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 A228369 Triangle read by rows in which row n lists the compositions (ordered partitions) of n in lexicographic order. 28
 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 (list; graph; refs; listen; history; text; internal format)
 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: A254011 A002635 A275806 * A296773 A108244 A277824 Adjacent sequences:  A228366 A228367 A228368 * A228370 A228371 A228372 KEYWORD nonn,tabf AUTHOR Omar E. Pol, Aug 28 2013 STATUS approved

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Last modified October 23 22:04 EDT 2018. Contains 316541 sequences. (Running on oeis4.)