A329744 -id:A329744 - cp's OEIS frontend

A124767 Number of level runs for compositions in standard order.

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

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

For n > 0, a(n) is one more than the number of adjacent unequal terms in the n-th composition in standard order. Also the number of runs in the same composition. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; the level runs are 2; 1,1; so a(11) = 2.
The table starts:
  0
  1
  1 1
  1 2 2 1
  1 2 1 2 2 3 2 1
  1 2 2 2 2 2 3 2 2 3 2 3 2 3 2 1
  1 2 2 2 1 3 3 2 2 3 1 2 3 4 3 2 2 3 3 3 3 3 4 3 2 3 2 3 2 3 2 1
The 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1) with runs ((3),(2),(1),(2,2),(1),(2),(5),(1,1,1)), so a(1234567) = 8. - _Gus Wiseman_, Apr 08 2020

Row-lengths are A011782.
Compositions counted by number of runs are A238279 or A333755.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Weakly decreasing compositions are A114994.
- Adjacent equal pairs are counted by A124762.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767 (this sequence).
- Weakly increasing compositions are A225620.
- Strict compositions A233564.
- Constant compositions are A272919.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
- Anti-run compositions are A333489.
- Runs-resistance is A333628.
- Run-lengths are A333769 (triangle).
Cf. A029931, A048793, A066099, A106356, A228351, A318928, A329744, A333217, A333219, A333627.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Split[stc[n]]],{n,0,100}] (* Gus Wiseman, Apr 17 2020 *)

a(0) = 0, a(n) = 1 + Sum_{1<=i=1 0.

For n > 0, a(n) = A333382(n) + 1. - Gus Wiseman, Apr 08 2020

A233564 c-squarefree numbers: positive integers which in binary are concatenation of distinct parts of the form 10...0 with nonnegative number of zeros.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 12, 16, 17, 18, 20, 24, 32, 33, 34, 37, 38, 40, 41, 44, 48, 50, 52, 64, 65, 66, 68, 69, 70, 72, 80, 81, 88, 96, 98, 104, 128, 129, 130, 132, 133, 134, 137, 140, 144, 145, 152, 160, 161, 176, 192, 194, 196, 200, 208, 256, 257, 258, 260, 261

Offset: 1

Vladimir Shevelev, Dec 13 2013

Number of terms in interval [2^(n-1), 2^n) is the number of compositions of n with distinct parts (cf. A032020). For example, if n=6, then interval [2^5, 2^6) contains  11 terms {32,...,52}. This corresponds to 11 compositions with distinct parts of 6: 6, 5+1, 1+5, 4+2, 2+4, 3+2+1, 3+1+2, 2+3+1, 2+1+3, 1+3+2, 1+2+3.
From Gus Wiseman, Apr 06 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 numbers k such that the k-th composition in standard order is strict. For example, the sequence together with the corresponding strict compositions begins:
()          38: (3,1,2)     98: (1,4,2)
(1)         40: (2,4)      104: (1,2,4)
(2)         41: (2,3,1)    128: (8)
(3)         44: (2,1,3)    129: (7,1)
(2,1)       48: (1,5)      130: (6,2)
(1,2)       50: (1,3,2)    132: (5,3)
(4)         52: (1,2,3)    133: (5,2,1)
(3,1)       64: (7)        134: (5,1,2)
(1,3)       65: (6,1)      137: (4,3,1)
(5)         66: (5,2)      140: (4,1,3)
(4,1)       68: (4,3)      144: (3,5)
(3,2)       69: (4,2,1)    145: (3,4,1)
(2,3)       70: (4,1,2)    152: (3,1,4)
(1,4)       72: (3,4)      160: (2,6)
(6)         80: (2,5)      161: (2,5,1)
(5,1)       81: (2,4,1)    176: (2,1,5)
(4,2)       88: (2,1,4)    192: (1,7)
(3,2,1)     96: (1,6)      194: (1,5,2)
(End)

			49 in binary has the following parts of the form 10...0 with nonnegative number of  zeros: (1),(1000),(1). Two of them are the same. So it is not in the sequence. On the other hand, 50 has distinct parts (1)(100)(10), thus it is a term.

Index entries for sequences related to binary expansion of n

Cf. A032020, A124771, A233249, A233312, A233416, A233420, A233569, A233655.
A subset of A333489 and superset of A333218.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Weighted sum is A029931.
- Partial sums from the right are A048793.
- Sum is A070939.
- Runs are counted by A124767.
- Reversed initial intervals A164894.
- Initial intervals are A246534.
- Constant compositions are A272919.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Anti-runs are counted by A333381.
- Anti-runs are A333489.
Cf. A114994, 225620, A228351, A238279, A242882, A329739, A329744, A333217.

bitPatt[n_]:=bitPatt[n]=Split[IntegerDigits[n,2],#1>#2||#2==0&];
Select[Range[0,300],bitPatt[#]==DeleteDuplicates[bitPatt[#]]&] (* Peter J. C. Moses, Dec 13 2013 *)
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@stc[#]&] (* Gus Wiseman, Apr 04 2020 *)

More terms from Peter J. C. Moses, Dec 13 2013

0 prepended by Gus Wiseman, Apr 04 2020

A333213 Triangle read by rows where T(n,k) is the number of compositions of n with k adjacent terms that are equal or increasing (weak ascents) n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 4, 1, 1, 0, 3, 6, 5, 1, 1, 0, 4, 10, 10, 6, 1, 1, 0, 5, 17, 20, 13, 7, 1, 1, 0, 6, 27, 38, 31, 16, 8, 1, 1, 0, 8, 40, 69, 67, 42, 19, 9, 1, 1, 0, 10, 58, 123, 132, 101, 54, 22, 10, 1, 1

Offset: 0

Gus Wiseman, Mar 14 2020

A composition of n is a finite sequence of positive integers summing to n.
Also the number of compositions of n with k + 1 maximal strictly decreasing subsequences.
Also the number of compositions of n with k adjacent terms that are equal or decreasing (weak descents).

			Triangle begins:
   1
   0   1
   0   1   1
   0   2   1   1
   0   2   4   1   1
   0   3   6   5   1   1
   0   4  10  10   6   1   1
   0   5  17  20  13   7   1   1
   0   6  27  38  31  16   8   1   1
   0   8  40  69  67  42  19   9   1   1
   0  10  58 123 132 101  54  22  10   1   1
   0  12  86 202 262 218 139  67  25  11   1   1
   0  15 121 332 484 467 324 182  81  28  12   1   1
Row n = 6 counts the following compositions:
  (6)    (15)    (114)   (1113)   (11112)  (111111)
  (42)   (24)    (123)   (1122)
  (51)   (33)    (222)   (11121)
  (321)  (132)   (1131)  (11211)
         (141)   (1212)  (12111)
         (213)   (1221)  (21111)
         (231)   (1311)
         (312)   (2112)
         (411)   (2211)
         (2121)  (3111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Compositions by length are A007318.
The case of reversed partitions (instead of compositions) is A008284.
The version counting equal adjacencies is A106356.
The case of partitions (instead of compositions) is A133121.
The version counting unequal adjacencies is A238279.
The strict/strong version is A238343.
Cf. A072704, A107429, A124764, A124769, A329744, A332875, A333230.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Split[#,#1>#2&]]==k&]],{n,0,12},{k,0,n}]

T(n)={my(M=matrix(n+1, n+1)); M[1,1]=x; for(n=1, n, for(k=1, n, M[1+n,1+k] = M[1+n,1+k-1] + x*M[1+n-k, 1+n-k] + (1-x)*M[1+n-k, 1+min(k-1, n-k)])); M[1,1]=1; vector(n+1, i, Vecrev(M[i,i]))}
{ my(A=T(12)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 19 2023

A335456 Number of normal patterns matched by compositions of n.

Original entry on oeis.org

1, 2, 5, 12, 32, 84, 211, 556, 1446, 3750, 9824, 25837, 67681, 178160, 468941, 1233837, 3248788, 8554709

Offset: 0

Gus Wiseman, Jun 16 2020

A composition of n is a finite sequence of positive integers summing to n.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The 8 compositions of 4 together with the a(4) = 32 patterns they match:
  4:   31:   13:   22:   211:   121:   112:   1111:
-----------------------------------------------------
  ()   ()    ()    ()    ()     ()     ()     ()
  (1)  (1)   (1)   (1)   (1)    (1)    (1)    (1)
       (21)  (12)  (11)  (11)   (11)   (11)   (11)
                         (21)   (12)   (12)   (111)
                         (211)  (21)   (112)  (1111)
                                (121)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

