A351013 -id:A351013 - cp's OEIS frontend

A351018 Number of integer compositions of n with all distinct even-indexed parts and all distinct odd-indexed parts.

1, 1, 2, 3, 6, 9, 18, 27, 46, 77, 122, 191, 326, 497, 786, 1207, 1942, 2905, 4498, 6703, 10574, 15597, 23754, 35043, 52422, 78369, 115522, 169499, 248150, 360521, 532466, 768275, 1116126, 1606669, 2314426, 3301879, 4777078, 6772657, 9677138, 13688079, 19406214

Offset: 0

Gus Wiseman, Feb 09 2022

nonn

Also the number of binary words of length n starting with 1 and having all distinct runs (ranked by A175413, counted by A351016).

			The a(1) = 1 through a(6) = 18 compositions:
  (1)  (2)    (3)    (4)      (5)      (6)
       (1,1)  (1,2)  (1,3)    (1,4)    (1,5)
              (2,1)  (2,2)    (2,3)    (2,4)
                     (3,1)    (3,2)    (3,3)
                     (1,1,2)  (4,1)    (4,2)
                     (2,1,1)  (1,1,3)  (5,1)
                              (1,2,2)  (1,1,4)
                              (2,2,1)  (1,2,3)
                              (3,1,1)  (1,3,2)
                                       (2,1,3)
                                       (2,3,1)
                                       (3,1,2)
                                       (3,2,1)
                                       (4,1,1)
                                       (1,1,2,2)
                                       (1,2,2,1)
                                       (2,1,1,2)
                                       (2,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The case of partitions is A000726.
The version for run-lengths instead of runs is A032020.
These words are ranked by A175413.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A116608 counts compositions by number of distinct parts.
A238130 and A238279 count compositions by number of runs.
A242882 counts compositions with distinct multiplicities.
A297770 counts distinct runs in binary expansion.
A325545 counts compositions with distinct differences.
A329738 counts compositions with equal run-lengths.
A329744 counts compositions by runs-resistance.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A003242, A025047, A098504, A098859, A106356, A212322, A328592, A329740, A334028, A349054, A350952, A351205.

Table[Length[Select[Tuples[{0,1},n],#=={}||First[#]==1&&UnsameQ@@Split[#]&]],{n,0,10}]

P(n)=prod(k=1, n, 1 + y*x^k + O(x*x^n));
seq(n)=my(p=P(n)); Vec(sum(k=0, n, polcoef(p,k\2,y)*(k\2)!*polcoef(p,(k+1)\2,y)*((k+1)\2)!)) \\ Andrew Howroyd, Feb 11 2022

a(n>0) = A351016(n)/2.

G.f.: Sum_{k>=0} floor(k/2)! * ceiling(k/2)! * ([y^floor(k/2)] P(x,y)) * ([y^ceiling(k/2)] P(x,y)), where P(x,y) = Product_{k>=1} 1 + y*x^k. - Andrew Howroyd, Feb 11 2022

Terms a(21) and beyond from Andrew Howroyd, Feb 11 2022

A351291 Numbers k such that the k-th composition in standard order does not have all distinct runs.

Original entry on oeis.org

13, 22, 25, 45, 46, 49, 53, 54, 59, 76, 77, 82, 89, 91, 93, 94, 97, 101, 102, 105, 108, 109, 110, 115, 118, 141, 148, 150, 153, 156, 162, 165, 166, 173, 177, 178, 180, 181, 182, 183, 187, 189, 190, 193, 197, 198, 201, 204, 205, 209, 210, 213, 214, 216, 217

Offset: 1

Gus Wiseman, Feb 12 2022

nonn

The n-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 n, 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:
  13:     1101  (1,2,1)
  22:    10110  (2,1,2)
  25:    11001  (1,3,1)
  45:   101101  (2,1,2,1)
  46:   101110  (2,1,1,2)
  49:   110001  (1,4,1)
  53:   110101  (1,2,2,1)
  54:   110110  (1,2,1,2)
  59:   111011  (1,1,2,1,1)
  76:  1001100  (3,1,3)
  77:  1001101  (3,1,2,1)
  82:  1010010  (2,3,2)
  89:  1011001  (2,1,3,1)
  91:  1011011  (2,1,2,1,1)
  93:  1011101  (2,1,1,2,1)
  94:  1011110  (2,1,1,1,2)

