A353744 -id:A353744 - cp's OEIS frontend

A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.

0, 1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 31, 32, 36, 39, 42, 46, 59, 60, 63, 64, 127, 128, 136, 138, 143, 168, 170, 175, 187, 238, 248, 250, 255, 256, 292, 316, 487, 511, 512, 528, 543, 682, 750, 955, 1008, 1023, 1024, 2047, 2048, 2080, 2084, 2090, 2111, 2184

Offset: 0

Gus Wiseman, May 30 2022

nonn

Every sequence can be uniquely split into non-overlapping runs, 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 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 terms together with their binary expansions and corresponding compositions begin:
     0:       0  ()
     1:       1  (1)
     2:      10  (2)
     3:      11  (1,1)
     4:     100  (3)
     7:     111  (1,1,1)
     8:    1000  (4)
    10:    1010  (2,2)
    11:    1011  (2,1,1)
    14:    1110  (1,1,2)
    15:    1111  (1,1,1,1)
    16:   10000  (5)
    31:   11111  (1,1,1,1,1)
    32:  100000  (6)
    36:  100100  (3,3)
    39:  100111  (3,1,1,1)
    42:  101010  (2,2,2)
    46:  101110  (2,1,1,2)
    59:  111011  (1,1,2,1,1)
    60:  111100  (1,1,1,3)
For example:
- The 59th composition in standard order is (1,1,2,1,1), with run-sums (2,2,2), so 59 is in the sequence.
- The 2298th composition in standard order is (4,1,1,1,1,2,2), with run-sums (4,4,4), so 2298 is in the sequence.
- The 2346th composition in standard order is (3,3,2,2,2), with run-sums (6,6), so 2346 is in the sequence.

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

Standard compositions are listed by A066099.
For equal lengths instead of sums we have A353744, counted by A329738.
The version for partitions is A353833, counted by A304442.
These compositions are counted by A353851.
The distinct instead of equal version is A353852, counted by A353850.
The run-sums themselves are listed by A353932, with A353849 distinct terms.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sum transformation for compositions.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
A353863 counts run-sum-complete partitions.
Cf. A003242, A047966, A106356, A140690, A238279, A274174, A333381, A333489, A333755, A353832, A353864.

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

A353849(a(n)) = 1.

A353849 Number of distinct positive run-sums of the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 30 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 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.

			Composition 462903 in standard order is (1,1,4,7,1,2,1,1,1), with run-sums (2,4,7,1,2,3), of which a(462903) = 5 are distinct.

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

Counting repeated runs also gives A124767.
Positions of first appearances are A246534.
For distinct runs instead of run-sums we have A351014 (firsts A351015).
A version for partitions is A353835, weak A353861.
Positions of 1's are A353848, counted by A353851.
The version for binary expansion is A353929 (firsts A353930).
The run-sums themselves are listed by A353932, with A353849 distinct terms.
For distinct run-lengths instead of run-sums we have A354579.
A005811 counts runs in binary expansion.
A066099 lists compositions in standard order.
A165413 counts distinct run-lengths in binary expansion.
A297770 counts distinct runs in binary expansion, firsts A350952.
A353847 represents the run-sum transformation for compositions.
A353853-A353859 pertain to composition run-sum trajectory.
Distinct runs: A032020, A175413, A351013, A351018, A329739, A351290.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A003242, A044813, A071625, A238279, A329738, A333381, A333489, A333755, A353744, A353832, A353850, A353852, A353866.

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

A382857 Number of ways to permute the prime indices of n so that the run-lengths are all equal.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 09 2025

nonn

The first x with a(x) > 1 but A382771(x) > 0 is a(216) = 4, A382771(216) = 4.

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 prime indices of 216 are {1,1,1,2,2,2} and we have permutations:
  (1,1,1,2,2,2)
  (1,2,1,2,1,2)
  (2,1,2,1,2,1)
  (2,2,2,1,1,1)
so a(216) = 4.
The prime indices of 25920 are {1,1,1,1,1,1,2,2,2,2,3} and we have permutations:
  (1,2,1,2,1,2,1,2,1,3,1)
  (1,2,1,2,1,2,1,3,1,2,1)
  (1,2,1,2,1,3,1,2,1,2,1)
  (1,2,1,3,1,2,1,2,1,2,1)
  (1,3,1,2,1,2,1,2,1,2,1)
so a(25920) = 5.

The restriction to signature representatives (A181821) is A382858, distinct A382773.
The restriction to factorials is A335407, distinct A382774.
For distinct instead of equal run-lengths we have A382771.
For run-sums instead of run-lengths we have A382877, distinct A382876.
Positions of first appearances are A382878.
Positions of 0 are A382879.
Positions of terms > 1 are A383089.
Positions of 1 are A383112.
A003963 gives product of prime indices.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294.
A304442 counts partitions with equal run-sums, ranks A353833.
A164707 lists numbers whose binary expansion has all equal run-lengths, distinct A328592.
A353744 ranks compositions with equal run-lengths, counted by A329738.
Cf. A000720, A000961, A001221, A001222, A003242, A008480, A047966, A238130, A238279, A351201, A351293, A351295.

Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]], SameQ@@Length/@Split[#]&]],{n,0,100}]

A382879 Positions of 0 in A382857 (permutations of prime indices with equal run-lengths).

Original entry on oeis.org

24, 40, 48, 54, 56, 80, 88, 96, 104, 112, 135, 136, 152, 160, 162, 176, 184, 189, 192, 208, 224, 232, 240, 248, 250, 272, 288, 296, 297, 304, 320, 328, 336, 344, 351, 352, 368, 375, 376, 384, 405, 416, 424, 448, 459, 464, 472, 480, 486, 488, 496, 513, 528, 536

Offset: 1

Gus Wiseman, Apr 09 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:
   24: {1,1,1,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   80: {1,1,1,1,3}
   88: {1,1,1,5}
   96: {1,1,1,1,1,2}
  104: {1,1,1,6}
  112: {1,1,1,1,4}
  135: {2,2,2,3}
  136: {1,1,1,7}
  152: {1,1,1,8}
  160: {1,1,1,1,1,3}

For distinct instead of equal the complement is A351294, counted by A239455.
For distinct instead of equal we have A351295, counted by A351293.
For run-sums instead of run-lengths we have A383100, zeros of A382877, distinct A382876.
Positions of 0 in A382857 (firsts A382878), by signature A382858 (distinct A382773).
For prime signature instead of prime indices we have A382914.
Partitions of this type are counted by A382915.
The complement is counted by A383013.
A005811 counts runs in binary expansion.
A056239 adds up prime indices, row sums of A112798.
A297770 counts distinct runs in binary expansion.
A164707 lists numbers whose binary form has equal runs of ones, distinct A328592.
A304442 counts partitions with equal run-sums, ranks A353833.
A329739 counts compositions with distinct run-lengths, ranks A351290.
A353744 ranks compositions with equal run-lengths, distinct A351596 (complement A351291).
Cf. A000720, A000961, A001221, A001222, A003242, A008480, A047966, A130091, A238279, A351201.

Select[Range[100], Select[Permutations[Join@@ConstantArray@@@FactorInteger[#]], SameQ@@Length/@Split[#]&]=={}&]

A382771 Number of ways to permute the prime indices of n so that the run-lengths are all different.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 07 2025

nonn

The first x with a(x) > 0 but A382857(x) > 1 is a(216) = 4, A382857(216) = 4.

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(96) = 4 permutations are:
  (1,1,1,1,1,2)
  (1,1,1,2,1,1)
  (1,1,2,1,1,1)
  (2,1,1,1,1,1)
The a(216) = 4 permutations are:
  (1,1,2,2,2,1)
  (1,2,2,2,1,1)
  (2,1,1,1,2,2)
  (2,2,1,1,1,2)
The a(360) = 6 permutations are:
  (1,1,1,2,2,3)
  (1,1,1,3,2,2)
  (2,2,1,1,1,3)
  (2,2,3,1,1,1)
  (3,1,1,1,2,2)
  (3,2,2,1,1,1)

Positions of 1 are A000961.
Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433, equal A382879.
Sorted positions of first appearances are A382772, equal A382878.
For prescribed signature we have A382773, equal A382858.
The restriction to factorials is A382774, equal A335407.
For equal instead of distinct run-lengths we have A382857.
For run-sums instead of run-lengths we have A382876, equal A382877.
Positions of terms > 1 are A383113.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, complement A351293.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
Cf. A000720, A001221, A001222, A003242, A048767, A051903, A051904, A130091, A238130, A351013, A351202.

Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],UnsameQ@@Length/@Split[#]&]],{n,30}]

a(A181821(n)) = a(A304660(n)) = A382773(n).

a(n!) = A382774(n).

A383013 Number of integer partitions of n having a permutation with all equal run-lengths.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 18, 21, 31, 38, 56, 67, 94, 121, 162, 199, 265, 330, 438, 543, 693, 859, 1103, 1353, 1702, 2097, 2619, 3194, 3972, 4821, 5943, 7206, 8796, 10632, 12938, 15536, 18794, 22539, 27133, 32374, 38827, 46175, 55134, 65421, 77751, 91951, 109011, 128482

Offset: 0

Gus Wiseman, Apr 12 2025

nonn

A partition of n counts towards a(n) if and only if #p + g >= 2*L where #p is the number of parts counted with multiplicity of the partition, g is the gcd of all the frequencies of every distinct part and L is the largest frequency of a part. - David A. Corneth, Apr 27 2025

