A351201 -id:A351201 - cp's OEIS frontend

A239455 Number of Look-and-Say partitions of n; see Comments.

0, 1, 2, 2, 4, 5, 7, 10, 13, 16, 21, 28, 33, 45, 55, 65, 83, 105, 121, 155, 180, 217, 259, 318, 362, 445, 512, 614, 707, 850, 958, 1155, 1309, 1543, 1754, 2079, 2327, 2740, 3085, 3592, 4042, 4699, 5253, 6093, 6815, 7839, 8751, 10069, 11208, 12832, 14266, 16270

Offset: 0

Clark Kimberling and Peter J. C. Moses, Mar 19 2014

nonn

Suppose that p = x(1) >= x(2) >= ... >= x(k) is a partition of n. Let y(1) > y(2) > ... > y(h) be the distinct parts of p, and let m(i) be the multiplicity of y(i) for 1 <= i <= h. Then we can "look" at p as "m(1) y(1)'s and m(2) y(2)'s and ... m(h) y(h)'s". Reversing the m's and y's, we can then "say" the Look-and-Say partition of p, denoted by LS(p). The name "Look-and-Say" follows the example of Look-and-Say integer sequences (e.g., A005150). As p ranges through the partitions of n, LS(p) ranges through all the Look-and-Say partitions of n. The number of these is A239455(n).
The Look-and-Say array is distinct from the Wilf array, described at A098859; for example, the number of Look-and-Say partitions of 9 is A239455(9) = 16, whereas the number of Wilf partitions of 9 is A098859(9) = 15. The Look-and-Say partition of 9 which is not a Wilf partition of 9 is [2,2,2,1,1,1].
Conjecture: 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 all distinct, so y is counted under a(9). - Gus Wiseman, Aug 11 2025
Also the number of integer partitions y of n such that there is a pairwise disjoint way to choose a strict integer partition of each multiplicity (or run-length) of y. - Gus Wiseman, Aug 11 2025

			The 11 partitions of 6 generate 7 Look-and-Say partitions as follows:
6 -> 111111
51 -> 111111
42 -> 111111
411 -> 21111
33 -> 222
321 -> 111111
3111 -> 3111
222 -> 33
2211 -> 222
21111 -> 411
111111 -> 6,
so that a(6) counts these 7 partitions: 111111, 21111, 222, 3111, 33, 411, 6.

These include all Wilf partitions, counted by A098859, ranked by A130091.
These partitions are listed by A239454 in graded reverse-lex order.
Non-Wilf partitions are counted by A336866, ranked by A130092.
A variant for runs is A351204, complement A351203.
The complement is counted by A351293, apparently ranked by A351295, conjugate A381433.
These partitions appear to be ranked by A351294, conjugate A381432.
The non-Wilf case is counted by A351592.
For normal multisets we appear to have A386580, complement A386581.
A000110 counts set partitions, ordered A000670.
A000569 = graphical partitions, complement A339617.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A181819 = Heinz number of the prime signature of n (prime shadow).
A279790 counts disjoint families on strongly normal multisets.
A329738 = compositions with all equal run-lengths.
A386583 counts separable partitions, sums A325534, ranks A335433.
A386584 counts inseparable partitions, sums A325535, ranks A335448.
A386585 counts separable type partitions, sums A336106, ranks A335127.
A386586 counts inseparable type partitions, sums A386638 or A025065, ranks A335126.
Counting words with all distinct run-lengths:
- A032020 = binary expansions, for runs A351018, ranked by A044813.
- A329739 = compositions, for runs A351013, ranked by A351596.
- A351017 = binary words, for runs A351016.
- A351292 = patterns, for runs A351200.
Cf. A000041, A008284, A047966, A182857, A225485, A297770, A304660, A305563, A329746, A351201, A351202, A351291.

LS[part_List] := Reverse[Sort[Flatten[Map[Table[#[[2]], {#[[1]]}] &, Tally[part]]]]]; LS[n_Integer] := #[[Reverse[Ordering[PadRight[#]]]]] &[DeleteDuplicates[Map[LS, IntegerPartitions[n]]]]; TableForm[t = Map[LS[#] &, Range[10]]](*A239454,array*)
Flatten[t](*A239454,sequence*)
Map[Length[LS[#]] &, Range[25]](*A239455*)
(* Peter J. C. Moses, Mar 18 2014 *)
disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
Table[Length[Select[IntegerPartitions[n],Length[disjointFamilies[#]]>0&]],{n,0,10}] (* Gus Wiseman, Aug 11 2025 *)

A351294 Numbers whose multiset of prime factors has at least one permutation with all distinct run-lengths.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109

Offset: 1

Gus Wiseman, Feb 15 2022

nonn

First differs from A130091 (Wilf partitions) in having 216.
See A239455 for the definition of Look-and-Say partitions.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
      1: ()            20: (3,1,1)         47: (15)
      2: (1)           23: (9)             48: (2,1,1,1,1)
      3: (2)           24: (2,1,1,1)       49: (4,4)
      4: (1,1)         25: (3,3)           50: (3,3,1)
      5: (3)           27: (2,2,2)         52: (6,1,1)
      7: (4)           28: (4,1,1)         53: (16)
      8: (1,1,1)       29: (10)            54: (2,2,2,1)
      9: (2,2)         31: (11)            56: (4,1,1,1)
     11: (5)           32: (1,1,1,1,1)     59: (17)
     12: (2,1,1)       37: (12)            61: (18)
     13: (6)           40: (3,1,1,1)       63: (4,2,2)
     16: (1,1,1,1)     41: (13)            64: (1,1,1,1,1,1)
     17: (7)           43: (14)            67: (19)
     18: (2,2,1)       44: (5,1,1)         68: (7,1,1)
     19: (8)           45: (3,2,2)         71: (20)
For example, the prime indices of 216 are {1,1,1,2,2,2}, and there are four permutations with distinct run-lengths: (1,1,2,2,2,1), (1,2,2,2,1,1), (2,1,1,1,2,2), (2,2,1,1,1,2); so 216 is in the sequence. It is the Heinz number of the Look-and-Say partition of (3,3,2,1).

The Wilf case (distinct multiplicities) is A130091, counted by A098859.
The complement of the Wilf case is A130092, counted by A336866.
These partitions appear to be counted by A239455.
A variant for runs is A351201, counted by A351203 (complement A351204).
The complement is A351295, counted by A351293.
A032020 = number of binary expansions with distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A056239 = sum of prime indices, row sums of A112798.
A165413 = number of run-lengths in binary expansion, for all runs A297770.
A181819 = Heinz number of prime signature (prime shadow).
A182850/A323014 = frequency depth, counted by A225485/A325280.
A320922 ranks graphical partitions, complement A339618, counted by A000569.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A333489 ranks anti-runs, complement A348612.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
A351292 = patterns with all distinct run-lengths, for all runs A351200.
Cf. A000961, A001221, A001222, A047966, A175413, A182857, A304660, A320924, A328592, A329747, A351202, A351290, A351592.

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

Name edited by Gus Wiseman, Aug 13 2025

A351293 Number of non-Look-and-Say partitions of n. Number of integer partitions of n such that there is no way to choose a disjoint strict integer partition of each multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 21, 28, 44, 56, 80, 111, 148, 192, 264, 335, 447, 575, 743, 937, 1213, 1513, 1924, 2396, 3011, 3715, 4646, 5687, 7040, 8600, 10556, 12804, 15650, 18897, 22930, 27593, 33296, 39884, 47921, 57168, 68360, 81295, 96807, 114685

Offset: 0

Gus Wiseman, Feb 16 2022

nonn

First differs from A336866 (non-Wilf partitions) at a(9) = 14, A336866(9) = 15, the difference being the partition (2,2,2,1,1,1).

See A239455 for the definition of Look-and-Say partitions.

			The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)     (54)
              (41)  (51)    (52)    (62)     (63)
                    (321)   (61)    (71)     (72)
                    (2211)  (421)   (431)    (81)
                            (3211)  (521)    (432)
                                    (3221)   (531)
                                    (3311)   (621)
                                    (4211)   (3321)
                                    (32111)  (4221)
                                             (4311)
                                             (5211)
                                             (32211)
                                             (42111)
                                             (321111)

The complement is counted by A239455, ranked by A351294.
These are all non-Wilf partitions (counted by A336866, ranked by A130092).
A variant for runs is A351203, complement A351204, ranked by A351201.
These partitions appear to be ranked by A351295.
Non-Wilf partitions in the complement are counted by A351592.
A000569 = graphical partitions, complement A339617.
A032020 = number of binary expansions with all distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A098859 = Wilf partitions (distinct multiplicities), ranked by A130091.
A181819 = Heinz number of the prime signature of n (prime shadow).
A329738 = compositions with all equal run-lengths.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
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, A047966, A182857, A225485, A238130, A297770, A304660, A305563, A329740, A329746, A351202, A351291.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
Table[Length[Select[IntegerPartitions[n],Length[disjointFamilies[#]]==0&]],{n,0,15}] (* Gus Wiseman, Aug 13 2025 *)

a(n) = A000041(n) - A239455(n).

Edited by Gus Wiseman, Aug 12 2025

A351295 Numbers whose multiset of prime factors has no permutation with all distinct 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, Feb 16 2022

nonn

First differs from A130092 (non-Wilf partitions) in lacking 216.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
      6: (2,1)         46: (9,1)         84: (4,2,1,1)
     10: (3,1)         51: (7,2)         85: (7,3)
     14: (4,1)         55: (5,3)         86: (14,1)
     15: (3,2)         57: (8,2)         87: (10,2)
     21: (4,2)         58: (10,1)        90: (3,2,2,1)
     22: (5,1)         60: (3,2,1,1)     91: (6,4)
     26: (6,1)         62: (11,1)        93: (11,2)
     30: (3,2,1)       65: (6,3)         94: (15,1)
     33: (5,2)         66: (5,2,1)       95: (8,3)
     34: (7,1)         69: (9,2)        100: (3,3,1,1)
     35: (4,3)         70: (4,3,1)      102: (7,2,1)
     36: (2,2,1,1)     74: (12,1)       105: (4,3,2)
     38: (8,1)         77: (5,4)        106: (16,1)
     39: (6,2)         78: (6,2,1)      110: (5,3,1)
     42: (4,2,1)       82: (13,1)       111: (12,2)
For example, the prime indices of 150 are {1,2,3,3}, with permutations and run-lengths (right):
  (3,3,2,1) -> (2,1,1)
  (3,3,1,2) -> (2,1,1)
  (3,2,3,1) -> (1,1,1,1)
  (3,2,1,3) -> (1,1,1,1)
  (3,1,3,2) -> (1,1,1,1)
  (3,1,2,3) -> (1,1,1,1)
  (2,3,3,1) -> (1,2,1)
  (2,3,1,3) -> (1,1,1,1)
  (2,1,3,3) -> (1,1,2)
  (1,3,3,2) -> (1,2,1)
  (1,3,2,3) -> (1,1,1,1)
  (1,2,3,3) -> (1,1,2)
Since none have all distinct run-lengths, 150 is in the sequence.

Wilf partitions are counted by A098859, ranked by A130091.
Non-Wilf partitions are counted by A336866, ranked by A130092.
A variant for runs is A351201, counted by A351203 (complement A351204).
These partitions appear to be counted by A351293.
The complement is A351294, apparently counted by A239455.
A032020 = number of binary expansions with distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A056239 = sum of prime indices, row sums of A112798.
A165413 = number of distinct run-lengths in binary expansion.
A181819 = Heinz number of prime signature (prime shadow).
A182850/A323014 = frequency depth, counted by A225485/A325280.
A297770 = number of distinct runs in binary expansion.
A320922 ranks graphical partitions, complement A339618, counted by A000569.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A329747 = runs-resistance, counted by A329746.
A333489 ranks anti-runs, complement A348612.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
Cf. A000961, A001221, A001222, A175413, A182857, A304660, A320924, A328592, A351202, A351290.