References found in the link are not all included here.
The version for standard compositions is A335454.
The contiguous case is A335457.
The version for Heinz numbers of partitions is A335549.
Patterns are counted by A000670 and ranked by A333217.
The n-th composition has A124771(n) distinct consecutive subsequences.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A333257(n) distinct subsequence-sums.
The n-th composition has A334299(n) distinct subsequences.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A106356, A108917, A124770, A269134, A329744, A333224, A335458.

mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Sum[Length[Union[mstype/@Subsets[y]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,8}]

a(14)-a(16) from Jinyuan Wang, Jun 26 2020

a(17) from John Tyler Rascoe, Mar 14 2025

A318928 Runs-resistance of binary representation of n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Sep 09 2018

Following Lenormand (2003), we define the "runs-resistance" of a finite list L to be the number of times the RUNS transformation must be applied to L in order to reduce L to a list with a single element.
Here it is immaterial whether we read the binary representation of n from left to right or right to left.
The RUNS transformation must be applied at least once, in order to obtain a list, so a(n) >= 1.

			11 in binary is [1, 0, 1, 1],
which has runs of lengths [1, 1, 2],
which has runs of lengths [2, 1],
which has runs of lengths [1, 1],
which has a single run of length [2].
This took four steps, so a(11) = 4.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003.
N. J. A. Sloane, Transforms

See A319103 for an inverse, and A319417 and A319418 for records.
Ignoring the first digit gives A329870.
Cuts-resistance is A319416.
Compositions counted by runs-resistance are A329744.
Binary words counted by runs-resistance are A319411 and A329767.
Cf. A107907, A319420, A329860, A329861, A329865, A329867.

with(transforms);
# compute Lenormand's "resistance" of a list
resist:=proc(a) local ct,i,b;
if whattype(a) <> list then ERROR("input must be a list"); fi:
ct:=0; b:=a; for i from 1 to 100 do
if nops(b)=1 then return(ct); fi;
b:=RUNS(b); ct:=ct+1; od; end;
a:=[1];
for n from 2 to 100 do
b:=convert(n,base,2);
r:=resist(b);
a:=[op(a),r];
od:

Table[If[n == 1, 1, Length[NestWhileList[Length/@Split[#] &, IntegerDigits[n, 2], Length[#] > 1 &]] - 1], {n, 50}] (* Gus Wiseman, Nov 25 2019 *)

a(1) corrected by N. J. A. Sloane, Sep 20 2018

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).

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

A332834 Number of compositions of n that are neither weakly increasing nor weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 14, 36, 88, 199, 432, 914, 1900, 3896, 7926, 16036, 32311, 64944, 130308, 261166, 523040, 1046996, 2095152, 4191796, 8385466, 16773303, 33549564, 67102848, 134210298, 268426328, 536859712, 1073728142, 2147466956, 4294947014, 8589909976

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(4) = 1 through a(6) = 14 compositions:
  (121)  (131)   (132)
         (212)   (141)
         (1121)  (213)
         (1211)  (231)
                 (312)
                 (1131)
                 (1212)
                 (1221)
                 (1311)
                 (2112)
                 (2121)
                 (11121)
                 (11211)
                 (12111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Unimodal Sequence.

The version for unsorted prime signature is A332831.
The version for run-lengths of compositions is A332833.
The complement appears to be counted by A329398.
Unimodal compositions are A001523.
Compositions that are not unimodal are A115981.
Partitions with weakly increasing or decreasing run-lengths are A332745.
Compositions with weakly increasing or decreasing run-lengths are A332835.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are neither unimodal nor is their negation are A332870.
Cf. A007052, A072704, A107429, A328509, A329744, A332281, A332284, A332578, A332640, A332641, A332643, A332669, A332746.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Or[LessEqual@@#,GreaterEqual@@#]&]],{n,0,10}]

a(n)={if(n==0, 0, 2^(n-1) - 2*numbpart(n) + numdiv(n))} \\ Andrew Howroyd, Dec 30 2020

a(n) = 2^(n - 1) - 2 * A000041(n) + A000005(n).

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

A124767 Number of level runs for compositions in standard order.

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A233564 c-squarefree numbers: positive integers which in binary are concatenation of distinct parts of the form 10...0 with nonnegative number of zeros.

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A333213 Triangle read by rows where T(n,k) is the number of compositions of n with k adjacent terms that are equal or increasing (weak ascents) n >= 0, 0 <= k <= n.

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A335456 Number of normal patterns matched by compositions of n.

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A318928 Runs-resistance of binary representation of n.

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

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A332834 Number of compositions of n that are neither weakly increasing nor weakly decreasing.

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