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

Author

Omar E. Pol, Aug 28 2013

Keywords

• 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 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

Examples

			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];
...
		

Links

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

Crossrefs

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

Programs

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

Mathematica

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

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

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