A124735 -id:A124735 - cp's OEIS frontend

A066099 Triangle read by rows, in which row n lists the compositions of n in reverse lexicographic order.

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

Offset: 1

Alford Arnold, Dec 30 2001

The representation of the compositions (for fixed n) is as lists of parts, the order between individual compositions (for the same n) is (list-)reversed lexicographic; see the example by Omar E. Pol. - Joerg Arndt, Sep 03 2013
This is the standard ordering for compositions in this database; it is similar to the Mathematica ordering for partitions (A080577). Other composition orderings include A124734 (similar to the Abramowitz & Stegun ordering for partitions, A036036), A108244 (similar to the Maple partition ordering, A080576), etc (see crossrefs).
Factorize each term in A057335; sequence records the values of the resulting exponents. It also runs through all possible permutations of multiset digits.
This can be regarded as a table in two ways: with each composition as a row, or with the compositions of each integer as a row. The first way has A000120 as row lengths and A070939 as row sums; the second has A001792 as row lengths and A001788 as row sums. - Franklin T. Adams-Watters, Nov 06 2006
This sequence includes every finite sequence of positive integers. - Franklin T. Adams-Watters, Nov 06 2006
Compositions (or ordered partitions) are also generated in sequence A101211. - Alford Arnold, Dec 12 2006
The equivalent sequence for partitions is A228531. - Omar E. Pol, Sep 03 2013
The sole partition of zero has no components, not a single component of length one. Hence the first nonempty row is row 1. - Franklin T. Adams-Watters, Apr 02 2014 [Edited by Andrey Zabolotskiy, May 19 2018]
See sequence A261300 for another version where the terms of each composition are concatenated to form one single integer: (0, 1, 2, 11, 3, 21, 12, 111,...). This also shows how the terms can be obtained from the binary numbers A007088, cf. Arnold's first Example. - M. F. Hasler, Aug 29 2015
The k-th composition in the list 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 is described as the standard ordering used in the OEIS, although the sister sequence A228351 is also sometimes considered to be canonical. Both sequences define a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, May 19 2020
First differences of A030303 = positions of bits 1 in the concatenation A030190 (= A030302) of numbers written in binary (A007088). - Indices of record values (= first occurrence of n) are given by A005183: a(A005183(n)) = n, cf. FORMULA for more. - M. F. Hasler, Oct 12 2020
The geometric mean approaches the Somos constant (A112302). - Jwalin Bhatt, Feb 10 2025

			A057335 begins 1 2 4 6 8 12 18 30 16 24 36 ... so we can write
  1 2 1 3 2 1 1 4 3 2 2 1 1 1 1 ...
  . . 1 . 1 2 1 . 1 2 1 3 2 1 1 ...
  . . . . . . 1 . . . 1 . 1 2 1 ...
  . . . . . . . . . . . . . . 1 ...
and the columns here gives the rows of the triangle, which begins
  1
  2; 1 1
  3; 2 1; 1 2; 1 1 1
  4; 3 1; 2 2; 2 1 1; 1 3; 1 2 1; 1 1 2; 1 1 1 1
  ...
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms:
  -----------------------------------
  n  j       Diagram   Composition j
  -----------------------------------
  .               _
  1  1           |_|   1;
  .             _ _
  2  1         |  _|   2,
  2  2         |_|_|   1, 1;
  .           _ _ _
  3  1       |    _|   3,
  3  2       |  _|_|   2, 1,
  3  3       | |  _|   1, 2,
  3  4       |_|_|_|   1, 1, 1;
  .         _ _ _ _
  4  1     |      _|   4,
  4  2     |    _|_|   3, 1,
  4  3     |   |  _|   2, 2,
  4  4     |  _|_|_|   2, 1, 1,
  4  5     | |    _|   1, 3,
  4  6     | |  _|_|   1, 2, 1,
  4  7     | | |  _|   1, 1, 2,
  4  8     |_|_|_|_|   1, 1, 1, 1;
(End)

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..5120 (through compositions of 10)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Lists of compositions of integers: this sequence (reverse lexicographic order; minus one gives A108730), A228351 (reverse colexicographic order - every composition is reversed; minus one gives A163510), A228369 (lexicographic), A228525 (colexicographic), A124734 (length, then lexicographic; minus one gives A124735), A296774 (length, then reverse lexicographic), A337243 (length, then colexicographic), A337259 (length, then reverse colexicographic), A296773 (decreasing length, then lexicographic), A296772 (decreasing length, then reverse lexicographic), A337260 (decreasing length, then colexicographic), A108244 (decreasing length, then reverse colexicographic), also A101211 and A227736 (run lengths of bits).
Cf. row length and row sums for different splittings into rows: A000120, A070939, A001792, A001788.
Cf. A228531, A096903, A065120, A057335, A055932, A005811, A261300, A007088.
Cf. lists of partitions of integers, or multisets of integers: A026791 and crosserfs therein, A112798 and crossrefs therein.
See link for additional crossrefs pertaining to standard compositions.
A related ranking of finite sets is A048793/A272020.
Cf. A005183, A030190, A030302, A030303, A035327, A036036, A080576, A080577, A106356, A112302, A238279, A333219.

a066099 = (!!) a066099_list
a066099_list = concat a066099_tabf
a066099_tabf = map a066099_row [1..]
a066099_row n = reverse $ a228351_row n
-- (each composition as a row)
-- Peter Kagey, Aug 25 2016

