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

Author

Alford Arnold, Dec 30 2001

Keywords

• easy
• nice
• nonn
• tabf

Comments

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

Examples

			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)
		

Links

• 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

Crossrefs

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.

Programs

Haskell

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

Mathematica

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 *)

PARI

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

Python

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

Sage

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

Formula

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)

Extensions

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

0th row removed by Andrey Zabolotskiy, May 19 2018