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

The version for Heinz numbers of partitions is A130092, complement A130091.
Normal multisets with a permutation of this type appear to be A283353.
Partitions w/o permutations of this type are A351204, complement A351203.
The version using binary expansions is A351205, complement A175413.
The complement is A351290, counted by A351013.
A005811 counts runs in binary expansion, distinct A297770.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has all distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612, counted by A003242.
A345167 ranks alternating compositions, counted by A025047.
Counting words with all distinct runs:
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Selected statistics of standard compositions (A066099, reverse A228351):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767, distinct A351014.
- 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. A098859, A106356, A113835, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351015, A351201.

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

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}]

A351201 Numbers whose multiset of prime factors has a permutation without all distinct runs.

Original entry on oeis.org

12, 18, 20, 28, 36, 44, 45, 48, 50, 52, 60, 63, 68, 72, 75, 76, 80, 84, 90, 92, 98, 99, 100, 108, 112, 116, 117, 120, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 162, 164, 168, 171, 172, 175, 176, 180, 188, 192, 196, 198, 200, 204, 207, 208, 212, 216

Offset: 1

Gus Wiseman, Feb 12 2022

nonn

			The prime factors of 80 are {2,2,2,2,5} and the permutation (2,2,5,2,2) has runs (2,2), (5), and (2,2), which are not all distinct, so 80 is in the sequence. On the other hand, 24 has prime factors {2,2,2,3}, and all four permutations (3,2,2,2), (2,3,2,2), (2,2,3,2), (2,2,2,3) have distinct runs, so 24 is not in the sequence.
The terms and their prime indices begin:
     12: (2,1,1)         76: (8,1,1)        132: (5,2,1,1)
     18: (2,2,1)         80: (3,1,1,1,1)    140: (4,3,1,1)
     20: (3,1,1)         84: (4,2,1,1)      144: (2,2,1,1,1,1)
     28: (4,1,1)         90: (3,2,2,1)      147: (4,4,2)
     36: (2,2,1,1)       92: (9,1,1)        148: (12,1,1)
     44: (5,1,1)         98: (4,4,1)        150: (3,3,2,1)
     45: (3,2,2)         99: (5,2,2)        153: (7,2,2)
     48: (2,1,1,1,1)    100: (3,3,1,1)      156: (6,2,1,1)
     50: (3,3,1)        108: (2,2,2,1,1)    162: (2,2,2,2,1)
     52: (6,1,1)        112: (4,1,1,1,1)    164: (13,1,1)
     60: (3,2,1,1)      116: (10,1,1)       168: (4,2,1,1,1)
     63: (4,2,2)        117: (6,2,2)        171: (8,2,2)
     68: (7,1,1)        120: (3,2,1,1,1)    172: (14,1,1)
     72: (2,2,1,1,1)    124: (11,1,1)       175: (4,3,3)
     75: (3,3,2)        126: (4,2,2,1)      176: (5,1,1,1,1)

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

The version for run-lengths instead of runs is A024619.
These permutations are counted by A351202.
These rank the partitions counted by A351203, complement A351204.
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.
A283353 counts normal multisets with a permutation w/o all distinct runs.
A297770 counts distinct runs in binary expansion.
A333489 ranks anti-runs, complement A348612.
A351014 counts distinct runs in standard compositions, firsts A351015.
A351291 ranks compositions without all distinct runs.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
Cf. A001055, A001221, A001222, A061395, A098859, A106356, A106529.

Select[Range[100],Select[Permutations[Join@@ ConstantArray@@@FactorInteger[#]],!UnsameQ@@Split[#]&]!={}&]

A351204 Number of integer partitions of n such that every permutation has all distinct runs.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 9, 11, 14, 18, 20, 25, 28, 34, 41, 47, 53, 64, 72, 84, 98, 113, 128, 148, 169, 194, 223, 255, 289, 333, 377, 428, 488, 554, 629, 715, 807, 913, 1033, 1166, 1313, 1483, 1667, 1875, 2111, 2369, 2655, 2977, 3332, 3729, 4170, 4657, 5195, 5797, 6459