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

Name edited by Gus Wiseman, Aug 13 2025

A351014 Number of distinct runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 07 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 number 3310 has binary expansion 110011101110 and standard composition (1,3,1,1,2,1,1,2), with runs (1), (3), (1,1), (2), (1,1), (2), of which 4 are distinct, so a(3310) = 4.

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

Counting not necessarily distinct runs gives A124767.
Using binary expansions instead of standard compositions gives A297770.
Positions of first appearances are A351015.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612.
A345167 ranks alternating compositions, counted by A025047.
A351204 counts partitions where every permutation has 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.
- A351202 = permutations of prime factors.
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. A098859, A106356, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351201.

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

A351013 Number of integer compositions of n with all distinct runs.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 48, 88, 161, 294, 512, 970, 1634, 2954, 5156, 9119, 15618, 27354, 46674, 80130, 138078, 232286, 394966, 665552, 1123231, 1869714, 3146410, 5186556, 8620936, 14324366, 23529274, 38564554, 63246744, 103578914, 167860584, 274465845

Offset: 0

Gus Wiseman, Feb 09 2022

nonn

			The a(1) = 1 through a(5) = 14 compositions:
  (1)  (2)    (3)      (4)        (5)
       (1,1)  (1,2)    (1,3)      (1,4)
              (2,1)    (2,2)      (2,3)
              (1,1,1)  (3,1)      (3,2)
                       (1,1,2)    (4,1)
                       (2,1,1)    (1,1,3)
                       (1,1,1,1)  (1,2,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,1,2)
                                  (1,1,2,1)
                                  (1,2,1,1)
                                  (2,1,1,1)
                                  (1,1,1,1,1)
For example, the composition c = (3,1,1,1,1,2,1,1,3,4,1,1) has runs (3), (1,1,1,1), (2), (1,1), (3), (4), (1,1), and since (3) and (1,1) both appear twice, c is not counted under a(20).

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 A329739, normal A329740.
These compositions are ranked by A351290, complement A351291.
A000005 counts constant compositions, ranked by A272919.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A059966 counts binary 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.
A329744 counts compositions by runs-resistance.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct runs:
- 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. A003242, A025047, A044813, A098504, A098859, A106356, A329738, A328592, A334028, A351015, A351201, A351204.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Split[#]&]],{n,0,10}]

\\ here LahI is A111596 as row polynomials.
LahI(n,y) = {sum(k=1, n, y^k*(-1)^(n-k)*(n!/k!)*binomial(n-1, k-1))}
S(n) = {my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); 1 + sum(i=1, (sqrtint(8*n+1)-1)\2, polcoef(p,i,y)*LahI(i,y))}
seq(n)={my(q=S(n)); [subst(serlaplace(p),y,1) | p<-Vec(prod(k=1, n, subst(q + O(x*x^(n\k)), x, x^k)))]} \\ Andrew Howroyd, Feb 12 2022

Terms a(26) and beyond from Andrew Howroyd, Feb 12 2022

A351202 Number of permutations of the multiset of prime factors of n (or ordered prime factorizations of n) with all distinct runs.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 13 2022

nonn

			The a(36) = 2 permutations are (1,1,2,2), (2,2,1,1). Missing are: (1,2,1,2), (1,2,2,1), (2,1,1,2), (2,1,2,1). Here we use prime indices instead of factors.

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

The maximum number of possible permutations is A008480.
Positions less than A008480 are A351201.
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.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A283353 counts normal multisets with a permutation without distinct runs.
A297770 counts distinct runs in binary expansion.
A351014 counts distinct runs in standard compositions, firsts A351015.
A351204 = partitions whose perms. have distinct runs, complement A351203.
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, A003242, A061395, A116608, A238130 or A238279, A328592, A351291.

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

A351015 Smallest k such that the k-th composition in standard order has n distinct runs.

Original entry on oeis.org

0, 1, 5, 27, 155, 1655, 18039, 281975

Offset: 0

Gus Wiseman, Feb 10 2022

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.

