A334028 -id:A334028 - cp's OEIS frontend

A350952 The smallest number whose binary expansion has exactly n distinct runs.

0, 1, 2, 11, 38, 311, 2254, 36079, 549790, 17593311, 549687102, 35179974591, 2225029922430, 284803830071167, 36240869367020798, 9277662557957324543, 2368116566113212692990, 1212475681849964898811391, 619877748107024946567312382, 634754814061593545284927880191

Offset: 0

Gus Wiseman, Feb 14 2022

nonn

Positions of first appearances in A297770 (with offset 0).

The binary expansion of terms for n > 0 starts with 1, then floor(n/2) 0's, then alternates runs of increasing numbers of 1's, and decreasing numbers of 0's; see Python code. Thus, for n even, terms have n*(n/2+1)/2 binary digits, and for n odd, ((n+1) + (n-1)*((n-1)/2+1))/2 binary digits. - Michael S. Branicky, Feb 14 2022

			The terms and their binary expansions begin:
       0:                   ()
       1:                    1
       2:                   10
      11:                 1011
      38:               100110
     311:            100110111
    2254:         100011001110
   36079:     1000110011101111
  549790: 10000110001110011110
For example, 311 has binary expansion 100110111 with 5 distinct runs: 1, 00, 11, 0, 111.

Michael S. Branicky, Table of n, a(n) for n = 0..114
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Runs in binary expansion are counted by A005811, distinct A297770.
The version for run-lengths instead of runs is A165933, for A165413.
Subset of A175413 (binary expansion has distinct runs), for lengths A044813.
The version for standard compositions is A351015.
A000120 counts binary weight.
A011782 counts integer compositions.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A334028 counts distinct parts in standard compositions.
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.
- A351018 = binary expansions, for run-lengths A032020.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A003242, A070939, A085207, A098859, A233564, A325545, A328592, A333489, A351203, A351291.

q=Table[Length[Union[Split[If[n==0,{},IntegerDigits[n,2]]]]],{n,0,1000}];Table[Position[q,i][[1,1]]-1,{i,Union[q]}]

a(n)={my(t=0); for(k=1, (n+1)\2, t=((t<Andrew Howroyd, Feb 15 2022

def a(n): # returns term by construction
    if n == 0: return 0
    q, r = divmod(n, 2)
    if r == 0:
        s = "".join("1"*i + "0"*(q-i+1) for i in range(1, q+1))
        assert len(s) == n*(n//2+1)//2
    else:
        s = "1" + "".join("0"*(q-i+2) + "1"*i for i in range(2, q+2))
        assert len(s) == ((n+1) + (n-1)*((n-1)//2+1))//2
    return int(s, 2)
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 14 2022

a(9)-a(19) from Michael S. Branicky, Feb 14 2022

A349051 Numbers k such that the k-th composition in standard order is an alternating permutation of {1..k} for some k.

Original entry on oeis.org

0, 1, 5, 6, 38, 41, 44, 50, 553, 562, 582, 593, 610, 652, 664, 708, 788, 808, 16966, 17036, 17048, 17172, 17192, 17449, 17458, 17542, 17676, 17712, 17940, 18000, 18513, 18530, 18593, 18626, 18968, 18992, 19496, 19536, 20625, 20676, 20769, 20868, 21256, 22600

Offset: 1

Gus Wiseman, Nov 08 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 together with the corresponding compositions begins:
        0: ()
        1: (1)
        5: (2,1)
        6: (1,2)
       38: (3,1,2)
       41: (2,3,1)
       44: (2,1,3)
       50: (1,3,2)
      553: (4,2,3,1)
      562: (4,1,3,2)
      582: (3,4,1,2)
      593: (3,2,4,1)
      610: (3,1,4,2)
      652: (2,4,1,3)
      664: (2,3,1,4)
      708: (2,1,4,3)
      788: (1,4,2,3)
      808: (1,3,2,4)
    16966: (5,3,4,1,2)
    17036: (5,2,4,1,3)

