

A066099


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


381



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
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OFFSET

1,2


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. AdamsWatters, Nov 06 2006
Compositions (or ordered partitions) are also generated in sequence A101211.  Alford Arnold, Dec 12 2006
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 kth 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


LINKS

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].


FORMULA



EXAMPLE

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
...
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 0's following each 1 in 11001 (the binary representation of the number 25).  Alford Arnold, Mar 05 2006
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[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 *)


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);
(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)
(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


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. 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.


KEYWORD

easy,nice,nonn,tabf


AUTHOR



EXTENSIONS



STATUS

approved