Table[FactorInteger[Apply[Times, Map[Prime, Accumulate @ IntegerDigits[n, 2]]]][[All, -1]], {n, 41}] // Flatten (* Michael De Vlieger, Jul 11 2017 *)
stc[n_] := Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]] // Reverse;
Table[stc[n], {n, 0, 20}] // Flatten (* Gus Wiseman, May 19 2020 *)
Table[Reverse @ LexicographicSort @ Flatten[Permutations /@ Partitions[n], 1], {n, 10}] // Flatten (* Eric W. Weisstein, Jun 26 2023 *)

arow(n) = {local(v=vector(n),j=0,k=0);
   while(n>0,k++; if(n%2==1,v[j++]=k;k=0);n\=2);
   vector(j,i,v[j-i+1])} \\ returns empty for n=0. - Franklin T. Adams-Watters, Apr 02 2014

from itertools import islice
from itertools import accumulate, count, groupby, islice
def A066099_gen():
    for i in count(1):
        yield [len(list(g)) for _,g in groupby(accumulate(int(b) for b in bin(i)[2:]))]
A066099 = list(islice(A066099_gen(), 120))  # Jwalin Bhatt, Feb 28 2025

def a_row(n): return list(reversed(Compositions(n)))
flatten([a_row(n) for n in range(1,6)]) # Peter Luschny, May 19 2018

From M. F. Hasler, Oct 12 2020: (Start)
a(n) = A030303(n+1) - A030303(n).
a(A005183(n)) = n; a(A005183(n)+1) = n-1 (n>1); a(A005183(n)+2) = 1. (End)

Edited with additional terms by Franklin T. Adams-Watters, Nov 06 2006

0th row removed by Andrey Zabolotskiy, May 19 2018

A124734 Table with all compositions sorted first by total, then by length and finally lexicographically.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Nov 06 2006

This is similar to the Abramowitz and Stegun ordering for partitions (see A036036). The standard ordering for compositions is A066099, which is more similar to the Mathematica partition ordering (A080577).
This can be regarded as a table in two ways: with each composition as a row, or with the compositions of each integer as a row. The first way has A124736 as row lengths and A070939 as row sums; the second has A001792 as row lengths and A001788 as row sums.
This sequence includes every finite sequence of positive integers.

			The table starts:
1
2; 1 1
3; 1 2; 2 1; 1 1 1
4; 1 3; 2 2; 3 1; 1 1 2; 1 2 1; 2 1 1; 1 1 1 1;

Alois P. Heinz, Rows n = 1..11, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A001788, A001792, A036036, A066099, A070939, A080577, A124735, A124736.

Table[Sort@Flatten[Permutations /@ IntegerPartitions@n, 1], {n, 8}] // Flatten (* Robert Price, Jun 13 2020 *)

A108730 Triangle read by rows: row n gives the list of the number of zeros following each 1 in the binary representation of n.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Jun 22 2005

This is probably the simplest method for putting the nonnegative integers into one-to-one correspondence with the finite sequences of nonnegative integers and is the standard ordering for such sequences in this database.
This sequence contains every finite sequence of nonnegative integers.
This can be regarded as a table in two ways: with each weak composition as a row, or with the weak compositions of each integer as a row. The first way has A000120 as row lengths and A080791 as row sums; the second has A001792 as row lengths and A001787 as row sums. - Franklin T. Adams-Watters, Nov 06 2006
Concatenate the base-two positive integers (A030190 less the initial zero). Left to right and disallowing leading zeros, reorganize the digits into the smallest possible numbers. These will be the base-two powers-of-two of A108730. - Hans Havermann, Nov 14 2009
T(2^(n-1),0) = n-1 and T(m,k) < n-1 for all m < 2^n, k <= A000120(m). When the triangle is seen as a flattened list, each n occurs first at position n*2^(n-1)+1, cf. A005183. - Reinhard Zumkeller, Feb 26 2012
Equals A066099-1, elementwise. - Andrey Zabolotskiy, May 18 2018

			Triangle begins:
  0
  1
  0,0
  2
  1,0
  0,1
  0,0,0
  3
For example, 25 = 11001_2; following the 1's are 0, 2 and 0 zeros, so row 25 is 0, 2, 0.

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..5120 (through 10 bit numbers)

Cf. A066099 (main entry for compositions), A007088, A000120, A080791, A001792, A001787, A124735.

import Data.List (unfoldr, group)
a108730 n k = a108730_tabf !! (n-1) !! (k-1)
a108730_row = f . group . reverse . unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2) where
   f [] = []
   f [os] = replicate (length os) 0
   f (os:zs:dss) = replicate (length os - 1) 0 ++ [length zs] ++ f dss
a108730_tabf = map a108730_row [1..]
a108730_list = concat a108730_tabf
-- Reinhard Zumkeller, Feb 26 2012

row[n_] := (id = IntegerDigits[n, 2]; sp = Split[id]; f[run_List] := If[First[run] == 0, run, Sequence @@ Table[{}, {Length[run] - 1}]]; len = Length /@ f /@ sp; If[Last[id] == 0, len, Append[len, 0]]); Flatten[ Table[row[n], {n, 1, 41}]] (* Jean-François Alcover, Jul 13 2012 *)

row(n)=my(v=vector(hammingweight(n)),t=n); for(i=0,#v-1,v[#v-i] = valuation(t,2); t>>=v[#v-i]+1); v \\ Charles R Greathouse IV, Sep 14 2015

Edited by Franklin T. Adams-Watters, Nov 06 2006

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A066099 Triangle read by rows, in which row n lists the compositions of n in reverse lexicographic order.

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A124734 Table with all compositions sorted first by total, then by length and finally lexicographically.

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A108730 Triangle read by rows: row n gives the list of the number of zeros following each 1 in the binary representation of n.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions