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A066099 Triangle read by rows, in which row n lists the compositions of n in reverse lexicographic order. 52
0, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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) and A108244 (similar to the Maple partition ordering, A080576).

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 (except for the initial 0) and A070939 as row sums; the second has A001792 as row lengths (again, except for the initial 0) 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 initial zero in this sequence is arbitrary; the sole partition of zero has no components, not a single component of length one. - Franklin T. Adams-Watters, Apr 02 2014

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

LINKS

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..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].

EXAMPLE

The 25th row is associated with the Quet number 162 = 2^1 * 3^3 * 5^1 so the exponents for the ordered prime signature form the vector (1,3,1). Following the method described in A108730 we subtract one from each cell yielding (0,2,0) which gives the number of zeros following each 1 in 11001 (the binary representation of the number 25).

- Alford Arnold, Mar 05 2006

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)

MATHEMATICA

Table[If[n == 0, {0}, FactorInteger[Apply[Times, Map[Prime, Accumulate@ IntegerDigits[n, 2]]]][[All, -1]]], {n, 0, 41}] // Flatten (* Michael De Vlieger, Jul 11 2017 *)

PROG

(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

(Haskell)

a066099 = (!!) a066099_list

a066099_list = concat a066099_tabf

a066099_tabf = map a066099_row [0..]

a066099_row 0 = [0]

a066099_row n = reverse $ a228351_row n

-- (each composition as a row)

-- Peter Kagey, Aug 25 2016

CROSSREFS

Cf. A065120, A057335, A055932. Other versions of this triangle are in A108244, A108730 and A124734.

Cf. A096903, A000120, A070939, A001792, A001788.

Cf. A005811, A101211.

Cf. A261300, A007088.

Sequence in context: A263633 A171850 A087782 * A254111 A234246 A006375

Adjacent sequences:  A066096 A066097 A066098 * A066100 A066101 A066102

KEYWORD

easy,nice,nonn,tabf

AUTHOR

Alford Arnold, Dec 30 2001

EXTENSIONS

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

STATUS

approved

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Last modified August 20 06:32 EDT 2017. Contains 290824 sequences.