th composition in standard order is not the sequence of run-sums of any other composition.

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A354905 First position of n in A354578, where A354578(k) is the number of integer compositions whose run-sums constitute the k-th composition in standard order (graded reverse-lexicographic, A066099).

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A354907 Number of distinct sums of contiguous constant subsequences (partial runs) of the n-th composition in standard order.

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