Offset: 0

Gus Wiseman, Feb 15 2022

nonn

Partitions enumerated by this sequence include those in which all parts are either the same or distinct as well as partitions with an even number of parts all of which except one are the same. - Andrew Howroyd, Feb 15 2022

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (2111)   (51)      (61)       (62)
                            (11111)  (222)     (421)      (71)
                                     (321)     (2221)     (431)
                                     (3111)    (4111)     (521)
                                     (111111)  (211111)   (2222)
                                               (1111111)  (5111)
                                                          (311111)
                                                          (11111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for run-lengths instead of runs is A000005.
The version for normal multisets is 2^(n-1) - A283353(n-3).
The complement is counted by A351203, ranked by A351201.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A238130 and A238279 count compositions by number of runs.
A297770 counts distinct runs in binary expansion.
A003242 counts anti-run compositions.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A000041, A035363, A047993, A116608, A144300, A329746, A351291.

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

\\ here Q(n) is A000009.
Q(n)={polcoef(prod(k=1, n, 1 + x^k + O(x*x^n)), n)}
a(n)={Q(n) + if(n, numdiv(n) - 1) + sum(k=1, (n-1)\3, sum(j=3, (n-1)\k, j%2==1 && n-k*j<>k))} \\ Andrew Howroyd, Feb 15 2022

Terms a(26) and beyond from Andrew Howroyd, Feb 15 2022

A350952 The smallest number whose binary expansion has exactly n distinct runs.

Original entry on oeis.org

0, 1, 2, 11, 38, 311, 2254, 36079, 549790, 17593311, 549687102, 35179974591, 2225029922430, 284803830071167, 36240869367020798, 9277662557957324543, 2368116566113212692990, 1212475681849964898811391, 619877748107024946567312382, 634754814061593545284927880191

Offset: 0

Gus Wiseman, Feb 14 2022

nonn

Positions of first appearances in A297770 (with offset 0).

The binary expansion of terms for n > 0 starts with 1, then floor(n/2) 0's, then alternates runs of increasing numbers of 1's, and decreasing numbers of 0's; see Python code. Thus, for n even, terms have n*(n/2+1)/2 binary digits, and for n odd, ((n+1) + (n-1)*((n-1)/2+1))/2 binary digits. - Michael S. Branicky, Feb 14 2022

			The terms and their binary expansions begin:
       0:                   ()
       1:                    1
       2:                   10
      11:                 1011
      38:               100110
     311:            100110111
    2254:         100011001110
   36079:     1000110011101111
  549790: 10000110001110011110
For example, 311 has binary expansion 100110111 with 5 distinct runs: 1, 00, 11, 0, 111.

Michael S. Branicky, Table of n, a(n) for n = 0..114
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Runs in binary expansion are counted by A005811, distinct A297770.
The version for run-lengths instead of runs is A165933, for A165413.
Subset of A175413 (binary expansion has distinct runs), for lengths A044813.
The version for standard compositions is A351015.
A000120 counts binary weight.
A011782 counts integer compositions.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A334028 counts distinct parts in standard compositions.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A003242, A070939, A085207, A098859, A233564, A325545, A328592, A333489, A351203, A351291.

q=Table[Length[Union[Split[If[n==0,{},IntegerDigits[n,2]]]]],{n,0,1000}];Table[Position[q,i][[1,1]]-1,{i,Union[q]}]

a(n)={my(t=0); for(k=1, (n+1)\2, t=((t<Andrew Howroyd, Feb 15 2022

def a(n): # returns term by construction
    if n == 0: return 0
    q, r = divmod(n, 2)
    if r == 0:
        s = "".join("1"*i + "0"*(q-i+1) for i in range(1, q+1))
        assert len(s) == n*(n//2+1)//2
    else:
        s = "1" + "".join("0"*(q-i+2) + "1"*i for i in range(2, q+2))
        assert len(s) == ((n+1) + (n-1)*((n-1)//2+1))//2
    return int(s, 2)
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 14 2022

a(9)-a(19) from Michael S. Branicky, Feb 14 2022

A351203 Number of integer partitions of n of whose permutations do not all have distinct runs.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 6, 11, 16, 24, 36, 52, 73, 101, 135, 184, 244, 321, 418, 543, 694, 889, 1127, 1427, 1789, 2242, 2787, 3463, 4276, 5271, 6465, 7921, 9655, 11756, 14254, 17262, 20830, 25102, 30152, 36172, 43270, 51691, 61594, 73300, 87023, 103189, 122099, 144296, 170193, 200497

Offset: 0

Gus Wiseman, Feb 12 2022

nonn

			The a(4) = 1 through a(9) = 16 partitions:
  (211)  (221)  (411)    (322)    (332)      (441)
         (311)  (2211)   (331)    (422)      (522)
                (21111)  (511)    (611)      (711)
                         (3211)   (3221)     (3321)
                         (22111)  (3311)     (4221)
                         (31111)  (4211)     (4311)
                                  (22211)    (5211)
                                  (32111)    (22221)
                                  (41111)    (32211)
                                  (221111)   (33111)
                                  (2111111)  (42111)
                                             (51111)
                                             (222111)
                                             (321111)
                                             (2211111)
                                             (3111111)
For example, the partition x = (2,1,1,1,1) has the permutation (1,1,2,1,1), with runs (1,1), (2), (1,1), which are not all distinct, so x is counted under a(6).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for run-lengths instead of runs is A144300.
The version for normal multisets is A283353.
The Heinz numbers of these partitions are A351201.
The complement is counted by A351204.
A005811 counts runs in binary expansion.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A297770 counts distinct runs in binary expansion.
A003242 counts anti-run compositions, ranked by A333489.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A000041, A035363, A047993, A116608, A238130 or A238279, A325545, A329746, A350842, A351003, A351004, A351291.

Table[Length[Select[IntegerPartitions[n],MemberQ[Permutations[#],_?(!UnsameQ@@Split[#]&)]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
from itertools import permutations, groupby
from collections import Counter
def A351203(n):
    c = 0
    for s, p in partitions(n,size=True):
        for q in permutations(Counter(p).elements(),s):
            if max(Counter(tuple(g) for k, g in groupby(q)).values(),default=0) > 1:
                c += 1
                break
    return c # Chai Wah Wu, Oct 16 2023

a(n) = A000041(n) - A351204(n). - Andrew Howroyd, Jan 27 2024

a(26) onwards from Andrew Howroyd, Jan 27 2024

A351592 Number of Look-and-Say partitions (A239455) of n without distinct multiplicities, i.e., those that are not Wilf partitions (A098859).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 5, 2, 8, 9, 8, 6, 21, 14, 20, 26, 31, 24, 53, 35, 60, 68, 78, 76, 140, 115, 163, 183, 232, 218, 343, 301, 433, 432, 565, 542, 774, 728, 958, 977, 1251, 1220, 1612, 1561, 2053, 2090, 2618, 2609, 3326, 3378

Offset: 0

Gus Wiseman, Feb 16 2022

nonn

A partition is Look-and-Say iff it has a permutation with all distinct run-lengths. For example, the partition y = (2,2,2,1,1,1) has the permutation (2,2,1,1,1,2), with run-lengths (2,3,1), which are distinct, so y is counted under A239455(9).
A partition is Wilf iff it has distinct multiplicities of parts. For example, (2,2,2,1,1,1) has multiplicities (3,3), so is not counted under A098859(9).
The Heinz numbers of these partitions are given by A351294 \ A130091.
Is a(17) = 0 the last zero of the sequence?

			The a(9) = 1 through a(18) = 5 partitions are (empty columns not shown):
  n=9:      n=12:       n=15:         n=16:       n=18:
  --------------------------------------------------------------
  (222111)  (333111)    (333222)      (33331111)  (444222)
            (22221111)  (444111)                  (555111)
                        (2222211111)              (3322221111)
                                                  (32222211111)
                                                  (222222111111)

Wilf partitions are counted by A098859, ranked by A130091.
Look-and-Say partitions are counted by A239455, ranked by A351294.
Non-Wilf partitions are counted by A336866, ranked by A130092.
Non-Look-and-Say partitions are counted by A351293, ranked by A351295.
A000569 = number of graphical partitions, complement A339617.
A032020 = number of binary expansions with all distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A225485/A325280 = frequency depth, ranked by A182850/A323014.
A329738 = compositions with all equal run-lengths.
A329739 = compositions with all distinct run-lengths
A351013 = compositions with all distinct runs.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
A351292 = patterns with all distinct run-lengths, for all runs A351200.
Cf. A000041, A008284, A018783, A047966, A181819, A182857, A238130, A305563, A319149, A351203, A351204, A351290.

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

a(n) = A239455(n) - A098859(n). Here we assume A239455(0) = 1.

More terms from Jinyuan Wang, Feb 14 2025

A353390 Number of compositions of n whose own run-lengths are a subsequence (not necessarily consecutive).

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 3, 2, 2, 8, 17, 26, 43, 77, 129, 210, 351, 569

Offset: 0

Gus Wiseman, May 15 2022

			The a(0) = 1 through a(9) = 8 compositions (empty columns indicated by dots):
  ()  (1)  .  .  (22)  (122)  (1122)  (11221)  (21122)  (333)
                       (221)  (1221)  (12211)  (22112)  (22113)
                              (2211)                    (22122)
                                                        (31122)
                                                        (121122)
                                                        (122112)
                                                        (211221)
                                                        (221121)
For example, the composition y = (2,2,3,3,1) has run-lengths (2,2,1), which form a (non-consecutive) subsequence, so y is counted under a(11).

The version for partitions is A325702.
The recursive version is A353391, ranked by A353431.
The consecutive case is A353392, ranked by A353432.
These compositions are ranked by A353402.
The reverse version is A353403.
The recursive consecutive version is A353430.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A047966 counts uniform partitions, compositions A329738.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223, partitions A108917.
A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-lengths, for runs A351013.
A353400 counts compositions with all run-lengths > 2.
Cf. A005811, A103295, A114901, A181591, A238279, A242882, A324572, A333755, A351017, A353401, A353426.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MemberQ[Subsets[#],Length/@Split[#]]&]],{n,0,15}]

A353391 Number of compositions of n that are empty, a singleton, or whose run-lengths are a subsequence that is already counted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 5, 7, 9, 11, 15, 22, 38, 45, 87, 93

Offset: 0

Gus Wiseman, May 15 2022

			The a(9) = 4 through a(14) = 15 compositions (A..E = 10..14):
  (9)       (A)       (B)       (C)       (D)       (E)
  (333)     (2233)    (141122)  (2244)    (161122)  (2255)
  (121122)  (3322)    (221123)  (4422)    (221125)  (5522)
  (221121)  (131122)  (221132)  (151122)  (221134)  (171122)
            (221131)  (221141)  (221124)  (221143)  (221126)
                      (231122)  (221142)  (221152)  (221135)
                      (321122)  (221151)  (221161)  (221153)
                                (241122)  (251122)  (221162)
                                (421122)  (341122)  (221171)
                                          (431122)  (261122)
                                          (521122)  (351122)
                                                    (531122)
                                                    (621122)
                                                    (122121122)
                                                    (221121221)

The non-recursive version is A353390, ranked by A353402.
The non-recursive consecutive version is A353392, ranked by A353432.
The non-recursive reverse version is A353403.
The unordered version is A353426, ranked by A353393 (nonprime A353389).
The consecutive version is A353430.
These compositions are ranked by A353431.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A329738 counts uniform compositions, partitions A047966.
A114901 counts compositions with no runs of length 1.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223.
A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-length.
Cf. A005811, A032020, A103295, A114640, A165413, A181591, A242882, A324572, A325702, A333755, A351013, A353401.

yosQ[y_]:=Length[y]<=1||MemberQ[Subsets[y],Length/@Split[y]]&&yosQ[Length/@Split[y]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],yosQ]],{n,0,15}]

cp's OEIS Frontend

A351018 Number of integer compositions of n with all distinct even-indexed parts and all distinct odd-indexed parts.

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A351291 Numbers k such that the k-th composition in standard order does not have all distinct runs.

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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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A351201 Numbers whose multiset of prime factors has a permutation without all distinct runs.

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A351204 Number of integer partitions of n such that every permutation has all distinct runs.

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A350952 The smallest number whose binary expansion has exactly n distinct runs.

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A351203 Number of integer partitions of n of whose permutations do not all have distinct runs.

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A351592 Number of Look-and-Say partitions (A239455) of n without distinct multiplicities, i.e., those that are not Wilf partitions (A098859).