Wikipedia, Alternating permutation

These permutations are counted by A001250, complement A348615.
Compositions of this type are counted by A025047, complement A345192.
Subset of A333218, which ranks permutations of initial intervals.
Subset of A345167, which ranks alternating compositions, complement A345168.
A003242 counts Carlitz (anti-run) compositions.
A345163 counts normal partitions with an alternating permutation.
A345164 counts alternating permutations of prime indices.
A345170 counts partitions with an alternating permutation.
Compositions in standard order are the rows of A066099:
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
- GCD and LCM are given by A326674 and A333226.
- Maximal runs and anti-runs are counted by A124767 and A333381.
- Heinz number is given by A333219.
- Runs-resistance is given by A333628.
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Carlitz (anti-run) compositions are ranked by A333489.
Cf. A025048, A025049, A344604, A344615, A344653, A344742, A348377, A348379, A345165.

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[0,1000],Sort[stc[#]]==Range[Length[stc[#]]]&&wigQ[stc[#]]&]

Equals A333218 (permutation) /\ A345167 (alternating).

A095684 Triangle read by rows. There are 2^(m-1) rows of length m, for m = 1, 2, 3, ... The rows are in lexicographic order. The rows have the property that the first entry is 1, the second distinct entry (reading from left to right) is 2, the third distinct entry is 3, etc.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 25 2004

Row k is the unique multiset that covers an initial interval of positive integers and has multiplicities equal to the parts of the k-th composition in standard order (graded reverse-lexicographic, A066099). This composition 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 example, the 13th composition is (1,2,1), so row 13 is {1,2,2,3}. - Gus Wiseman, Apr 26 2020

			1, 11, 12, 111, 112, 122, 123, 1111, 1112, 1122, 1123, 1222, 1223, 1233, ...
The 8 strings of length 4 are 1111, 1112, 1122, 1123, 1222, 1223, 1233, 1234.
From _Gus Wiseman_, Apr 26 2020: (Start)
The triangle read by columns begins:
  1:{1}  2:{1,1}  4:{1,1,1}   8:{1,1,1,1}  16:{1,1,1,1,1}
         3:{1,2}  5:{1,1,2}   9:{1,1,1,2}  17:{1,1,1,1,2}
                  6:{1,2,2}  10:{1,1,2,2}  18:{1,1,1,2,2}
                  7:{1,2,3}  11:{1,1,2,3}  19:{1,1,1,2,3}
                             12:{1,2,2,2}  20:{1,1,2,2,2}
                             13:{1,2,2,3}  21:{1,1,2,2,3}
                             14:{1,2,3,3}  22:{1,1,2,3,3}
                             15:{1,2,3,4}  23:{1,1,2,3,4}
                                           24:{1,2,2,2,2}
                                           25:{1,2,2,2,3}
                                           26:{1,2,2,3,3}
                                           27:{1,2,2,3,4}
                                           28:{1,2,3,3,3}
                                           29:{1,2,3,3,4}
                                           30:{1,2,3,4,4}
                                           31:{1,2,3,4,5}
(End)

J. C. Kieffer, W. Szpankowski and E.-H. Yang, Problems on sequences: information theory and computer science interface, IEEE Trans. Inform. Theory, 50 (No. 7, 2004), 1385-1392.

See A096299 for another version.
The number of distinct parts in row n is A000120(n), also the maximum part.
Row sums are A029931.
Heinz numbers of rows are A057335.
Row lengths are A070939.
Row products are A284001.
The version for prime indices is A305936.
There are A333942(n) multiset partitions of row n.
Multisets of compositions are counted by A034691.
Combinatory separations of normal multisets are A269134.
All of the following pertain to compositions in standard order (A066099):
- Necklaces are A065609.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Dealings are counted by A333939.
- Distinct parts are counted by A334028.
Cf. A318284, A333764, A333940, A334030.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ptnToNorm[y_]:=Join@@Table[ConstantArray[i,y[[i]]],{i,Length[y]}];
Table[ptnToNorm[stc[n]],{n,15}] (* Gus Wiseman, Apr 26 2020 *)

A334029 Length of the co-Lyndon factorization of the k-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 14 2020

nonn

We define the co-Lyndon product of two or more finite sequences to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (2,3,1) with (2,1,3) is (2,1,2,3,1,3), the product of (2,2,1) with (2,1,3) is (2,1,2,2,1,3), and the product of (1,2,2) with (2,1,2,1) is (1,2,1,2,1,2,2). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1,0,0,1) has co-Lyndon factorization {(1),(1,0,0)}.

