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%I #63 Sep 14 2013 12:46:36
%S 1,1,1,2,1,1,1,2,1,1,2,3,1,1,1,1,2,1,1,1,2,1,3,1,1,1,2,2,2,1,3,4,1,1,
%T 1,1,1,2,1,1,1,1,2,1,1,3,1,1,1,1,2,1,2,2,1,1,3,1,4,1,1,1,1,2,2,1,2,1,
%U 2,2,3,2,1,1,3,2,3,1,4,5,1,1,1,1,1,1
%N Triangle read by rows in which row n lists the compositions (ordered partitions) of n in colexicographic order.
%C The representation of the compositions (for fixed n) is as lists of parts, the order between individual compositions (for the same n) is co-lexicographic. [_Joerg Arndt_, Sep 02 2013]
%C The equivalent sequence for partitions is A211992.
%C Row n has length A001792(n-1).
%C Row sums give A001787, n >= 1.
%H Joerg Arndt, <a href="/A228525/b228525.txt">Table of n, a(n) for n = 1..10000</a>
%e Illustration of initial terms:
%e ---------------------------------
%e n j Diagram Composition
%e ---------------------------------
%e . _
%e 1 1 |_| 1;
%e . _ _
%e 2 1 |_| | 1, 1,
%e 2 2 |_ _| 2;
%e . _ _ _
%e 3 1 |_| | | 1, 1, 1,
%e 3 2 |_ _| | 2, 1,
%e 3 3 |_| | 1, 2,
%e 3 4 |_ _ _| 3;
%e . _ _ _ _
%e 4 1 |_| | | | 1, 1, 1, 1,
%e 4 2 |_ _| | | 2, 1, 1,
%e 4 3 |_| | | 1, 2, 1,
%e 4 4 |_ _ _| | 3, 1,
%e 4 5 |_| | | 1, 1, 2,
%e 4 6 |_ _| | 2, 2,
%e 4 7 |_| | 1, 3,
%e 4 8 |_ _ _ _| 4;
%e .
%e Triangle begins:
%e [1];
%e [1,1],[2];
%e [1,1,1],[2,1],[1,2],[3];
%e [1,1,1,1],[2,1,1],[1,2,1],[3,1],[1,1,2],[2,2],[1,3],[4];
%e [1,1,1,1,1],[2,1,1,1],[1,2,1,1],[3,1,1],[1,1,2,1],[2,2,1],[1,3,1],[4,1],[1,1,1,2],[2,1,2],[1,2,2],[3,2],[1,1,3],[2,3],[1,4],[5];
%o (PARI)
%o gen_comp(n)=
%o { /* Generate compositions of n as lists of parts (order is lex): */
%o my(ct = 0);
%o my(m, z, pt);
%o \\ init:
%o my( a = vector(n, j, 1) );
%o m = n;
%o while ( 1,
%o ct += 1;
%o pt = vector(m, j, a[j]);
%o \\ /* for A228369 print composition: */
%o \\ for (j=1, m, print1(pt[j],", ") );
%o /* for A228525 print reversed (order is colex): */
%o forstep (j=m, 1, -1, print1(pt[j],", ") );
%o if ( m<=1, return(ct) ); \\ current is last
%o a[m-1] += 1;
%o z = a[m] - 2;
%o a[m] = 1;
%o m += z;
%o );
%o return(ct);
%o }
%o for(n=1, 12, gen_comp(n) );
%o \\ _Joerg Arndt_, Sep 02 2013
%Y Cf. A066099, A211992, A228351, A228369.
%K nonn,tabf
%O 1,4
%A _Omar E. Pol_, Aug 24 2013
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