A333379 -id:A333379 - cp's OEIS frontend

A333217 Numbers k such that the k-th composition in standard order covers an initial interval of positive integers.

0, 1, 3, 5, 6, 7, 11, 13, 14, 15, 21, 22, 23, 26, 27, 29, 30, 31, 37, 38, 41, 43, 44, 45, 46, 47, 50, 52, 53, 54, 55, 58, 59, 61, 62, 63, 75, 77, 78, 83, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 101, 102, 105, 106, 107, 108, 109, 110, 111, 114, 116, 117, 118

Offset: 1

Gus Wiseman, Mar 15 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 the corresponding compositions begins:
    0: ()              37: (3,2,1)           75: (3,2,1,1)
    1: (1)             38: (3,1,2)           77: (3,1,2,1)
    3: (1,1)           41: (2,3,1)           78: (3,1,1,2)
    5: (2,1)           43: (2,2,1,1)         83: (2,3,1,1)
    6: (1,2)           44: (2,1,3)           85: (2,2,2,1)
    7: (1,1,1)         45: (2,1,2,1)         86: (2,2,1,2)
   11: (2,1,1)         46: (2,1,1,2)         87: (2,2,1,1,1)
   13: (1,2,1)         47: (2,1,1,1,1)       89: (2,1,3,1)
   14: (1,1,2)         50: (1,3,2)           90: (2,1,2,2)
   15: (1,1,1,1)       52: (1,2,3)           91: (2,1,2,1,1)
   21: (2,2,1)         53: (1,2,2,1)         92: (2,1,1,3)
   22: (2,1,2)         54: (1,2,1,2)         93: (2,1,1,2,1)
   23: (2,1,1,1)       55: (1,2,1,1,1)       94: (2,1,1,1,2)
   26: (1,2,2)         58: (1,1,2,2)         95: (2,1,1,1,1,1)
   27: (1,2,1,1)       59: (1,1,2,1,1)      101: (1,3,2,1)
   29: (1,1,2,1)       61: (1,1,1,2,1)      102: (1,3,1,2)
   30: (1,1,1,2)       62: (1,1,1,1,2)      105: (1,2,3,1)
   31: (1,1,1,1,1)     63: (1,1,1,1,1,1)    106: (1,2,2,2)

Robert Price, Table of n, a(n) for n = 1..1008

Sequences covering an initial interval are counted by A000670.
Composition in standard order are A066099.
The case of strictly increasing initial intervals is A164894.
The case of strictly decreasing initial intervals is A246534.
The case of permutations is A333218.
The weakly increasing version is A333379.
The weakly decreasing version is A333380.
Cf. A000120, A029931, A048793, A070939, A225620, A228351, A233564, A272919, A333219, A333220.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],normQ[stc[#]]&]

A225620 Indices of partitions in the table of compositions of A228351.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 14, 15, 16, 20, 24, 26, 28, 30, 31, 32, 36, 40, 42, 48, 52, 56, 58, 60, 62, 63, 64, 72, 80, 84, 96, 100, 104, 106, 112, 116, 120, 122, 124, 126, 127, 128, 136, 144, 160, 164, 168, 170, 192, 200, 208, 212, 224, 228, 232, 234, 240, 244, 248, 250, 252, 254, 255

Offset: 1

Omar E. Pol, Aug 03 2013

Also triangle read by rows in which T(n,k) is the decimal representation of a binary number whose mirror represents the k-th partition of n according with the list of juxtaposed reverse-lexicographically ordered partitions of the positive integers (A026792).
In order to construct this sequence as a triangle we use the following rules:
- In the list of A026792 we replace each part of size j of the k-th partition of n by concatenation of j - 1 zeros and only one 1.
- Then replace this new set of parts by the concatenation of its parts.
- Then replace this string by its mirror version which is a binary number.
T(n,k) is the decimal value of this binary number, which represents the k-th partition of n (see example).
The partitions of n are represented by a subsequence with A000041(n) integers starting with 2^(n-1) and ending with 2^n - 1, n >= 1. The odd numbers of the sequence are in A000225.
First differs from A065609 at a(23).
Conjecture: this sequence is a sorted version of b(n) where b(2^k) = 2^k for k >= 0, b(n) = A080100(n)*(2*b(A053645(n)) + 1) otherwise. - Mikhail Kurkov, Oct 21 2023

			T(6,8) = 58 because 58 in base 2 is 111010 whose mirror is 010111 which is the concatenation of 01, 01, 1, 1, whose number of digits are 2, 2, 1, 1, which are also the 8th partition of 6.
Illustration of initial terms:
The sequence represents a table of partitions (see below):
--------------------------------------------------------
.            Binary                        Partitions
n  k  T(n,k) number  Mirror   Diagram       (A026792)
.                                          1 2 3 4 5 6
--------------------------------------------------------
.                             _
1  1     1       1    1        |           1,
.                             _ _
1  1     2      10    01      _  |           2,
2  2     3      11    11       | |         1,1,
.                             _ _ _
3  1     4     100    001     _ _  |           3,
3  2     6     110    011     _  | |         2,1,
3  3     7     111    111      | | |       1,1,1,
.                             _ _ _ _
4  1     8    1000    0001    _ _    |           4,
4  2    10    1010    0101    _ _|_  |         2,2,
4  3    12    1100    0011    _ _  | |         3,1,
4  4    14    1110    0111    _  | | |       2,1,1,
4  5    15    1111    1111     | | | |     1,1,1,1,
.                             _ _ _ _ _
5  1    16   10000    00001   _ _ _    |           5,
5  2    20   10100    00101   _ _ _|_  |         3,2,
5  3    24   11000    00011   _ _    | |         4,1,
5  4    26   11010    01011   _ _|_  | |       2,2,1,
5  5    28   11100    00111   _ _  | | |       3,1,1,
5  6    30   11110    01111   _  | | | |     2,1,1,1,
5  7    31   11111    11111    | | | | |   1,1,1,1,1,
.                             _ _ _ _ _ _
6  1    32  100000    000001  _ _ _      |           6
6  2    36  100100    001001  _ _ _|_    |         3,3,
6  3    40  101000    000101  _ _    |   |         4,2,
6  4    42  101010    010101  _ _|_ _|_  |       2,2,2,
6  5    48  110000    000011  _ _ _    | |         5,1,
6  6    52  110100    001011  _ _ _|_  | |       3,2,1,
6  7    56  111000    000111  _ _    | | |       4,1,1,
6  8    58  111010    010111  _ _|_  | | |     2,2,1,1,
6  9    60  111100    001111  _ _  | | | |     3,1,1,1,
6  10   62  111110    011111  _  | | | | |   2,1,1,1,1,
6  11   63  111111    111111   | | | | | | 1,1,1,1,1,1,
.
Triangle begins:
  1;
  2,   3;
  4,   6,  7;
  8,  10, 12, 14, 15;
  16, 20, 24, 26, 28, 30, 31;
  32, 36, 40, 42, 48, 52, 56, 58, 60, 62, 63;
  ...
From _Gus Wiseman_, Apr 01 2020: (Start)
Using the encoding of A066099, this sequence ranks all finite nonempty multisets, as follows.
   1: {1}
   2: {2}
   3: {1,1}
   4: {3}
   6: {1,2}
   7: {1,1,1}
   8: {4}
  10: {2,2}
  12: {1,3}
  14: {1,1,2}
  15: {1,1,1,1}
  16: {5}
  20: {2,3}
  24: {1,4}
  26: {1,2,2}
  28: {1,1,3}
  30: {1,1,1,2}
  31: {1,1,1,1,1}
(End)

Column 1 is A000079. Row n has length A000041(n). Right border gives A000225.
Cf. A026792, A065609, A135010, A141285, A186114, A194446, A194546, A206437, A207779, A211978, A225600, A225610, A228351.
The case covering an initial interval is A333379 or A333380.
All of the following pertain to compositions in the order of A066099.
- The weakly increasing version is this sequence.
- The weakly decreasing version is A114994.
- The strictly increasing version is A333255.
- The strictly decreasing version is A333256.
- The unequal version is A233564.
- The equal version is A272919.
- The case covering an initial interval is A333217.
- Initial intervals are ranked by A164894.
- Reversed initial intervals are ranked by A246534.
Cf. A000120, A029931, A048793, A070939, A124766, A233249, A333219, A333220.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],LessEqual@@stc[#]&] (* Gus Wiseman, Apr 01 2020 *)

Conjecture: a(A000070(m) - k) = 2^m - A228354(k) for m > 0, 0 < k <= A000041(m). - Mikhail Kurkov, Oct 20 2023

A124766 Number of monotonically increasing runs for compositions in standard order.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 3, 2, 2, 1, 2, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 3, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 1, 2, 2, 2, 1, 3, 2, 2, 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 weakly increasing runs in this composition. Alternatively, a(n) is one plus the number of strict descents in the same composition. For example, the weakly increasing runs of the 1234567th composition are ((3),(2),(1,2,2),(1,2,5),(1,1,1)), so a(1234567) = 5. The 4 strict descents together with the weak 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 increasing runs are 2; 1,1; so a(11) = 2.
The table starts:
  0
  1
  1 1
  1 2 1 1
  1 2 1 2 1 2 1 1
  1 2 2 2 1 2 2 2 1 2 1 2 1 2 1 1
  1 2 2 2 1 3 2 2 1 2 1 2 2 3 2 2 1 2 2 2 1 2 2 2 1 2 1 2 1 2 1 1

Cf. A066099, A124761, A011782 (row lengths).
Compositions of n with k strict descents are A238343.
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 (this sequence).
- 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.
- Initial intervals are A246534.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Permutations are A333218.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Runs-resistance is A333628.
Cf. A124760, A124763, A124764, A333213, A333220, A333379.

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

a(0) = 0, a(n) = A124761(n) + 1 for n > 0.

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[#]&]

A164894 Base-10 representation of the binary string formed by appending 10, 100, 1000, 10000, ..., etc., to 1.

Original entry on oeis.org

1, 6, 52, 840, 26896, 1721376, 220336192, 56406065280, 28879905423616, 29573023153783296, 60565551418948191232, 248076498612011791288320, 2032242676629600594233921536, 33296264013899376135928570454016, 1091051979207454757222107396637212672

Offset: 1

Gil Broussard, Aug 29 2009

These numbers are half the sum of powers of 2 indexed by differences of a triangular number and each smaller triangular number (e.g., 21 - 15 = 6, 21 - 10 = 11, ..., 21 - 0 = 21).
This suggests another way to think about these numbers: consider the number triangle formed by the characteristic function of the triangular numbers (A010054), join together the first n rows (the very first row is row 0) as a single binary string and that gives the (n + 1)th term of this sequence. - Alonso del Arte, Nov 15 2013
Numbers k such that the k-th composition in standard order (row k of A066099) is an initial interval. - Gus Wiseman, Apr 02 2020

			a(1) = 1, also 1 in binary.
a(2) = 6, or 110 in binary.
a(3) = 52, or 110100 in binary.
a(4) = 840, or 1101001000 in binary.

Harvey P. Dale, Table of n, a(n) for n = 1..81

The version for prime (rather than binary) indices is A002110.
The non-strict generalization is A225620.
The reversed version is A246534.
Standard composition numbers of permutations are A333218.
Standard composition numbers of strict increasing compositions are A333255.
Cf. A000120, A029931, A048793, A066099, A070939, A124768, A233564, A272919, A333217, A333220, A333379.

Table[Sum[2^((n^2 + n)/2 - (k^2 + k)/2 - 1), {k, 0, n - 1}], {n, 25}] (* Alonso del Arte, Nov 14 2013 *)
Module[{nn=15,t},t=Table[10^n,{n,0,nn}];Table[FromDigits[Flatten[IntegerDigits/@Take[t,k]],2],{k,nn}]] (* Harvey P. Dale, Jan 16 2024 *)

def a(n): return int("".join("1"+"0"*i for i in range(n)), 2)
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Jul 05 2021

def A164894(n): return sum(1<<(k*((n<<1)-k-1)>>1)+n-1 for k in range(n)) # Chai Wah Wu, Jul 11 2025

a(n) = Sum_{k=0..n-1} 2^((n^2 + n)/2 - (k^2 + k)/2 - 1). - Alonso del Arte, Nov 15 2013
Intersection of A333255 and A333217. - Gus Wiseman, Apr 02 2020
a(n) = Sum_{k=0..n-1} 2^(k*(2*n-k-1)/2+n-1). - Chai Wah Wu, Jul 11 2025

A335515 Number of patterns of length n matching the pattern (1,2,3).

Original entry on oeis.org

0, 0, 0, 1, 19, 257, 3167, 38909, 498235, 6811453, 100623211, 1612937661, 28033056683, 526501880989, 10639153638795, 230269650097469, 5315570416909995, 130370239796988957, 3385531348514480651, 92801566389186549245, 2677687663571344712043, 81124824154544921317597

Offset: 0

Gus Wiseman, Jun 19 2020

nonn

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 a(3) = 1 through a(4) = 19 patterns:
  (1,2,3)  (1,1,2,3)
           (1,2,1,3)
           (1,2,2,3)
           (1,2,3,1)
           (1,2,3,2)
           (1,2,3,3)
           (1,2,3,4)
           (1,2,4,3)
           (1,3,2,3)
           (1,3,2,4)
           (1,3,4,2)
           (1,4,2,3)
           (2,1,2,3)
           (2,1,3,4)
           (2,3,1,4)
           (2,3,4,1)
           (3,1,2,3)
           (3,1,2,4)
           (4,1,2,3)

Andrew Howroyd, Table of n, a(n) for n = 0..200
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The complement A226316 is the avoiding version.
Compositions matching this pattern are counted by A335514 and ranked by A335479.
Permutations of prime indices matching this pattern are counted by A335520.
Patterns are counted by A000670 and ranked by A333217.
Patterns matching the pattern (1,1) are counted by A019472.
Permutations matching (1,2,3) are counted by A056986.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A158005, A158009, A238279, A333218, A333379, A333942.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],MatchQ[#,{_,x_,_,y_,_,z_,_}/;x

seq(n)=Vec( serlaplace(1/(2-exp(x + O(x*x^n)))) - 1/2 - 1/(1+sqrt(1-8*x+8*x^2 + O(x*x^n))), -(n+1)) \\ Andrew Howroyd, Jan 28 2024

a(n) = A000670(n) - A226316(n). - Andrew Howroyd, Jan 28 2024

a(9) onwards from Andrew Howroyd, Jan 28 2024

A124761 Number of falls for compositions in standard order.

Original entry on oeis.org

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

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 one fewer than the number of maximal weakly increasing runs in this composition. Alternatively, a(n) is the number of strict descents in the same composition. For example, the weakly increasing of runs of the 1234567th composition are ((3),(2),(1,2,2),(1,2,5),(1,1,1)), so a(1234567) = 5 - 1 = 4. The 4 strict descents together with the weak 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; 2>1<=1, so a(11) = 1.
The table starts:
  0
  0
  0 0
  0 1 0 0
  0 1 0 1 0 1 0 0
  0 1 1 1 0 1 1 1 0 1 0 1 0 1 0 0
  0 1 1 1 0 2 1 1 0 1 0 1 1 2 1 1 0 1 1 1 0 1 1 1 0 1 0 1 0 1 0 0

Cf. A066099, A124760, A124763, A124764, A011782 (row lengths), A045883 (row sums), A333213, A333220, A333379.
Positions of zeros are A225620.
Compositions of n with k strict descents are A238343.
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.
- 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.
- Initial intervals are A246534.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Permutations are A333218.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Runs-resistance is A333628.

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

For a composition b(1),...,b(k), a(n) = Sum_{1<=i=1b(i+1)} 1.

For n > 0, a(n) = A124766(n) - 1. - Gus Wiseman, Apr 08 2020

A333380 Numbers k such that the k-th composition in standard order is weakly decreasing and covers an initial interval of positive integers.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 15, 21, 23, 31, 37, 43, 47, 63, 75, 85, 87, 95, 127, 149, 151, 171, 175, 191, 255, 293, 299, 303, 341, 343, 351, 383, 511, 549, 587, 597, 599, 607, 683, 687, 703, 767, 1023, 1099, 1173, 1175, 1195, 1199, 1215, 1365, 1367, 1375, 1407, 1535

Offset: 1

Gus Wiseman, Mar 21 2020

nonn

			The sequence of terms together with the corresponding compositions begins:
    0: ()               127: (1,1,1,1,1,1,1)
    1: (1)              149: (3,2,2,1)
    3: (1,1)            151: (3,2,1,1,1)
    5: (2,1)            171: (2,2,2,1,1)
    7: (1,1,1)          175: (2,2,1,1,1,1)
   11: (2,1,1)          191: (2,1,1,1,1,1,1)
   15: (1,1,1,1)        255: (1,1,1,1,1,1,1,1)
   21: (2,2,1)          293: (3,3,2,1)
   23: (2,1,1,1)        299: (3,2,2,1,1)
   31: (1,1,1,1,1)      303: (3,2,1,1,1,1)
   37: (3,2,1)          341: (2,2,2,2,1)
   43: (2,2,1,1)        343: (2,2,2,1,1,1)
   47: (2,1,1,1,1)      351: (2,2,1,1,1,1,1)
   63: (1,1,1,1,1,1)    383: (2,1,1,1,1,1,1,1)
   75: (3,2,1,1)        511: (1,1,1,1,1,1,1,1,1)
   85: (2,2,2,1)        549: (4,3,2,1)
   87: (2,2,1,1,1)      587: (3,3,2,1,1)
   95: (2,1,1,1,1,1)    597: (3,2,2,2,1)

Sequences covering an initial interval are counted by A000670.
Compositions in standard order are A066099.
Weakly decreasing runs are counted by A124765.
Removing the covering condition gives A114994.
Removing the ordering condition gives A333217.
The strictly decreasing case is A246534.
The unequal version is A333218.
The weakly increasing version is A333379.
Cf. A000120, A000225, A029931, A048793, A070939, A164894, A225620, A228351, A233564, A272919, A333219.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],normQ[stc[#]]&&GreaterEqual@@stc[#]&]

Intersection of A333217 and A114994.

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A333217 Numbers k such that the k-th composition in standard order covers an initial interval of positive integers.

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A225620 Indices of partitions in the table of compositions of A228351.

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A124766 Number of monotonically increasing runs for compositions in standard order.

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

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A164894 Base-10 representation of the binary string formed by appending 10, 100, 1000, 10000, ..., etc., to 1.

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A335515 Number of patterns of length n matching the pattern (1,2,3).

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A124761 Number of falls for compositions in standard order.

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A333380 Numbers k such that the k-th composition in standard order is weakly decreasing and covers an initial interval of positive integers.

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