A329744 -id:A329744 - cp's OEIS frontend

A329747 Runs-resistance of the sequence of prime indices of n.

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

Offset: 1

Gus Wiseman, Nov 21 2019

nonn

First differs from A304455 at a(90) = 3, A304455(90) = 4.
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.
A prime index of n is a number m such that prime(m) divides n. The sequence of prime indices of n is row n of A112798.

			We have (1,2,2,3) -> (1,2,1) -> (1,1,1) -> (3), so a(90) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.
Index entries for sequences related to prime indices in the factorization of n.

The version for partitions is A329746.
The version for compositions is A329744.
The version for binary words is A329767.
The version for binary expansion is A318928.
Cf. A008578 (positions of 0's), A056239, A112798, A329745, A329750.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[runsres[primeMS[n]],{n,50}]

pis_to_runs(n) = { my(runs=List([]), f=factor(n)); for(i=1,#f~,while(f[i,2], listput(runs,primepi(f[i,1])); f[i,2]--)); (runs); };
runlengths(lista) = if(!#lista, lista, if(1==#lista, List([1]), my(runs=List([]), rl=1); for(i=1, #lista, if((i< #lista) && (lista[i]==lista[i+1]), rl++, listput(runs,rl); rl=1)); (runs)));
A329747(n) = { my(runs=pis_to_runs(n)); for(i=0,oo,if(#runs<=1, return(i), runs = runlengths(runs))); }; \\ Antti Karttunen, Jan 20 2025

More terms from Antti Karttunen, Jan 20 2025

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

Original entry on oeis.org

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

A319411 Triangle read by rows: T(n,k) = number of binary vectors of length n with runs-resistance k (1 <= k <= n).

Original entry on oeis.org

2, 2, 2, 2, 2, 4, 2, 4, 6, 4, 2, 2, 12, 12, 4, 2, 6, 30, 18, 8, 0, 2, 2, 44, 44, 32, 4, 0, 2, 6, 82, 76, 74, 16, 0, 0, 2, 4, 144, 138, 172, 52, 0, 0, 0, 2, 6, 258, 248, 350, 156, 4, 0, 0, 0, 2, 2, 426, 452, 734, 404, 28, 0, 0, 0, 0, 2, 10, 790, 752, 1500, 938, 104, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, Sep 20 2018

"Runs-resistance" is defined in A318928.
Row sums are 2,4,8,16,... (the binary vectors may begin with 0 or 1).
This is similar to A329767 but without the k = 0 column and with a different row n = 1. - Gus Wiseman, Nov 25 2019

			Triangle begins:
2,
2, 2,
2, 2, 4,
2, 4, 6, 4,
2, 2, 12, 12, 4,
2, 6, 30, 18, 8, 0,
2, 2, 44, 44, 32, 4, 0,
2, 6, 82, 76, 74, 16, 0, 0,
2, 4, 144, 138, 172, 52, 0, 0, 0,
2, 6, 258, 248, 350, 156, 4, 0, 0, 0,
2, 2, 426, 452, 734, 404, 28, 0, 0, 0, 0,
2, 10, 790, 752, 1500, 938, 104, 0, 0, 0, 0, 0,
...
Lenormand gives the first 20 rows.
The calculation of row 4 is as follows.
We may assume the first bit is a 0, and then double the answers.
vector / runs / steps to reach a single number:
0000 / 4 / 1
0001 / 31 -> 11 -> 2 / 3
0010 / 211 -> 12 -> 11 -> 2 / 4
0011 / 22 -> 2 / 2
0100 / 112 -> 21 -> 11 -> 2 / 4
0101 / 1111 -> 4 / 2
0110 / 121 -> 111 -> 3 / 3
0111 / 13 -> 11 -> 2 / 3
and we get 1 (once), 2 (twice), 3 (three times) and 4 (twice).
So row 4 is: 2,4,6,4.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..1830
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.

Row sums are A000079.
Column k = 2 is 2 * A032741 = A319410.
Column k = 3 is 2 * A329745 (because runs-resistance 2 for compositions corresponds to runs-resistance 3 for binary words).
The version for compositions is A329744.
The version for partitions is A329746.
The number of nonzero entries in row n > 0 is A319412(n).
The runs-resistance of the binary expansion of n is A318928.
Cf. A096365, A225485, A245563, A319413, A319414, A319415, A325280, A329747, A329750, A329767, A329870.

runsresist[q_]:=If[Length[q]==1,1,Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1];
Table[Length[Select[Tuples[{0,1},n],runsresist[#]==k&]],{n,10},{k,n}] (* Gus Wiseman, Nov 25 2019 *)

A332833 Number of compositions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 8, 27, 75, 185, 441, 1025, 2276, 4985, 10753, 22863, 48142, 100583, 208663, 430563, 884407, 1809546, 3690632, 7506774, 15233198, 30851271, 62377004, 125934437, 253936064, 511491634, 1029318958, 2069728850, 4158873540, 8351730223, 16762945432

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

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

			The a(6) = 3 and a(7) = 8 compositions:
  (1221)   (2113)
  (2112)   (3112)
  (11211)  (11311)
           (12112)
           (21112)
           (21121)
           (111211)
           (112111)

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

The case of partitions is A332641.
The version for unsorted prime signature is A332831.
The version for the compositions themselves (not run-lengths) is A332834.
The complement is counted by A332835.
Unimodal compositions are A001523.
Partitions with weakly increasing run-lengths are A100883.
Compositions that are not unimodal are A115981.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are unimodal are A332726.
Compositions whose run-lengths are not unimodal are A332727.
Partitions with weakly increasing or weakly decreasing run-lengths: A332745.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are neither unimodal nor is their negation are A332870.
Cf. A001462, A072704, A072706, A107429, A181819, A329398, A329744, A329746, A329766, A332273, A332640, A332746.

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

a(n) = 2^(n - 1) - 2 * A332836(n) + A329738(n).

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A332835 Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 29, 56, 101, 181, 327, 583, 1023, 1820, 3207, 5631, 9905, 17394, 30489, 53481, 93725, 164169, 287606, 503672, 881834, 1544018, 2703161, 4731860, 8283291, 14499392, 25379278, 44422866, 77754798, 136093756, 238204369, 416923752, 729728031

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

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

			The a(6) = 29 compositions:
  (6)    (141)  (213)   (1113)  (21111)
  (51)   (114)  (132)   (222)   (12111)
  (15)   (33)   (123)   (2211)  (11121)
  (42)   (321)  (3111)  (2121)  (11112)
  (24)   (312)  (1311)  (1212)  (111111)
  (411)  (231)  (1131)  (1122)
Missing are: (2112), (1221), (11211).

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

The version for the compositions themselves (not run-lengths) is A329398.
Compositions with equal run-lengths are A329738.
The case of partitions is A332745.
The version for unsorted prime signature is the complement of A332831.
The complement is counted by A332833.
Unimodal compositions are A001523.
Partitions with weakly decreasing run-lengths are A100882.
Partitions with weakly increasing run-lengths are A100883.
Compositions that are not unimodal are A115981.
Compositions whose negation is unimodal are A332578.
Compositions whose run-lengths are unimodal are A332726.
Neither weakly increasing nor weakly decreasing compositions are A332834.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are neither unimodal nor is their negation are A332870.
Cf. A001462, A181819, A329744, A329766, A332273, A332640, A332641, A332746.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,20}]

a(n) = 2 * A332836(n) - A329738(n).

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A319416 Cuts-resistance of n: number of applications of Lernormand's "raboter" map needed to transform the binary expansion of n to the empty string.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 21 2018

Here we are using Lenormand's "raboter" map in a stricter sense than in A318921 and A319419. If S is a binary string with successive runs of lengths b,c,d,e,..., the "raboter" map sends S to the binary string with successive runs of lengths b-1,c-1,d-1,e-1,... Runs of length 0 are omitted (they are indicated by dots in the examples below).
To get a(n), start with S equal to the binary expansion of n beginning with the most significant bit, and keep applying the map until we reach the empty string.
After the first step, the string may start with a string of 0's: this is acceptable because we are working with strings, not binary expansions of numbers.
For example, 34 = 100010 -> .00.. = 00 -> 0. = 0 -> . (the empty string), taking 3 steps, so a(34) = 3.
Note: this is not the same as the number of applications of the map k -> A318921(k) needed to reduce the binary expansion of n to zero (because A318921 does not distinguish between 0 and the empty string).
This is also not the same as the number of applications of the map k -> A319419(k) needed to reduce the binary expansion of n to -1 (because A319419 does not distinguish between a string of 0's and a single 0).
The value k appears for the first time when n = 2^k - 1.

			n: repeatedly applying the map / number of steps = a(n)
0: 0 -> . / 1
1: 1 -> . / 1
2: 10 -> . / 1
3: 11 -> 1 -> . / 2
4: 100 -> 0 -> . / 2
5: 101 -> . / 1
6: 110 -> 1 -> . / 2
7: 111 -> 11 -> 1 -> . / 3
8: 1000 -> 00 -> 0 -> . / 3
9: 1001 -> 0 -> . / 2
10: 1010 -> . / 1
11: 1011 -> 1 -> . / 2
12: 1100 -> 10 -> . / 2
...

Rémy Sigrist, Table of n, a(n) for n = 0..16384
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, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Positions of 1's are A000975.
Positions of 2's are A329862.
The version for runs-resistance is A318928.
The version for compositions is A329861.
Binary words counted by cuts-resistance are A319421 or A329860.
Cf. A318921, A319411, A319419, A319420, A329744, A329767, A329863, A329865.

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[degdep[IntegerDigits[n,2]],{n,0,50}] (* Gus Wiseman, Nov 25 2019 *)

a(n) = my (b=binary(n), w=#b); for (k=1, oo, my (ww=0); for (i=2, w, if (b[i-1]==b[i], b[ww++]=b[i])); if (ww==0, return (k), w=ww)) \\ Rémy Sigrist, Sep 23 2018

More terms from Rémy Sigrist, Sep 23 2018

A329745 Number of compositions of n with runs-resistance 2.

Original entry on oeis.org

0, 0, 2, 3, 6, 15, 22, 41, 72, 129, 213, 395, 660, 1173, 2031, 3582, 6188, 10927, 18977, 33333, 58153, 101954, 178044, 312080, 545475, 955317, 1670990, 2925711, 5118558, 8960938, 15680072, 27447344, 48033498, 84076139, 147142492, 257546234, 450748482, 788937188

Offset: 1

Gus Wiseman, Nov 21 2019

nonn

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.
These are non-constant compositions with equal run-lengths (A329738).

			The a(3) = 2 through a(6) = 15 compositions:
  (1,2)  (1,3)    (1,4)    (1,5)
  (2,1)  (3,1)    (2,3)    (2,4)
         (1,2,1)  (3,2)    (4,2)
                  (4,1)    (5,1)
                  (1,3,1)  (1,2,3)
                  (2,1,2)  (1,3,2)
                           (1,4,1)
                           (2,1,3)
                           (2,3,1)
                           (3,1,2)
                           (3,2,1)
                           (1,1,2,2)
                           (1,2,1,2)
                           (2,1,2,1)
                           (2,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Column k = 2 of A329744.
Column k = n - 2 of A329750.
Cf. A000740, A003242, A008965, A098504, A242882, A318928, A329743, A329746, A329747, A329767.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==2&]],{n,10}]

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

a(n) = A329738(n) - A000005(n).

a(n) = Sum_{d|n} (A003242(d) - 1). - Andrew Howroyd, Dec 30 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A329766 Number of compositions of n whose run-lengths cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 3, 6, 13, 21, 48, 89, 180, 355, 707, 1382, 2758, 5448, 10786, 21391, 42476, 84291, 167516, 333036, 662153, 1317687, 2622706, 5221951, 10400350, 20720877, 41288823, 82294979, 164052035, 327088649, 652238016, 1300788712, 2594486045, 5175378128, 10324522020

Offset: 0

Gus Wiseman, Nov 20 2019

nonn

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

			The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)  (3)    (4)      (5)
                (1,2)  (1,3)    (1,4)
                (2,1)  (3,1)    (2,3)
                       (1,1,2)  (3,2)
                       (1,2,1)  (4,1)
                       (2,1,1)  (1,1,3)
                                (1,2,2)
                                (1,3,1)
                                (2,1,2)
                                (2,2,1)
                                (3,1,1)
                                (1,1,2,1)
                                (1,2,1,1)

Looking at multiplicities instead of run-lengths gives A329741.
The complete case is A329749.
Complete compositions are A107429.
Cf. A000740, A008965, A242882, A244164, A274174, A329738, A329739, A329740, A329744, A329748.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[Length/@Split[#]]&]],{n,0,10}]

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

a(27)-a(35) from Alois P. Heinz, Jul 06 2020

A329767 Triangle read by rows where T(n,k) is the number of binary words of length n >= 0 with runs-resistance k, 0 <= k <= n.