			The partition (2,2,1,1,1,1) has permutation (1,1,2,2,1,1) with equal run-lengths (2,2,2) so is counted under a(8).
The a(1) = 1 through a(8) = 18 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (311)    (222)     (322)      (71)
                            (11111)  (321)     (331)      (332)
                                     (411)     (421)      (422)
                                     (2211)    (511)      (431)
                                     (111111)  (3211)     (521)
                                               (22111)    (611)
                                               (1111111)  (2222)
                                                          (3221)
                                                          (3311)
                                                          (4211)
                                                          (22211)
                                                          (32111)
                                                          (221111)
                                                          (11111111)

David A. Corneth, Table of n, a(n) for n = 0..82

For distinct instead of equal run-lengths we have A239455, ranked by A351294.
The complement for distinct run-lengths is A351293, ranked by A351295.
The complement is counted by A382915, ranked by A382879, by signature A382914.
A000041 counts integer partitions, strict A000009.
A304442 counts partitions with equal run-sums, ranks A353833.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A382857 counts permutations of prime indices with equal run-lengths, firsts A382878.
Cf. A003242, A047966, A164707, A238279, A351201, A351290, A351291, A353837, A353851, A382858.

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

More terms from Bert Dobbelaere, Apr 26 2025

A382915 Number of integer partitions of n having no permutation with all equal run-lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 4, 4, 9, 11, 18, 21, 34, 41, 55, 69, 98, 120, 160, 189, 249, 309, 396, 472, 605, 734, 913, 1099, 1371, 1632, 2021, 2406, 2937, 3514, 4251, 5039, 6101, 7221, 8646, 10205, 12209, 14347, 17086, 20041, 23713, 27807, 32803, 38262, 45043, 52477, 61471, 71496

Offset: 0

Gus Wiseman, Apr 12 2025

nonn

			The partition y = (2,2,1,1,1) has permutations and run-lengths:
  (2,2,1,1,1) (2,3)
  (2,1,2,1,1) (1,1,1,2)
  (2,1,1,2,1) (1,2,1,1)
  (2,1,1,1,2) (1,3,1)
  (1,2,2,1,1) (1,2,2)
  (1,2,1,2,1) (1,1,1,1,1)
  (1,2,1,1,2) (1,1,2,1)
  (1,1,2,2,1) (2,2,1)
  (1,1,2,1,2) (2,1,1,1)
  (1,1,1,2,2) (3,2)
Since (1,2,1,2,1) has all equal run-lengths (1,1,1,1,1), y is not counted under a(7).
The a(5) = 1 through a(10) = 11 partitions:
  (2111)  (3111)   (2221)    (5111)     (3222)      (3331)
          (21111)  (4111)    (41111)    (6111)      (4222)
                   (31111)   (311111)   (22221)     (7111)
                   (211111)  (2111111)  (51111)     (61111)
                                        (321111)    (421111)
                                        (411111)    (511111)
                                        (2211111)   (3211111)
                                        (3111111)   (4111111)
                                        (21111111)  (22111111)
                                                    (31111111)
                                                    (211111111)

The complement for distinct run-lengths is A239455, ranked by A351294.
For distinct instead of equal run-lengths we have A351293, ranked by A351295.
These partitions are ranked by A382879, by signature A382914.
The complement is counted by A383013.
A000041 counts integer partitions, strict A000009.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A329738 counts compositions with equal run-lengths, ranks A353744.
A382857 counts permutations of prime indices with equal run-lengths.
Cf. A003242, A047966, A238279, A329739, A351201, A351290, A351596, A382773.

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

More terms from Bert Dobbelaere, Apr 26 2025

A382773 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all different.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 0, 1, 0, 4, 4, 1, 0, 4, 4, 0, 0, 1, 6, 1, 0, 4, 6, 4, 0, 1, 6, 4, 0, 1, 6, 1, 0, 0, 8, 1, 0, 4, 0, 6, 0, 1, 0, 6, 0, 6, 8, 1, 0, 1, 10, 0, 0, 8, 6, 1, 0, 8, 6, 1, 0, 1, 10, 0, 0, 6, 6, 1, 0, 0, 12, 1, 0, 16

Offset: 1

Gus Wiseman, Apr 09 2025

nonn

This described multiset (row n of A305936, Heinz number A181821) is generally not the same as the multiset of prime indices of n (A112798). For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(n) partitions for n = 6, 21, 30, 46:
  (1,1,2)  (1,1,1,1,2,2)  (1,1,1,2,2,3)  (1,1,1,1,1,1,1,1,1,2)
  (2,1,1)  (1,1,1,2,2,1)  (1,1,1,3,2,2)  (1,1,1,1,1,1,1,2,1,1)
           (1,2,2,1,1,1)  (2,2,1,1,1,3)  (1,1,1,1,1,1,2,1,1,1)
           (2,2,1,1,1,1)  (2,2,3,1,1,1)  (1,1,1,1,1,2,1,1,1,1)
                          (3,1,1,1,2,2)  (1,1,1,1,2,1,1,1,1,1)
                          (3,2,2,1,1,1)  (1,1,1,2,1,1,1,1,1,1)
                                         (1,1,2,1,1,1,1,1,1,1)
                                         (2,1,1,1,1,1,1,1,1,1)

