A228369 Triangle read by rows in which row n lists the compositions (ordered partitions) of n in lexicographic order.

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

Offset

1,4

Comments

The representation of the compositions (for fixed n) is as lists of parts, the order between individual compositions (for the same n) is lexicographic. - Joerg Arndt, Sep 02 2013

The equivalent sequence for partitions is A026791.

Row n has length A001792(n-1).

Row sums give A001787, n >= 1.

The m-th composition has length A008687(m+1), m >= 1. - Andrey Zabolotskiy, Jul 19 2017

Links

Joerg Arndt, Table of n, a(n) for n = 1..10000

Example

Illustration of initial terms:
-----------------------------------
n  j       Diagram   Composition j
-----------------------------------
.               _
1  1           |_|   1;
.             _ _
2  1         | |_|   1, 1,
2  2         |_ _|   2;
.           _ _ _
3  1       | | |_|   1, 1, 1,
3  2       | |_ _|   1, 2,
3  3       |   |_|   2, 1,
3  4       |_ _ _|   3;
.         _ _ _ _
4  1     | | | |_|   1, 1, 1, 1,
4  2     | | |_ _|   1, 1, 2,
4  3     | |   |_|   1, 2, 1,
4  4     | |_ _ _|   1, 3,
4  5     |   | |_|   2, 1, 1,
4  6     |   |_ _|   2, 2,
4  7     |     |_|   3, 1,
4  8     |_ _ _ _|   4;
.
Triangle begins:
[1];
[1,1],[2];
[1,1,1],[1,2],[2,1],[3];
[1,1,1,1],[1,1,2],[1,2,1],[1,3],[2,1,1],[2,2],[3,1],[4];
[1,1,1,1,1],[1,1,1,2],[1,1,2,1],[1,1,3],[1,2,1,1],[1,2,2],[1,3,1],[1,4],[2,1,1,1],[2,1,2],[2,2,1],[2,3],[3,1,1],[3,2],[4,1],[5];
...

Mathematica

Table[Sort[Join@@Permutations/@IntegerPartitions[n], OrderedQ[PadRight[{#1, #2}]]&], {n, 5}] (* Gus Wiseman, Dec 14 2017 *)

Prog

(PARI)
gen_comp(n)=
{  /* Generate compositions of n as lists of parts (order is lex): */
    my(ct = 0);
    my(m, z, pt);
    \\ init:
    my( a = vector(n, j, 1) );
    m = n;
    while ( 1,
        ct += 1;
        pt = vector(m, j, a[j]);
        /* for A228369  print composition: */
        for (j=1, m, print1(pt[j], ", ") );
\\        /* for A228525 print reversed (order is colex): */
\\        forstep (j=m, 1, -1, print1(pt[j], ", ") );
        if ( m<=1,  return(ct) );  \\ current is last
        a[m-1] += 1;
        z = a[m] - 2;
        a[m] = 1;
        m += z;
    );
    return(ct);
}
for(n=1, 12, gen_comp(n) );
\\ Joerg Arndt, Sep 02 2013
(Haskell)
a228369 n = a228369_list !! (n - 1)
a228369_list = concatMap a228369_row [1..]
a228369_row 0 = []
a228369_row n
  | 2^k == 2 * n + 2 = [k - 1]
  | otherwise        = a228369_row (n `div` 2^k) ++ [k] where
    k = a007814 (n + 1) + 1
-- Peter Kagey, Jun 27 2016
(Python)
a = [[[]], [[1]]]
for s in range(2, 9):
    a.append([])
    for k in range(1, s+1):
        for ss in a[s-k]:
            a[-1].append([k]+ss)
print(a)
# Andrey Zabolotskiy, Jul 19 2017

Crossrefs

Cf. A001511, A026791, A066099, A101211, A124734, A228351, A228525, A281013.

Sequence in context: A002635 A275806 A362833 * A296773 A108244 A277824

Adjacent sequences: A228366 A228367 A228368 * A228370 A228371 A228372

Keyword

nonn,tabf

Author

Omar E. Pol, Aug 28 2013

Status

approved