A225620 -id:A225620 - cp's OEIS frontend

A333218 Numbers k such that the k-th composition in standard order is a permutation (of an initial interval).

0, 1, 5, 6, 37, 38, 41, 44, 50, 52, 549, 550, 553, 556, 562, 564, 581, 582, 593, 600, 610, 616, 649, 652, 657, 664, 708, 712, 786, 788, 802, 808, 836, 840, 16933, 16934, 16937, 16940, 16946, 16948, 16965, 16966, 16977, 16984, 16994, 17000, 17033, 17036, 17041

Offset: 1

Gus Wiseman, Mar 16 2020

nonn

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.

			The sequence of terms together with their corresponding compositions begins:
        0: ()             593: (3,2,4,1)      16937: (5,4,2,3,1)
        1: (1)            600: (3,2,1,4)      16940: (5,4,2,1,3)
        5: (2,1)          610: (3,1,4,2)      16946: (5,4,1,3,2)
        6: (1,2)          616: (3,1,2,4)      16948: (5,4,1,2,3)
       37: (3,2,1)        649: (2,4,3,1)      16965: (5,3,4,2,1)
       38: (3,1,2)        652: (2,4,1,3)      16966: (5,3,4,1,2)
       41: (2,3,1)        657: (2,3,4,1)      16977: (5,3,2,4,1)
       44: (2,1,3)        664: (2,3,1,4)      16984: (5,3,2,1,4)
       50: (1,3,2)        708: (2,1,4,3)      16994: (5,3,1,4,2)
       52: (1,2,3)        712: (2,1,3,4)      17000: (5,3,1,2,4)
      549: (4,3,2,1)      786: (1,4,3,2)      17033: (5,2,4,3,1)
      550: (4,3,1,2)      788: (1,4,2,3)      17036: (5,2,4,1,3)
      553: (4,2,3,1)      802: (1,3,4,2)      17041: (5,2,3,4,1)
      556: (4,2,1,3)      808: (1,3,2,4)      17048: (5,2,3,1,4)
      562: (4,1,3,2)      836: (1,2,4,3)      17092: (5,2,1,4,3)
      564: (4,1,2,3)      840: (1,2,3,4)      17096: (5,2,1,3,4)
      581: (3,4,2,1)    16933: (5,4,3,2,1)    17170: (5,1,4,3,2)
      582: (3,4,1,2)    16934: (5,4,3,1,2)    17172: (5,1,4,2,3)

A superset of A164894.
Also a superset of A246534.
Not requiring the parts to be distinct gives A333217.
Cf. A000120, A000142, A048793, A066099, A070939, A114994, A225620, A233564, A272919, A333219, A333221, A333255, A333256.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],#==0||UnsameQ@@stc[#]&&Max@@stc[#]==Length[stc[#]]&]

A124768 Number of strictly increasing runs for compositions in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 2, 4, 1, 2, 2, 3, 1, 3, 2, 4, 1, 2, 2, 3, 2, 3, 3, 5, 1, 2, 2, 3, 2, 3, 2, 4, 1, 2, 3, 4, 2, 3, 3, 5, 1, 2, 2, 3, 1, 3, 2, 4, 2, 3, 3, 4, 3, 4, 4, 6, 1, 2, 2, 3, 2, 3, 2, 4, 1, 3, 3, 4, 2, 3, 3, 5, 1, 2, 2, 3, 2, 4, 3, 5, 2, 3, 3, 4, 3, 4, 4, 6, 1, 2, 2, 3, 2, 3, 2, 4, 1

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

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. a(n) is the number of maximal strictly increasing runs in this composition. Alternatively, a(n) is one plus the number of weak descents in the same composition. For example, the strictly increasing runs of the 1234567th composition are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)), so a(1234567) = 8. The 7 weak descents together with the strict ascents are: 3 >= 2 >= 1 < 2 >= 2 >= 1 < 2 < 5 >= 1 >= 1 >= 1. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; the strictly increasing runs are 2; 1; 1; so a(11) = 3.
The table starts:
  0
  1
  1 2
  1 2 1 3
  1 2 2 3 1 2 2 4
  1 2 2 3 1 3 2 4 1 2 2 3 2 3 3 5
  1 2 2 3 2 3 2 4 1 2 3 4 2 3 3 5 1 2 2 3 1 3 2 4 2 3 3 4 3 4 4 6

Cf. A066099, A124763, A011782 (row lengths).
Compositions of n with k weak descents are A333213.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Partial sums from the right are A048793.
- 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.
- Strictly increasing runs are counted by A124768 (this sequence).
- Strictly decreasing runs are counted by A124769.
- Weakly increasing compositions are A225620.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Initial intervals are A246534.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Anti-runs are A333489.
Cf. A029931, A124760, A124761, A124764, A238343, A333220.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Split[stc[n],Less]],{n,0,100}] (* Gus Wiseman, Apr 08 2020 *)

a(0) = 0, a(n) = A124763(n) + 1 for n > 0.

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

			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).

A124765 Number of monotonically decreasing runs for compositions in standard order.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

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. a(n) is the number of maximal weakly decreasing runs in this composition. Alternatively, a(n) is one plus the number of strict ascents in the same composition. For example, the weakly decreasing runs of the 1234567th composition are ((3,2,1),(2,2,1),(2),(5,1,1,1)), so a(1234567) = 4. The 3 strict ascents together with the weak descents are: 3 >= 2 >= 1 < 2 >= 2 >= 1 < 2 < 5 >= 1 >= 1 >= 1. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; the decreasing runs are 2,1,1; so a(11) = 1.
The table starts:
  0
  1
  1 1
  1 1 2 1
  1 1 1 1 2 2 2 1
  1 1 1 1 2 1 2 1 2 2 2 2 2 2 2 1
  1 1 1 1 1 1 2 1 2 2 1 1 2 2 2 1 2 2 2 2 3 2 3 2 2 2 2 2 2 2 2 1

Antti Karttunen, Table of n, a(n) for n = 0..16383

Cf. A066099, A124760, A011782 (row lengths).
Compositions of n with k strict ascents are A238343.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Weakly decreasing compositions are A114994.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Weakly increasing compositions are A225620.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
- Anti-runs are A333489.
- Runs-resistance is A333628.
Cf. A124761, A124763, A124764, A233249, A333213, A333380.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Split[stc[n],GreaterEqual]],{n,0,100}] (* Gus Wiseman, Apr 08 2020 *)

a(0) = 0, a(n) = A124760(n) + 1 for n > 0.

A124769 Number of strictly decreasing runs for compositions in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 2, 2, 2, 3, 4, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 4, 5, 1, 1, 1, 2, 2, 1, 2, 3, 2, 2, 3, 3, 2, 2, 3, 4, 2, 2, 2, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4, 4, 5, 6, 1, 1, 1, 2, 1, 1, 2, 3, 2, 2, 2, 2, 2, 2, 3, 4, 2, 2, 2, 3, 3, 3, 3, 4, 2, 2, 3, 3, 3, 3, 4, 5, 2, 2, 2, 3, 3, 2, 3, 4, 3

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

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. a(n) is the number of maximal strictly decreasing runs in this composition. Alternatively, a(n) is one plus the number of weak ascents in the same composition. For example, the strictly decreasing runs of the 1234567th composition are ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)), so a(1234567) = 7. The 6 weak ascents together with the strict descents are: 3 > 2 > 1 <= 2 <= 2 > 1 <= 2 <= 5 > 1 <= 1 <= 1. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; the strictly increasing runs are 2,1; 1; so a(11) = 2.
The table starts:
  0
  1
  1 2
  1 1 2 3
  1 1 2 2 2 2 3 4
  1 1 1 2 2 2 2 3 2 2 3 3 3 3 4 5
  1 1 1 2 2 1 2 3 2 2 3 3 2 2 3 4 2 2 2 3 3 3 3 4 3 3 4 4 4 4 5 6

Cf. A066099, A124764, A011782 (row lengths).
Compositions of n with k weak ascents are A333213.
Positions of ones are A333256.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Partial sums from the right are A048793 (triangle).
- Sum is A070939.
- Weakly decreasing compositions are A114994.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769 (this sequence).
- Reversed initial intervals A164894.
- Weakly increasing compositions are A225620.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Anti-runs are A333489.
Cf. A124760, A124761, A124763, A233249, A238343.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Split[stc[n],Greater]],{n,0,100}] (* Gus Wiseman, Apr 08 2020 *)

a(0) = 0, a(n) = A124764(n) + 1 for n > 0.

A087207 A binary representation of the primes that divide a number, shown in decimal.

Original entry on oeis.org

0, 1, 2, 1, 4, 3, 8, 1, 2, 5, 16, 3, 32, 9, 6, 1, 64, 3, 128, 5, 10, 17, 256, 3, 4, 33, 2, 9, 512, 7, 1024, 1, 18, 65, 12, 3, 2048, 129, 34, 5, 4096, 11, 8192, 17, 6, 257, 16384, 3, 8, 5, 66, 33, 32768, 3, 20, 9, 130, 513, 65536, 7, 131072, 1025, 10, 1, 36, 19, 262144, 65, 258

Offset: 1

Mitch Cervinka (puritan(AT)planetkc.com), Oct 26 2003

The binary representation of a(n) shows which prime numbers divide n, but not the multiplicities. a(2)=1, a(3)=10, a(4)=1, a(5)=100, a(6)=11, a(10)=101, a(30)=111, etc.
For n > 1, a(n) gives the (one-based) index of the column where n is located in array A285321. A008479 gives the other index. - Antti Karttunen, Apr 17 2017
From Antti Karttunen, Jun 18 & 20 2017: (Start)
A268335 gives all n such that a(n) = A248663(n); the squarefree numbers (A005117) are all the n such that a(n) = A285330(n) = A048675(n).
For all n > 1 for which the value of A285331(n) is well-defined, we have A285331(a(n)) <= floor(A285331(n)/2), because then n is included in the binary tree A285332 and a(n) is one of its ancestors (in that tree), and thus must be at least one step nearer to its root than n itself.
Conjecture: Starting at any n and iterating the map n -> a(n), we will always reach 0 (see A288569). This conjecture is equivalent to the conjecture that at any n that is neither a prime nor a power of two, we will eventually hit a prime number (which then becomes a power of two in the next iteration). If this conjecture is false then sequence A285332 cannot be a permutation of natural numbers. On the other hand, if the conjecture is true, then A285332 must be a permutation of natural numbers, because all primes and powers of 2 occur in definite positions in that tree. This conjecture also implies the conjectures made in A019565 and A285320 that essentially claim that there are neither finite nor infinite cycles in A019565.
If there are any 2-cycles in this sequence, then both terms of the cycle should be present in A286611 and the larger one should be present in A286612.
(End)
Binary rank of the distinct prime indices of n, where the binary rank of an integer partition y is given by Sum_i 2^(y_i-1). For all prime indices (with multiplicity) we have A048675. - Gus Wiseman, May 25 2024

			a(38) = 129 because 38 = 2*19 = prime(1)*prime(8) and 129 = 2^0 + 2^7 (in binary 10000001).
a(140) = 13, binary 1101 because 140 is divisible by the first, third and fourth primes and 2^(1-1) + 2^(3-1) + 2^(4-1) = 13.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 [First 1000 terms from _T. D. Noe_]
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

For partial sums see A288566.
Cf. A000040, A000120, A001221, A005117, A008479, A019565, A055396, A285320, A285321, A285329, A285330, A285332.
Sequences with related definitions: A007947, A008472, A027748, A048675, A248663, A276379 (same sequence shown in base 2), A288569, A289271, A297404.
Cf. A286608 (numbers n for which a(n) < n), A286609 (n for which a(n) > n), and also A286611, A286612.
A003986, A003961, A059896 are used to express relationship between terms of this sequence.
Related to A267116 via A225546.
Positions of particular values are: A000079\{1} (1), A000244\{1} (2), A033845 (3), A000351\{1} (4), A033846 (5), A033849 (6), A143207 (7), A000420\{1} (8), A033847 (9), A033850 (10), A033851 (12), A147576 (14), A147571 (15), A001020\{1} (16), A033848 (17).
A048675 gives binary rank of prime indices.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices (listed A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- sum A029931, product A096111
- max A029837 or A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000720, A005940, A018819, A023506, A071814, A225620, A277319, A277905, A304818, A372689, A372890.

a087207 = sum . map ((2 ^) . (subtract 1) . a049084) . a027748_row
-- Reinhard Zumkeller, Jul 16 2013

a[n_] := Total[ 2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]; a[1] = 0; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Dec 12 2011 *)

a(n) = {if (n==1, 0, my(f=factor(n), v = []); forprime(p=2, vecmax(f[,1]), v = concat(v, vecsearch(f[,1], p)!=0);); fromdigits(Vecrev(v), 2));} \\ Michel Marcus, Jun 05 2017

A087207(n)=vecsum(apply(p->1<M. F. Hasler, Jun 23 2017

from sympy import factorint, primepi
def a(n):
    return sum(2**primepi(i - 1) for i in factorint(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

(definec (A087207 n) (if (= 1 n) 0 (+ (A000079 (+ -1 (A055396 n))) (A087207 (A028234 n))))) ;; This uses memoization-macro definec
(define (A087207 n) (A048675 (A007947 n))) ;; Needs code from A007947 and A048675. - Antti Karttunen, Jun 19 2017

Additive with a(p^e) = 2^(i-1) where p is the i-th prime. - Vladeta Jovovic, Oct 29 2003
a(n) gives the m such that A019565(m) = A007947(n). - Naohiro Nomoto, Oct 30 2003
A000120(a(n)) = A001221(n); a(n) = Sum(2^(A049084(p)-1): p prime-factor of n). - Reinhard Zumkeller, Nov 30 2003
G.f.: Sum_{k>=1} 2^(k-1)*x^prime(k)/(1-x^prime(k)). - Franklin T. Adams-Watters, Sep 01 2009
From Antti Karttunen, Apr 17 2017, Jun 19 2017 & Dec 06 2018: (Start)
a(n) = A048675(A007947(n)).
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A028234(n)).
A000035(a(n)) = 1 - A000035(n). [a(n) and n are of opposite parity.]
A248663(n) <= a(n) <= A048675(n). [XOR-, OR- and +-variants.]
a(A293214(n)) = A218403(n).
a(A293442(n)) = A267116(n).
A069010(a(n)) = A287170(n).
A007088(a(n)) = A276379(n).
A038374(a(n)) = A300820(n) for n >= 1.
(End)
From Peter Munn, Jan 08 2020: (Start)
a(A059896(n,k)) = a(n) OR a(k) = A003986(a(n), a(k)).
a(A003961(n)) = 2*a(n).
a(n^2) = a(n).
a(n) = A267116(A225546(n)).
a(A225546(n)) = A267116(n).
(End)

More terms from Don Reble, Ray Chandler and Naohiro Nomoto, Oct 28 2003

Name clarified by Antti Karttunen, Jun 18 2017

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}]

A345168 Numbers k such that the k-th composition in standard order is not alternating.

Original entry on oeis.org

3, 7, 10, 11, 14, 15, 19, 21, 23, 26, 27, 28, 29, 30, 31, 35, 36, 37, 39, 42, 43, 46, 47, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 69, 71, 73, 74, 75, 78, 79, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 99, 100, 101, 103, 104, 105, 106, 107, 110

Offset: 1

Gus Wiseman, Jun 15 2021

nonn

The k-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 k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The sequence of terms together with their binary indices begins:
     3: (1,1)          35: (4,1,1)        59: (1,1,2,1,1)
     7: (1,1,1)        36: (3,3)          60: (1,1,1,3)
    10: (2,2)          37: (3,2,1)        61: (1,1,1,2,1)
    11: (2,1,1)        39: (3,1,1,1)      62: (1,1,1,1,2)
    14: (1,1,2)        42: (2,2,2)        63: (1,1,1,1,1,1)
    15: (1,1,1,1)      43: (2,2,1,1)      67: (5,1,1)
    19: (3,1,1)        46: (2,1,1,2)      69: (4,2,1)
    21: (2,2,1)        47: (2,1,1,1,1)    71: (4,1,1,1)
    23: (2,1,1,1)      51: (1,3,1,1)      73: (3,3,1)
    26: (1,2,2)        52: (1,2,3)        74: (3,2,2)
    27: (1,2,1,1)      53: (1,2,2,1)      75: (3,2,1,1)
    28: (1,1,3)        55: (1,2,1,1,1)    78: (3,1,1,2)
    29: (1,1,2,1)      56: (1,1,4)        79: (3,1,1,1,1)
    30: (1,1,1,2)      57: (1,1,3,1)      83: (2,3,1,1)
    31: (1,1,1,1,1)    58: (1,1,2,2)      84: (2,2,3)

Wikipedia, Alternating permutation

The complement is A345167.
These compositions are counted by A345192.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A025047 counts alternating or wiggly compositions, directed A025048, A025049.
A344604 counts alternating compositions with twins.
A345194 counts alternating patterns (with twins: A344605).
A345164 counts alternating permutations of prime indices (with twins: A344606).
A345165 counts partitions without a alternating permutation, ranked by A345171.
A345170 counts partitions with a alternating permutation, ranked by A345172.
A348610 counts alternating ordered factorizations, complement A348613.
Statistics of standard compositions:
- Length is A000120.
- Constant runs are A124767.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Runs-resistance is A333628.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Weakly decreasing compositions (partitions) are A114994.
- Weakly increasing compositions (multisets) are A225620.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Anti-run compositions are A333489.
- Non-anti-run compositions are A348612.
Cf. A001222, A005649, A008965, A059893, A106356, A238279, A344615, A345162, A345163, A345166, A345169, A348377, A348380.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[0,100],Not@*wigQ@*stc]

A333256 Numbers k such that the k-th composition in standard order is strictly decreasing.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 16, 17, 18, 32, 33, 34, 37, 64, 65, 66, 68, 69, 128, 129, 130, 132, 133, 137, 256, 257, 258, 260, 261, 264, 265, 274, 512, 513, 514, 516, 517, 520, 521, 529, 530, 549, 1024, 1025, 1026, 1028, 1029, 1032, 1033, 1040, 1041, 1042, 1058, 1061

Offset: 1

Gus Wiseman, Mar 20 2020

nonn

			The sequence of positive terms together with the corresponding compositions begins:
     1: (1)         128: (8)         517: (7,2,1)
     2: (2)         129: (7,1)       520: (6,4)
     4: (3)         130: (6,2)       521: (6,3,1)
     5: (2,1)       132: (5,3)       529: (5,4,1)
     8: (4)         133: (5,2,1)     530: (5,3,2)
     9: (3,1)       137: (4,3,1)     549: (4,3,2,1)
    16: (5)         256: (9)        1024: (11)
    17: (4,1)       257: (8,1)      1025: (10,1)
    18: (3,2)       258: (7,2)      1026: (9,2)
    32: (6)         260: (6,3)      1028: (8,3)
    33: (5,1)       261: (6,2,1)    1029: (8,2,1)
    34: (4,2)       264: (5,4)      1032: (7,4)
    37: (3,2,1)     265: (5,3,1)    1033: (7,3,1)
    64: (7)         274: (4,3,2)    1040: (6,5)
    65: (6,1)       512: (10)       1041: (6,4,1)
    66: (5,2)       513: (9,1)      1042: (6,3,2)
    68: (4,3)       514: (8,2)      1058: (5,4,2)
    69: (4,2,1)     516: (7,3)      1061: (5,3,2,1)

Strictly increasing runs are counted by A124768.
The normal case is A246534.
The weakly decreasing version is A114994.
The weakly increasing version is A225620.
The unequal version is A233564.
The equal version is A272919.
The strictly increasing version is A333255.
Cf. A000120, A029931, A048793, A066099, A070939, A124769, A228351, A233249, A333217, A333256, A333380.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Greater@@stc[#]&]

A333255 Numbers k such that the k-th composition in standard order is strictly increasing.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 16, 20, 24, 32, 40, 48, 52, 64, 72, 80, 96, 104, 128, 144, 160, 192, 200, 208, 256, 272, 288, 320, 328, 384, 400, 416, 512, 544, 576, 640, 656, 768, 784, 800, 832, 840, 1024, 1056, 1088, 1152, 1280, 1296, 1312, 1536, 1568, 1600, 1664, 1680

Offset: 1

Gus Wiseman, Mar 20 2020

nonn

			The sequence of positive terms together with the corresponding compositions begins:
     1: (1)         128: (8)         656: (2,3,5)
     2: (2)         144: (3,5)       768: (1,9)
     4: (3)         160: (2,6)       784: (1,4,5)
     6: (1,2)       192: (1,7)       800: (1,3,6)
     8: (4)         200: (1,3,4)     832: (1,2,7)
    12: (1,3)       208: (1,2,5)     840: (1,2,3,4)
    16: (5)         256: (9)        1024: (11)
    20: (2,3)       272: (4,5)      1056: (5,6)
    24: (1,4)       288: (3,6)      1088: (4,7)
    32: (6)         320: (2,7)      1152: (3,8)
    40: (2,4)       328: (2,3,4)    1280: (2,9)
    48: (1,5)       384: (1,8)      1296: (2,4,5)
    52: (1,2,3)     400: (1,3,5)    1312: (2,3,6)
    64: (7)         416: (1,2,6)    1536: (1,10)
    72: (3,4)       512: (10)       1568: (1,4,6)
    80: (2,5)       544: (4,6)      1600: (1,3,7)
    96: (1,6)       576: (3,7)      1664: (1,2,8)
   104: (1,2,4)     640: (2,8)      1680: (1,2,3,5)

Strictly increasing runs are counted by A124768.
The normal case is A164894.
The weakly decreasing version is A114994.
The weakly increasing version is A225620.
The unequal version is A233564.
The equal version is A272919.
The strictly decreasing version is A333256.
Cf. A000120, A029931, A048793, A066099, A070939, A124762, A228351, A333217, A333220, A333379.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Less@@stc[#]&]

cp's OEIS Frontend

A333218 Numbers k such that the k-th composition in standard order is a permutation (of an initial interval).

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A124768 Number of strictly increasing runs for compositions in standard order.

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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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A124765 Number of monotonically decreasing runs for compositions in standard order.

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A124769 Number of strictly decreasing runs for compositions in standard order.

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A087207 A binary representation of the primes that divide a number, shown in decimal.

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

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A345168 Numbers k such that the k-th composition in standard order is not alternating.