Positions of 1 are A008578.
For anti-run permutations we have A335125.
For just prime indices we have A382771, firsts A382772, equal A382857.
These permutations for factorials are counted by A382774, equal A335407.
For equal instead of distinct run-lengths we have A382858.
Positions of 0 are A382912, complement A382913.
A044813 lists numbers whose binary expansion has distinct run-lengths, equal A140690.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000670, A000720, A003242, A048767, A130091, A181821, A238130, A305936, A351013, A351202, A382876, A382879.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Length[Select[Permutations[nrmptn[n]],UnsameQ@@Length/@Split[#]&]],{n,100}]

a(n) = A382771(A181821(n)) = A382771(A304660(n)).

A383089 Numbers whose prime indices have more than one permutation with all equal run-lengths.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140

Offset: 1

Gus Wiseman, Apr 18 2025

nonn

First differs from A362606 (complement A359178 with 1) in having 180 and lacking 240.
First differs from A130092 (complement A130091) in having 360 and lacking 240.
First differs from A351295 (complement A351294) in having 216 and lacking 240.
Includes all squarefree numbers A005117 except the primes A000040.
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 prime indices of 36 are {1,1,2,2}, and we have 4 permutations each having all equal run-lengths: (1,1,2,2), (1,2,1,2), (2,2,1,1), (2,1,2,1), so 36 is in the sequence.
The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}
   46: {1,9}
   51: {2,7}
   55: {3,5}
   57: {2,8}
   58: {1,10}
   60: {1,1,2,3}

Positions of terms > 1 in A382857 (distinct A382771), zeros A382879, ones A383112.
For run-sums instead of lengths we have A383015, counted by A383097.
Partitions of this type are counted by A383090.
The complement is A383091, counted by A383092, just zero A382915, just one A383094.
For distinct instead of equal run-sums we have A383113.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A047966 counts partitions with equal run-lengths, compositions A329738.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct run-lengths, ranks A130091.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000720, A000961, A001221, A001222, A048767, A353744, A353833, A381541, A381871, A382877, A383014, A383100.

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

The complement is A383091 = A382879 \/ A383112, counted by A382915 + A383094.

A383094 Number of integer partitions of n having exactly one permutation with all equal run-lengths.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 5, 6, 9, 7, 11, 10, 13, 12, 17, 14, 21, 16, 21, 18, 27, 22, 29, 22, 34, 25, 35, 28, 41, 28, 43, 30, 48, 38, 47, 38, 55, 36, 53, 46, 64, 40, 67, 42, 69, 54, 65, 46, 84, 51, 75, 62, 83, 52, 86, 62, 94, 70, 83, 58, 111, 60, 89, 80, 106, 74, 115, 66, 111

Offset: 0

Gus Wiseman, Apr 20 2025

nonn

			The partition (222211) has exactly one permutation with all equal run-lengths: (221122), so is counted under a(10).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (11111)  (411)     (511)      (422)
                                     (111111)  (22111)    (611)
                                               (1111111)  (2222)
                                                          (22211)
                                                          (221111)
                                                          (11111111)

The complement is ranked by A382879 \/ A383089.
For no choices we have A382915, ranks A382879.
For at least one choice we have A383013, for run-sums A383098, ranks A383110.
For more than one choice we have A383090, ranks A383089.
For at most one choice we have A383092, ranks A383091.
For run-sums instead of lengths we have A383095, ranks A383099.
Partitions of this type are ranked by A383112 = positions of 1 in A382857.
Cf. A047966, A072774, A098859, A100471, A100881, A100882, A100883, A304442.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A047993, A362608, A381871, A382076, A382771, A383096, A383097, A383111.

Table[Length[Select[IntegerPartitions[n], Length[Select[Permutations[#], SameQ@@Length/@Split[#]&]]==1&]],{n,0,20}]

More terms from Bert Dobbelaere, Apr 26 2025

cp's OEIS Frontend

A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A353849 Number of distinct positive run-sums of the n-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A382857 Number of ways to permute the prime indices of n so that the run-lengths are all equal.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A382879 Positions of 0 in A382857 (permutations of prime indices with equal run-lengths).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A382771 Number of ways to permute the prime indices of n so that the run-lengths are all different.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A383013 Number of integer partitions of n having a permutation with all equal run-lengths.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A382915 Number of integer partitions of n having no permutation with all equal run-lengths.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions

A382773 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all different.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula