A098504 -id:A098504 - cp's OEIS frontend

A272919 Numbers of the form 2^(n-1)(2^(nm)-1)/(2^n-1), n >= 1, m >= 1.

1, 2, 3, 4, 7, 8, 10, 15, 16, 31, 32, 36, 42, 63, 64, 127, 128, 136, 170, 255, 256, 292, 511, 512, 528, 682, 1023, 1024, 2047, 2048, 2080, 2184, 2340, 2730, 4095, 4096, 8191, 8192, 8256, 10922, 16383, 16384, 16912, 18724, 32767, 32768, 32896, 34952, 43690, 65535, 65536, 131071

Offset: 1

Ivan Neretin, May 10 2016

nonn

In other words, numbers whose binary representation consists of one or more repeating blocks with only one 1 in each block.
Also, fixed points of the permutations A139706 and A139708.
Each a(n) is a term of A064896 multiplied by some power of 2. As such, this sequence must also be a subsequence of A125121.
Also the numbers that uniquely index a Haar graph (i.e., 5 and 6 are not in the sequence since H(5) is isomorphic to H(6)). - Eric W. Weisstein, Aug 19 2017
From Gus Wiseman, Apr 04 2020: (Start)
The k-th composition in standard order (row k of 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. This sequence lists all positive integers k such that the k-th composition in standard order is constant. For example, the sequence together with the corresponding constant compositions begins:
()                  136: (4,4)
(1)                 170: (2,2,2,2)
(2)                 255: (1,1,1,1,1,1,1,1)
(1,1)               256: (9)
(3)                 292: (3,3,3)
(1,1,1)             511: (1,1,1,1,1,1,1,1,1)
(4)                 512: (10)
(2,2)               528: (5,5)
(1,1,1,1)           682: (2,2,2,2,2)
(5)                1023: (1,1,1,1,1,1,1,1,1,1)
(1,1,1,1,1)        1024: (11)
(6)                2047: (1,1,1,1,1,1,1,1,1,1,1)
(3,3)              2048: (12)
(2,2,2)            2080: (6,6)
(1,1,1,1,1,1)      2184: (4,4,4)
(7)                2340: (3,3,3,3)
(1,1,1,1,1,1,1)    2730: (2,2,2,2,2,2)
(8)                4095: (1,1,1,1,1,1,1,1,1,1,1,1)
(End)

Ivan Neretin, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Haar Graph

Cf. A064896, A139708.
Cf. A137706 (smallest number indexing a new Haar graph).
Compositions in standard order are A066099.
Strict compositions are ranked by A233564.
Cf. A000120, A027750, A070939, A098504, A124767, A164894, A228351, A238279.

N:= 10^6: # to get all terms <= N
R:= select(`<=`,{seq(seq(2^(n-1)*(2^(n*m)-1)/(2^n-1), m = 1 .. ilog2(2*N)/n), n = 1..ilog2(2*N))},N):
sort(convert(R,list)); # Robert Israel, May 10 2016

Flatten@Table[d = Reverse@Divisors[n]; 2^(d - 1)*(2^n - 1)/(2^d - 1), {n, 17}]

From Gus Wiseman, Apr 04 2020: (Start)
A333381(a(n)) = A027750(n).
For n > 0, A124767(a(n)) = 1.
If n is a power of two, A333628(a(n)) = 0, otherwise = 1.
A333627(a(n)) is a power of 2.
(End)

A329739 Number of compositions of n whose run-lengths are all different.

Original entry on oeis.org

1, 1, 2, 2, 5, 8, 10, 20, 28, 41, 62, 102, 124, 208, 278, 426, 571, 872, 1158, 1718, 2306, 3304, 4402, 6286, 8446, 11725, 15644, 21642, 28636, 38956, 52296, 70106, 93224, 124758, 165266, 218916, 290583, 381706, 503174, 659160, 865020, 1124458, 1473912, 1907298

Offset: 0

Gus Wiseman, Nov 20 2019

nonn

A composition of n is a finite sequence of positive integers with sum n.