It would be very interesting to have a formula or general construction for a(n). - Gus Wiseman, Feb 12 2022

			The terms together with their binary expansions and corresponding compositions begin:
       0:                    0  ()
       1:                    1  (1)
       5:                  101  (2,1)
      27:                11011  (1,2,1,1)
     155:             10011011  (3,1,2,1,1)
    1655:          11001110111  (1,3,1,1,2,1,1,1)
   18039:      100011001110111  (4,1,3,1,1,2,1,1,1)
  281975:  1000100110101110111  (4,3,1,2,2,1,1,2,1,1,1)

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

The version for Heinz numbers and prime multiplicities is A006939.
Counting not necessarily distinct runs gives A113835 (up to zero).
Using binary expansions instead of standard compositions gives A350952.
These are the positions of first appearances in A351014.
A005811 counts runs in binary expansion, distinct A297770.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612.
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.
Selected statistics of standard compositions (A066099, reverse A228351):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767.
- Number of distinct parts is A334028.
Cf. A106356, A238279, A242882, A318928, A325545, A328592, A329745, A351201, A351204.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
s=Table[Length[Union[Split[stc[n]]]],{n,0,1000}];
Table[Position[s,k][[1,1]]-1,{k,Union[s]}]

A351016 Number of binary words of length n with all distinct runs.

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 36, 54, 92, 154, 244, 382, 652, 994, 1572, 2414, 3884, 5810, 8996, 13406, 21148, 31194, 47508, 70086, 104844, 156738, 231044, 338998, 496300, 721042, 1064932, 1536550, 2232252, 3213338, 4628852, 6603758, 9554156, 13545314, 19354276

Offset: 0

Gus Wiseman, Feb 07 2022

nonn

These are binary words where the runs of zeros have all distinct lengths and the runs of ones also have all distinct lengths. For n > 0 this is twice the number of terms of A175413 that have n digits in binary.

			The a(0) = 1 through a(4) = 12 binary words:
  ()   0    00    000    0000
       1    01    001    0001
            10    011    0010
            11    100    0011
                  110    0100
                  111    0111
                         1000
                         1011
                         1100
                         1101
                         1110
                         1111
For example, the word (1,1,0,1) has three runs (1,1), (0), (1), which are all distinct, so is counted under a(4).

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

The version for compositions is A351013, lengths A329739, ranked by A351290.
The version for [run-]lengths is A351017.
The version for expansions is A351018, lengths A032020, ranked by A175413.
The version for patterns is A351200, lengths A351292.
The version for permutations of prime factors is A351202.
A000120 counts binary weight.
A001037 counts binary Lyndon words, necklaces A000031, aperiodic A027375.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A242882 counts compositions with distinct multiplicities.
A297770 counts distinct runs in binary expansion.
A325545 counts compositions with distinct differences.
A329767 counts binary words by runs-resistance.
A351014 counts distinct runs in standard compositions.
A351204 counts partitions whose permutations all have all distinct runs.
Cf. A003242, A098859, A106356, A116608, A238130 or A238279, A328592, A329738, A329745, A334028, A351201.

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

from itertools import groupby, product
def adr(s):
    runs = [(k, len(list(g))) for k, g in groupby(s)]
    return len(runs) == len(set(runs))
def a(n):
    if n == 0: return 1
    return 2*sum(adr("1"+"".join(w)) for w in product("01", repeat=n-1))
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 08 2022

a(n>0) = 2 * A351018(n).

a(25)-a(32) from Michael S. Branicky, Feb 08 2022

a(33)-a(38) from David A. Corneth, Feb 08 2022

cp's OEIS Frontend

A239455 Number of Look-and-Say partitions of n; see Comments.

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A351294 Numbers whose multiset of prime factors has at least one permutation with all distinct run-lengths.

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A351293 Number of non-Look-and-Say partitions of n. Number of integer partitions of n such that there is no way to choose a disjoint strict integer partition of each multiplicity.

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A351295 Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.

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A351014 Number of distinct runs in the n-th composition in standard order.

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A351013 Number of integer compositions of n with all distinct runs.

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A351202 Number of permutations of the multiset of prime factors of n (or ordered prime factorizations of n) with all distinct runs.

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

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