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 441st composition in standard order is (1,2,1,1,3,1), with co-Lyndon factorization {(1),(3,1),(2,1,1)}, so a(441) = 3.

The dual version is A329312.
The version for binary expansion is (also) A329312.
The version for reversed binary expansion is A329326.
Binary Lyndon/co-Lyndon words are counted by A001037.
Necklaces covering an initial interval are A019536.
Lyndon/co-Lyndon compositions are counted by A059966
Length of Lyndon factorization of binomial expansion is A211100.
Numbers whose prime signature is a necklace are A329138.
Length of Lyndon factorization of reversed binary expansion is A329313.
A list of all binary co-Lyndon words is A329318.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Reversed co-necklaces are A328595.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Co-Lyndon factorizations are counted by A333765.
- Lyndon factorizations are counted by A333940.
- Reversed necklaces are A333943.
- Co-necklaces are A334028.
Cf. A034691, A060223, A102659, A211097, A292884, A296372, A328596, A329358, A329359, A329362, A329400, A329401, A333939.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#1],q}]=={RotateRight[q,#1],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],colynQ[Take[q,#1]]&]]]]
Table[Length[colynfac[stc[n]]],{n,0,100}]

A335239 Numbers k such that the k-th composition in standard-order (A066099) does not have all pairwise coprime parts, where a singleton is not coprime unless it is (1).

Original entry on oeis.org

0, 2, 4, 8, 10, 16, 21, 22, 26, 32, 34, 36, 40, 42, 43, 45, 46, 53, 54, 58, 64, 69, 70, 73, 74, 76, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 98, 100, 104, 106, 107, 109, 110, 117, 118, 122, 128, 130, 136, 138, 139, 141, 142, 146, 147, 148, 149, 150, 153

Offset: 1

Gus Wiseman, May 28 2020

nonn

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).

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.

			The sequence together with the corresponding compositions begins:
    0: ()            45: (2,1,2,1)     86: (2,2,1,2)
    2: (2)           46: (2,1,1,2)     87: (2,2,1,1,1)
    4: (3)           53: (1,2,2,1)     88: (2,1,4)
    8: (4)           54: (1,2,1,2)     90: (2,1,2,2)
   10: (2,2)         58: (1,1,2,2)     91: (2,1,2,1,1)
   16: (5)           64: (7)           93: (2,1,1,2,1)
   21: (2,2,1)       69: (4,2,1)       94: (2,1,1,1,2)
   22: (2,1,2)       70: (4,1,2)       98: (1,4,2)
   26: (1,2,2)       73: (3,3,1)      100: (1,3,3)
   32: (6)           74: (3,2,2)      104: (1,2,4)
   34: (4,2)         76: (3,1,3)      106: (1,2,2,2)
   36: (3,3)         81: (2,4,1)      107: (1,2,2,1,1)
   40: (2,4)         82: (2,3,2)      109: (1,2,1,2,1)
   42: (2,2,2)       84: (2,2,3)      110: (1,2,1,1,2)
   43: (2,2,1,1)     85: (2,2,2,1)    117: (1,1,2,2,1)

The complement is A333227.
The version without singletons is A335236.
Ignoring repeated parts gives A335238.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Numbers whose binary indices are pairwise coprime are A326675.
Coprime partitions are counted by A327516.
Non-coprime partitions are counted by A335240.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
- Coprime compositions are A333227.
- Compositions whose distinct parts are coprime are A333228.
- Number of distinct parts is A334028.
Cf. A007360, A048793, A101268, A291166, A302569, A335235, A335237.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!CoprimeQ@@stc[#]&]

A284001 a(n) = A005361(A283477(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 4, 6, 1, 2, 4, 6, 8, 12, 18, 24, 1, 2, 4, 6, 8, 12, 18, 24, 16, 24, 36, 48, 54, 72, 96, 120, 1, 2, 4, 6, 8, 12, 18, 24, 16, 24, 36, 48, 54, 72, 96, 120, 32, 48, 72, 96, 108, 144, 192, 240, 162, 216, 288, 360, 384, 480, 600, 720, 1, 2, 4, 6, 8, 12, 18, 24, 16, 24, 36, 48, 54, 72, 96, 120, 32, 48, 72, 96, 108, 144, 192, 240, 162, 216, 288, 360, 384, 480

Offset: 0

Antti Karttunen, Mar 18 2017

a(n) is the product of elements of the multiset that covers an initial interval of positive integers with multiplicities equal to the parts of the n-th composition in standard order (graded reverse-lexicographic, A066099). This composition 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. For example, the 13th composition is (1,2,1) giving the multiset {1,2,2,3} with product 12, so a(13) = 12. - Gus Wiseman, Apr 26 2020

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A005361, A283477, A284002, A284005.
Row products of A095684.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Weighted sum is A029931.
- Necklaces are A065609.
- Sum is A070939.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Distinct parts are counted by A334028.
Cf. A003963, A034691, A053645, A057335, A305936, A318284, A333764, A333940, A333942, A334030.

Table[Times @@ FactorInteger[#][[All, -1]] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]]], {n, 0, 93}] (* Michael De Vlieger, Mar 18 2017 *)

A005361(n) = factorback(factor(n)[, 2]); \\ From A005361
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A283477(n) = A108951(A019565(n));
A284001(n) = A005361(A283477(n));

(define (A284001 n) (A005361 (A283477 n)))

a(n) = A005361(A283477(n)).
a(n) = A003963(A057335(n)). - Gus Wiseman, Apr 26 2020
a(n) = A284005(A053645(n)) for n > 0 with a(0) = 1. - Mikhail Kurkov, Jun 05 2021 [verification needed]

A334265 Numbers k such that the k-th composition in standard order is a reversed Lyndon word.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 11, 16, 17, 18, 19, 21, 23, 32, 33, 34, 35, 37, 39, 41, 43, 47, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 87, 91, 95, 128, 129, 130, 131, 132, 133, 135, 137, 138, 139, 141, 143, 145, 146, 147, 149, 151, 155, 159, 161, 163

Offset: 1

Gus Wiseman, Apr 22 2020

nonn

Reversed Lyndon words are different from co-Lyndon words (A326774).
A Lyndon word is a finite sequence of positive integers that is lexicographically strictly less than all of its cyclic rotations.
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.

			The sequence of all reversed Lyndon words begins:
    0: ()            37: (3,2,1)         83: (2,3,1,1)
    1: (1)           39: (3,1,1,1)       85: (2,2,2,1)
    2: (2)           41: (2,3,1)         87: (2,2,1,1,1)
    4: (3)           43: (2,2,1,1)       91: (2,1,2,1,1)
    5: (2,1)         47: (2,1,1,1,1)     95: (2,1,1,1,1,1)
    8: (4)           64: (7)            128: (8)
    9: (3,1)         65: (6,1)          129: (7,1)
   11: (2,1,1)       66: (5,2)          130: (6,2)
   16: (5)           67: (5,1,1)        131: (6,1,1)
   17: (4,1)         68: (4,3)          132: (5,3)
   18: (3,2)         69: (4,2,1)        133: (5,2,1)
   19: (3,1,1)       71: (4,1,1,1)      135: (5,1,1,1)
   21: (2,2,1)       73: (3,3,1)        137: (4,3,1)
   23: (2,1,1,1)     74: (3,2,2)        138: (4,2,2)
   32: (6)           75: (3,2,1,1)      139: (4,2,1,1)
   33: (5,1)         77: (3,1,2,1)      141: (4,1,2,1)
   34: (4,2)         79: (3,1,1,1,1)    143: (4,1,1,1,1)
   35: (4,1,1)       81: (2,4,1)        145: (3,4,1)