Original entry on oeis.org

1, 2, 0, 0, 2, 2, 0, 2, 2, 4, 0, 2, 4, 6, 4, 0, 2, 2, 12, 12, 4, 0, 2, 6, 30, 18, 8, 0, 0, 2, 2, 44, 44, 32, 4, 0, 0, 2, 6, 82, 76, 74, 16, 0, 0, 0, 2, 4, 144, 138, 172, 52, 0, 0, 0, 0, 2, 6, 258, 248, 350, 156, 4, 0, 0, 0, 0, 2, 2, 426, 452, 734, 404, 28, 0, 0, 0, 0

Offset: 0

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.
Except for the k = 0 column and the n = 0 and n = 1 rows, this is the triangle appearing on page 3 of Lenormand, which is A319411. Unlike A318928, we do not here require that a(n) >= 1.
The n = 0 row is chosen to ensure that the row-sums are A000079, although the empty word arguably has indeterminate runs-resistance.

			Triangle begins:
   1
   2   0
   0   2   2
   0   2   2   4
   0   2   4   6   4
   0   2   2  12  12   4
   0   2   6  30  18   8   0
   0   2   2  44  44  32   4   0
   0   2   6  82  76  74  16   0   0
   0   2   4 144 138 172  52   0   0   0
   0   2   6 258 248 350 156   4   0   0   0
   0   2   2 426 452 734 404  28   0   0   0   0
For example, row n = 4 counts the following words:
  0000  0011  0001  0010
  1111  0101  0110  0100
        1010  0111  1011
        1100  1000  1101
              1001
              1110

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

Row sums are A000079.
Column k = 2 is A319410.
Column k = 3 is 2 * A329745.
The version for compositions is A329744.
The version for partitions is A329746.
The number of nonzero entries in row n > 0 is A319412(n).
The runs-resistance of the binary expansion of n is A318928.
Cf. A001037, A096365, A225485, A245563, A319411, A325280, A329747, A329750.

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

A332726 Number of compositions of n whose run-lengths are unimodal.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 61, 120, 228, 438, 836, 1580, 2976, 5596, 10440, 19444, 36099, 66784, 123215, 226846, 416502, 763255, 1395952, 2548444, 4644578, 8452200, 15358445, 27871024, 50514295, 91446810, 165365589, 298730375, 539127705, 972099072, 1751284617, 3152475368

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

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

			The only composition of 6 whose run-lengths are not unimodal is (1,1,2,1,1).

Andrew Howroyd, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.

Looking at the composition itself (not run-lengths) gives A001523.
The case of partitions is A332280, with complement counted by A332281.
The complement is counted by A332727.
Unimodal compositions are A001523.
Unimodal normal sequences appear to be A007052.
Non-unimodal compositions are A115981.
Compositions with normal run-lengths are A329766.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283, with complement A332284, with Heinz numbers A332287.
Compositions whose negated run-lengths are unimodal are A332578.
Compositions whose negated run-lengths are not unimodal are A332669.
Compositions whose run-lengths are weakly increasing are A332836.
Cf. A072706, A100883, A181819, A227038, A328509, A329744, A329746, A332642, A332670, A332741, A332833, A332835.

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],unimodQ[Length/@Split[#]]&]],{n,0,10}]

step(M, m)={my(n=matsize(M)[1]); for(p=m+1, n, my(v=vector((p-1)\m, i, M[p-i*m,i]), s=vecsum(v)); M[p,]+=vector(#M,i,s-if(i<=#v, v[i]))); M}
desc(M, m)={my(n=matsize(M)[1]); while(m>1, m--; M=step(M,m)); vector(n, i, vecsum(M[i,]))/(#M-1)}
seq(n)={my(M=matrix(n+1, n+1, i, j, i==1), S=M[,1]~); for(m=1, n, my(D=M); M=step(M, m); D=(M-D)[m+1..n+1,1..n-m+2]; S+=concat(vector(m), desc(D,m))); S} \\ Andrew Howroyd, Dec 31 2020

a(n) + A332727(n) = 2^(n - 1).

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

cp's OEIS Frontend

A329747 Runs-resistance of the sequence of prime indices of n.

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

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A319411 Triangle read by rows: T(n,k) = number of binary vectors of length n with runs-resistance k (1 <= k <= n).

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A332833 Number of compositions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

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A332835 Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.

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A319416 Cuts-resistance of n: number of applications of Lernormand's "raboter" map needed to transform the binary expansion of n to the empty string.

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A329745 Number of compositions of n with runs-resistance 2.

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