			The a(1) = 1 through a(7) = 20 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (113)    (33)      (115)
                    (112)   (122)    (114)     (133)
                    (211)   (221)    (222)     (223)
                    (1111)  (311)    (411)     (322)
                            (1112)   (1113)    (331)
                            (2111)   (3111)    (511)
                            (11111)  (11112)   (1114)
                                     (21111)   (1222)
                                     (111111)  (2221)
                                               (4111)
                                               (11113)
                                               (11122)
                                               (22111)
                                               (31111)
                                               (111112)
                                               (111211)
                                               (112111)
                                               (211111)
                                               (1111111)

The normal case is A329740.
The case of partitions is A098859.
Strict compositions are A032020.
Compositions with relatively prime run-lengths are A000740.
Compositions with distinct multiplicities are A242882.
Compositions with distinct differences are A325545.
Compositions with equal run-lengths are A329738.
Compositions with normal run-lengths are A329766.
Cf. A003242, A008965, A059966, A098504, A098859, A107429, A175342, A238130, A328592, A329745.

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

a(21)-a(26) from Giovanni Resta, Nov 22 2019

a(27)-a(43) from Alois P. Heinz, Jul 06 2020

A304442 Number of partitions of n in which the sequence of the sum of the same summands is constant.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 5, 2, 7, 3, 5, 2, 13, 2, 5, 4, 11, 2, 13, 2, 12, 4, 5, 2, 28, 3, 5, 5, 12, 2, 18, 2, 17, 4, 5, 4, 44, 2, 5, 4, 24, 2, 18, 2, 12, 10, 5, 2, 63, 3, 9, 4, 12, 2, 34, 4, 24, 4, 5, 2, 67, 2, 5, 10, 27, 4, 18, 2, 12, 4, 14, 2, 120, 2, 5, 7, 12, 4, 18, 2, 54

Offset: 0

Seiichi Manyama, May 12 2018

nonn

Said differently, these are partitions whose run-sums are all equal. - Gus Wiseman, Jun 25 2022

			a(72) = binomial(d(72),1) + binomial(d(36),2) + binomial(d(24),3) + binomial(d(18),4) + binomial(d(12),6) = 12 + 36 + 56 + 15 + 1 = 120, where d(n) is the number of divisors of n.
--+----------------------+-----------------------------------------
n |                      | Sequence of the sum of the same summands
--+----------------------+-----------------------------------------
1 | 1                    | 1
2 | 2                    | 2
  | 1+1                  | 2
3 | 3                    | 3
  | 1+1+1                | 3
4 | 4                    | 4
  | 2+2                  | 4
  | 2+1+1                | 2, 2
  | 1+1+1+1              | 4
5 | 5                    | 5
  | 1+1+1+1+1            | 5
6 | 6                    | 6
  | 3+3                  | 6
  | 3+1+1+1              | 3, 3
  | 2+2+2                | 6
  | 1+1+1+1+1+1          | 6

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000005 (d(n)), A304405, A304406, A304428, A304430.
All parts are divisors of n, see A018818, compositions A100346.
For run-lengths instead of run-sums we have A047966, compositions A329738.
These partitions are ranked by A353833.
The distinct instead of equal version is A353837, ranked by A353838, compositions A353850.
The version for compositions is A353851, ranked by A353848.
Cf. A098504, A098859, A275870, A353832, A353847, A353864, A353932.

Table[Length[Select[IntegerPartitions[n],SameQ@@Total/@Split[#]&]],{n,0,15}] (* Gus Wiseman, Jun 25 2022 *)

a(n) = if (n==0, 1, sumdiv(n, d, binomial(numdiv(n/d), d))); \\ Michel Marcus, May 13 2018

a(n) >= 2 for n > 1.

a(n) = Sum_{d|n} binomial(A000005(n/d), d) for n > 0.

A329738 Number of compositions of n whose run-lengths are all equal.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 19, 24, 45, 75, 133, 215, 401, 662, 1177, 2035, 3587, 6190, 10933, 18979, 33339, 58157, 101958, 178046, 312088, 545478, 955321, 1670994, 2925717, 5118560, 8960946, 15680074, 27447350, 48033502, 84076143, 147142496, 257546243, 450748484, 788937192

Offset: 0

Gus Wiseman, Nov 20 2019

nonn

A composition of n is a finite sequence of positive integers with sum n.

			The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (121)   (41)     (42)
                    (1111)  (131)    (51)
                            (212)    (123)
                            (11111)  (132)
                                     (141)
                                     (213)
                                     (222)
                                     (231)
                                     (312)
                                     (321)
                                     (1122)
                                     (1212)
                                     (2121)
                                     (2211)
                                     (111111)

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

Compositions with relatively prime run-lengths are A000740.
Compositions with equal multiplicities are A098504.
Compositions with equal differences are A175342.
Compositions with distinct run-lengths are A329739.
Cf. A003242, A008965, A107429, A164707, A238130, A242882, A274174, A329745, A329766.

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

seq(n)={my(b=Vec(1/(1 - sum(k=1, n, x^k/(1+x^k) + O(x*x^n)))-1)); concat([1], vector(n, k, sumdiv(k, d, b[d])))} \\ Andrew Howroyd, Dec 30 2020

a(n) = Sum_{d|n} A003242(d).

a(n) = A329745(n) + A000005(n).

A333627 The a(n)-th composition in standard order is the sequence of run-lengths of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 3, 4, 1, 3, 2, 6, 3, 7, 5, 8, 1, 3, 3, 6, 3, 5, 7, 12, 3, 7, 6, 14, 5, 11, 9, 16, 1, 3, 3, 6, 2, 7, 7, 12, 3, 7, 4, 10, 7, 15, 13, 24, 3, 7, 7, 14, 7, 13, 15, 28, 5, 11, 10, 22, 9, 19, 17, 32, 1, 3, 3, 6, 3, 7, 7, 12, 3, 5, 6, 14, 7, 15, 13

Offset: 0

Gus Wiseman, Mar 30 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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 standard compositions and their run-lengths:
       0 ~ () -> () ~ 0
      1 ~ (1) -> (1) ~ 1
      2 ~ (2) -> (1) ~ 1
     3 ~ (11) -> (2) ~ 2
      4 ~ (3) -> (1) ~ 1
     5 ~ (21) -> (11) ~ 3
     6 ~ (12) -> (11) ~ 3
    7 ~ (111) -> (3) ~ 4
      8 ~ (4) -> (1) ~ 1
     9 ~ (31) -> (11) ~ 3
    10 ~ (22) -> (2) ~ 2
   11 ~ (211) -> (12) ~ 6
    12 ~ (13) -> (11) ~ 3
   13 ~ (121) -> (111) ~ 7
   14 ~ (112) -> (21) ~ 5
  15 ~ (1111) -> (4) ~ 8
     16 ~ (5) -> (1) ~ 1
    17 ~ (41) -> (11) ~ 3
    18 ~ (32) -> (11) ~ 3
   19 ~ (311) -> (12) ~ 6

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Positions of first appearances are A333630.
All of the following pertain to compositions in standard order (A066099):
- The length is A000120.
- The partial sums from the right are A048793.
- The sum is A070939.
- Adjacent equal pairs are counted by A124762.
- Equal runs are counted by A124767.
- Strict compositions are ranked by A233564.
- The partial sums from the left are A272020.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Heinz number is A333219.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
- Runs-resistance is A333628.
- First appearances of run-resistances are A333629.
Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total[2^(Accumulate[Reverse[Length/@Split[stc[n]]]])]/2,{n,0,30}]

A000120(n) = A070939(a(n)).

A000120(a(n)) = A124767(n).

A329744 Triangle read by rows where T(n,k) is the number of compositions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 1, 1, 6, 6, 2, 1, 3, 15, 9, 4, 0, 1, 1, 22, 22, 16, 2, 0, 1, 3, 41, 38, 37, 8, 0, 0, 1, 2, 72, 69, 86, 26, 0, 0, 0, 1, 3, 129, 124, 175, 78, 2, 0, 0, 0, 1, 1, 213, 226, 367, 202, 14, 0, 0, 0, 0, 1, 5, 395, 376, 750, 469, 52, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 21 2019

A composition of n is a finite sequence of positive integers with sum n.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			Triangle begins:
   1
   1   1
   1   1   2
   1   2   3   2
   1   1   6   6   2
   1   3  15   9   4   0
   1   1  22  22  16   2   0
   1   3  41  38  37   8   0   0
   1   2  72  69  86  26   0   0   0
   1   3 129 124 175  78   2   0   0   0
   1   1 213 226 367 202  14   0   0   0   0
   1   5 395 376 750 469  52   0   0   0   0   0
Row n = 6 counts the following compositions:
  (6)  (33)      (15)    (114)    (1131)
       (222)     (24)    (411)    (1311)
       (111111)  (42)    (1113)   (11121)
                 (51)    (1221)   (12111)
                 (123)   (2112)
                 (132)   (3111)
                 (141)   (11112)
                 (213)   (11211)
                 (231)   (21111)
                 (312)
                 (321)
                 (1122)
                 (1212)
                 (2121)
                 (2211)

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Row sums are A000079.
Column k = 1 is A032741.
Column k = 2 is A329745.
Column k = n - 2 is A329743.
The version for partitions is A329746.
The version with rows reversed is A329750.
Cf. A000740, A008965, A098504, A242882, A318928, A329747.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==k&]],{n,10},{k,0,n-1}]

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

A353851 Number of integer compositions of n with all equal run-sums.

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 8, 2, 12, 5, 8, 2, 34, 2, 8, 8, 43, 2, 52, 2, 70, 8, 8, 2, 282, 5, 8, 18, 214, 2, 386, 2, 520, 8, 8, 8, 1957, 2, 8, 8, 2010, 2, 2978, 2, 3094, 94, 8, 2, 16764, 5, 340, 8, 12310, 2, 26514, 8, 27642, 8, 8, 2, 132938, 2, 8, 238, 107411, 8, 236258

Offset: 0

Gus Wiseman, May 31 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 a(0) = 1 through a(8) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                        (112)            (222)                (224)
                        (211)            (1113)               (422)
                        (1111)           (2112)               (2222)
                                         (3111)               (11114)
                                         (11211)              (41111)
                                         (111111)             (111122)
                                                              (112112)
                                                              (211211)
                                                              (221111)
                                                              (11111111)
For example:
  (1,1,2,1,1) has run-sums (2,2,2) so is counted under a(6).
  (4,1,1,1,1,2,2) has run-sums (4,4,4) so is counted under a(12).
  (3,3,2,2,2) has run-sums (6,6) so is counted under a(12).

David A. Corneth, Table of n, a(n) for n = 0..10000

The version for parts or runs instead of run-sums is A000005.
The version for multiplicities instead of run-sums is A098504.
All parts are divisors of n, see A100346.
The version for partitions is A304442, ranked by A353833.
The version for run-lengths instead of run-sums is A329738, ptns A047966.
These compositions are ranked by A353848.
The distinct instead of equal version is A353850.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A353847 represents the composition run-sum transformation.
For distinct instead of equal run-sums: A032020, A098859, A242882, A329739, A351013, A353837, ranked by A353838 (complement A353839), A353852, A354580, ranked by A354581.
Cf. A000005, A006881, A238279, A275870, A333755, A351014, A351016, A351017, A353832, A353834, A353849, A353853-A353859 (run-sum trajectory), A353860, A353863, A353864, A353932.

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

a(n) = {if(n <=1, return(1)); my(d = divisors(n), res = 0); for(i = 1, #d, nd = numdiv(d[i]); res+=(nd*(nd-1)^(n/d[i]-1)) ); res } \\ David A. Corneth, Jun 02 2022

From David A. Corneth, Jun 02 2022 (Start)
a(p) = 2 for prime p.
a(p*q) = 8 for distinct primes p and q (Cf. A006881).
a(n) = Sum_{d|n} tau(d)*(tau(d)-1) ^ (n/d - 1) where tau = A000005. (End)

More terms from David A. Corneth, Jun 02 2022

A333628 Runs-resistance of the n-th composition in standard order. Number of steps taking run-lengths to reduce the n-th composition in standard order to a singleton.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 1, 0, 2, 1, 3, 2, 2, 3, 1, 0, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 4, 3, 4, 3, 1, 0, 2, 2, 3, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 3, 3, 2, 2, 2, 4, 2, 3, 2, 4, 3, 4, 2, 3, 3, 4, 3, 1, 0, 2, 2, 3, 2, 2, 2, 3, 2, 3, 3, 4, 2, 2, 3, 3, 2, 2, 2, 4, 3, 3, 4

Offset: 1

Gus Wiseman, Mar 31 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			Starting with 13789 and repeatedly applying A333627 gives: 13789 -> 859 -> 110 -> 29 -> 11 -> 6 -> 3 -> 2, corresponding to the compositions: (1,2,2,1,1,2,1,1,2,1) -> (1,2,2,1,2,1,1) -> (1,2,1,1,2) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2), so a(13789) = 7.

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Number of times applying A333627 to reach a power of 2, starting with n.
Positions of first appearances are A333629.
All of the following pertain to compositions in standard order (A066099):
- The length is A000120.
- The partial sums from the right are A048793.
- The sum is A070939.
- Adjacent equal pairs are counted by A124762.
- Equal runs are counted by A124767.
- Strict compositions are ranked by A233564.
- The partial sums from the left are A272020.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Heinz number is A333219.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
- First appearances for specified run-lengths are A333630.
Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.

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

A329746 Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 0, 1, 3, 4, 3, 0, 0, 1, 1, 4, 8, 1, 0, 0, 1, 3, 6, 10, 2, 0, 0, 0, 1, 2, 8, 13, 6, 0, 0, 0, 0, 1, 3, 11, 20, 7, 0, 0, 0, 0, 0, 1, 1, 11, 29, 14, 0, 0, 0, 0, 0, 0, 1, 5, 19, 31, 20, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 21 2019

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			Triangle begins:
  1
  1  1
  1  1  1
  1  2  1  1
  1  1  2  3  0
  1  3  4  3  0  0
  1  1  4  8  1  0  0
  1  3  6 10  2  0  0  0
  1  2  8 13  6  0  0  0  0
  1  3 11 20  7  0  0  0  0  0
  1  1 11 29 14  0  0  0  0  0  0
  1  5 19 31 20  1  0  0  0  0  0  0
  1  1 17 50 30  2  0  0  0  0  0  0  0
  1  3 25 64 37  5  0  0  0  0  0  0  0  0
  1  3 29 74 62  7  0  0  0  0  0  0  0  0  0
Row n = 8 counts the following partitions:
  (8)  (44)        (53)    (332)      (4211)
       (2222)      (62)    (422)      (32111)
       (11111111)  (71)    (611)
                   (431)   (3221)
                   (521)   (5111)
                   (3311)  (22211)
                           (41111)
                           (221111)
                           (311111)
                           (2111111)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Row sums are A000041.
Column k = 1 is A032741.
Column k = 2 is A329745.
A similar invariant is frequency depth; see A323014, A325280.
The version for compositions is A329744.
The version for binary words is A329767.
Cf. A098504, A182850, A225485, A242882, A318928, A325410, A329747.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[IntegerPartitions[n],runsres[#]==k&]],{n,10},{k,0,n-1}]

\\ rr(p) gives runs resistance of partition.
rr(p)={my(r=0); while(#p > 1, my(L=List(), k=0); for(i=1, #p, if(i==#p||p[i]<>p[i+1], listput(L, i-k); k=i)); p=Vec(L); r++); r}
row(n)={my(v=vector(n)); forpart(p=n, v[1+rr(Vec(p))]++); v}
{ for(n=1, 10, print(row(n))) } \\ Andrew Howroyd, Jan 19 2023

cp's OEIS Frontend

A272919 Numbers of the form 2^(n-1)*(2^(n*m)-1)/(2^n-1), n >= 1, m >= 1.

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A329739 Number of compositions of n whose run-lengths are all different.

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A304442 Number of partitions of n in which the sequence of the sum of the same summands is constant.

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A329738 Number of compositions of n whose run-lengths are all equal.

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A333627 The a(n)-th composition in standard order is the sequence of run-lengths of the n-th composition in standard order.

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A329744 Triangle read by rows where T(n,k) is the number of compositions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

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

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A353851 Number of integer compositions of n with all equal run-sums.

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A272919 Numbers of the form 2^(n-1)(2^(nm)-1)/(2^n-1), n >= 1, m >= 1.