The non-reversed version is A275692.
The generalization to necklaces is A333943.
The dual version (reversed co-Lyndon words) is A328596.
The case that is also co-Lyndon is A334266.
Binary Lyndon words are counted by A001037.
Lyndon compositions are counted by A059966.
Normal Lyndon words are counted by A060223.
Numbers whose prime signature is a reversed Lyndon word are A334298.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Reversed Lyndon words are A334265 (this sequence).
- Co-Lyndon words are A326774.
- Reversed co-Lyndon words are A328596.
- Length of Lyndon factorization is A329312.
- Distinct rotations are counted by A333632.
- Lyndon factorizations are counted by A333940.
- Length of Lyndon factorization of reverse is A334297.
Cf. A000031, A027375, A138904, A211100, A328595, A329131, A329313, A333764, A334028, A334267.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
Select[Range[0,100],lynQ[Reverse[stc[#]]]&]

A351010 Numbers k such that the k-th composition in standard order is a concatenation of twins (x,x).

Original entry on oeis.org

0, 3, 10, 15, 36, 43, 58, 63, 136, 147, 170, 175, 228, 235, 250, 255, 528, 547, 586, 591, 676, 683, 698, 703, 904, 915, 938, 943, 996, 1003, 1018, 1023, 2080, 2115, 2186, 2191, 2340, 2347, 2362, 2367, 2696, 2707, 2730, 2735, 2788, 2795, 2810, 2815, 3600, 3619

Offset: 1

Gus Wiseman, Feb 01 2022

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.

			The terms together with their binary expansions and the corresponding compositions begin:
    0:         0  ()
    3:        11  (1,1)
   10:      1010  (2,2)
   15:      1111  (1,1,1,1)
   36:    100100  (3,3)
   43:    101011  (2,2,1,1)
   58:    111010  (1,1,2,2)
   63:    111111  (1,1,1,1,1,1)
  136:  10001000  (4,4)
  147:  10010011  (3,3,1,1)
  170:  10101010  (2,2,2,2)
  175:  10101111  (2,2,1,1,1,1)
  228:  11100100  (1,1,3,3)
  235:  11101011  (1,1,2,2,1,1)
  250:  11111010  (1,1,1,1,2,2)
  255:  11111111  (1,1,1,1,1,1,1,1)

The case of twins (binary weight 2) is A000120.
The Heinz numbers of these compositions are given by A000290.
All terms are evil numbers A001969.
Partitions of this type are counted by A035363, any length A351004.
These compositions are counted by A077957(n-2), see also A016116.
The strict case (distinct twins) is A351009, counted by A032020 with 0's.
The anti-run case is A351011, counted by A003242 interspersed with 0's.
A011782 counts integer compositions.
A085207/A085208 represent concatenation of standard compositions.
A333489 ranks anti-runs, complement A348612.
A345167/A350355/A350356 rank alternating compositions.
A351014 counts distinct runs in standard compositions.
Cf. A018819, A025047, A027383, A035457, A053738, A088218, A106356, A238279, A344604, A351012, A351015.
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.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],And@@EvenQ/@Length/@Split[stc[#]]&]

A350355 Numbers k such that the k-th composition in standard order is up/down.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 13, 16, 20, 24, 25, 32, 40, 41, 48, 49, 50, 54, 64, 72, 80, 81, 82, 96, 97, 98, 102, 108, 109, 128, 144, 145, 160, 161, 162, 166, 192, 193, 194, 196, 198, 204, 205, 216, 217, 256, 272, 288, 289, 290, 320, 321, 322, 324, 326, 332, 333, 384

Offset: 1

Gus Wiseman, Jan 15 2022

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 composition is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase. For example, the partition (3,2,2,2,1) has no up/down permutations, even though it does have the anti-run permutation (2,3,2,1,2).

			The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   6: (1,2)
   8: (4)
  12: (1,3)
  13: (1,2,1)
  16: (5)
  20: (2,3)
  24: (1,4)
  25: (1,3,1)
  32: (6)
  40: (2,4)
  41: (2,3,1)
  48: (1,5)
  49: (1,4,1)
  50: (1,3,2)
  54: (1,2,1,2)

The case of permutations is counted by A000111.
These compositions are counted by A025048, down/up A025049.
The strict case is counted by A129838, undirected A349054.
The weak version is counted by A129852, down/up A129853.
The version for anti-runs is A333489, a superset, complement A348612.
This is the up/down case of A345167, counted by A025047.
Counting patterns of this type gives A350354.
The down/up version is A350356.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A011782 counts compositions, unordered A000041.
A345192 counts non-alternating compositions, ranked by A345168.
A349052 counts weakly alternating compositions, complement A349053.
A349057 ranks non-weakly alternating compositions.
Statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Patterns are A333217.
Cf. A008965, A049774, A095684, A106356, A238279, A344604, A344614, A344615, A345169, A345170, A345172, A349051, A349799.

updoQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]>y[[m+1]],y[[m]]

A345167 = A350355 \/ A350356.

A350356 Numbers k such that the k-th composition in standard order is down/up.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 16, 17, 18, 22, 32, 33, 34, 38, 44, 45, 64, 65, 66, 68, 70, 76, 77, 88, 89, 128, 129, 130, 132, 134, 140, 141, 148, 152, 153, 176, 177, 178, 182, 256, 257, 258, 260, 262, 264, 268, 269, 276, 280, 281, 296, 297, 304, 305, 306, 310, 352, 353

Offset: 1

Gus Wiseman, Jan 15 2022

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 composition is down/up if it is alternately strictly increasing and strictly decreasing, starting with a decrease. For example, the partition (3,2,2,2,1) has no down/up permutations, even though it does have the anti-run permutation (2,1,2,3,2).

			The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   8: (4)
   9: (3,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  22: (2,1,2)
  32: (6)
  33: (5,1)
  34: (4,2)
  38: (3,1,2)
  44: (2,1,3)
  45: (2,1,2,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions
Wikipedia, Alternating permutation

The case of permutations is counted by A000111.
These compositions are counted by A025049, up/down A025048.
The strict case is counted by A129838, undirected A349054.
The weak version is counted by A129853, up/down A129852.
The version for anti-runs is A333489, a superset, complement A348612.
This is the down/up case of A345167, counted by A025047.
Counting patterns of this type gives A350354.
The up/down version is A350355.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A011782 counts compositions, unordered A000041.
A345192 counts non-alternating compositions, ranked by A345168.
A349052 counts weakly alternating compositions, complement A349053.
A349057 ranks non-weakly alternating compositions.
Statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Patterns are A333217.
Cf. A008965, A049774, A095684, A106356, A238279, A344604, A344614, A344615, A345169, A345170, A345172, A349051, A349799.

doupQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]y[[m+1]]],{m,1,Length[y]-1}];
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],doupQ[stc[#]]&]

A345167 = A350355 \/ A350356.

cp's OEIS Frontend

A350952 The smallest number whose binary expansion has exactly n distinct runs.

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A349051 Numbers k such that the k-th composition in standard order is an alternating permutation of {1..k} for some k.

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A095684 Triangle read by rows. There are 2^(m-1) rows of length m, for m = 1, 2, 3, ... The rows are in lexicographic order. The rows have the property that the first entry is 1, the second distinct entry (reading from left to right) is 2, the third distinct entry is 3, etc.

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A334029 Length of the co-Lyndon factorization of the k-th composition in standard order.

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A335239 Numbers k such that the k-th composition in standard-order (A066099) does not have all pairwise coprime parts, where a singleton is not coprime unless it is (1).

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A284001 a(n) = A005361(A283477(n)).

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A334265 Numbers k such that the k-th composition in standard order is a reversed Lyndon word.

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A351010 Numbers k such that the k-th composition in standard order is a concatenation of twins (x,x).

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