cp's OEIS Frontend

A228349 Triangle read by rows: T(j,k) is the k-th part in nondecreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

%I A228349 #19 May 24 2017 02:39:54
%S A228349 1,1,2,1,1,1,2,3,1,1,2,1,1,1,1,1,2,2,3,4,1,1,2,1,1,1,2,3,1,1,2,1,1,1,
%T A228349 1,1,1,1,1,1,2,2,2,2,3,3,4,5,1,1,2,1,1,1,2,3,1,1,2,1,1,1,1,1,2,2,3,4,
%U A228349 1,1,2,1,1,1,2,3,1,1,2,1,1,1,1,1,1,1
%N A228349 Triangle read by rows: T(j,k) is the k-th part in nondecreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).
%C A228349 Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A090996, n >= 1.
%C A228349 The equivalent sequence for partitions is A220482.
%H A228349 Michael De Vlieger, <a href="/A228349/b228349.txt">Table of n, a(n) for n = 1..13312</a> (rows 1 <= n <= 2^11 = 2048).
%e A228349 ----------------------------------------------------------
%e A228349 .             Diagram                Triangle
%e A228349 Compositions    of            of compositions (rows)
%e A228349 of 5          regions          and regions (columns)
%e A228349 ----------------------------------------------------------
%e A228349 .            _ _ _ _ _
%e A228349 5           |_        |                                 5
%e A228349 1+4         |_|_      |                               1 4
%e A228349 2+3         |_  |     |                             2   3
%e A228349 1+1+3       |_|_|_    |                           1 1   3
%e A228349 3+2         |_    |   |                         3       2
%e A228349 1+2+2       |_|_  |   |                       1 2       2
%e A228349 2+1+2       |_  | |   |                     2   1       2
%e A228349 1+1+1+2     |_|_|_|_  |                   1 1   1       2
%e A228349 4+1         |_      | |                 4               1
%e A228349 1+3+1       |_|_    | |               1 3               1
%e A228349 2+2+1       |_  |   | |             2   2               1
%e A228349 1+1+2+1     |_|_|_  | |           1 1   2               1
%e A228349 3+1+1       |_    | | |         3       1               1
%e A228349 1+2+1+1     |_|_  | | |       1 2       1               1
%e A228349 2+1+1+1     |_  | | | |     2   1       1               1
%e A228349 1+1+1+1+1   |_|_|_|_|_|   1 1   1       1               1
%e A228349 .
%e A228349 Written as an irregular triangle in which row n lists the parts of the n-th region the sequence begins:
%e A228349 1;
%e A228349 1,2;
%e A228349 1;
%e A228349 1,1,2,3;
%e A228349 1;
%e A228349 1,2;
%e A228349 1;
%e A228349 1,1,1,1,2,2,3,4;
%e A228349 1;
%e A228349 1,2;
%e A228349 1;
%e A228349 1,1,2,3;
%e A228349 1;
%e A228349 1,2;
%e A228349 1;
%e A228349 1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
%e A228349 ...
%e A228349 Alternative interpretation of this sequence:
%e A228349 Triangle read by rows in which row r lists the regions of the last section of the set of compositions of r:
%e A228349 [1];
%e A228349 [1,2];
%e A228349 [1],[1,1,2,3];
%e A228349 [1],[1,2],[1],[1,1,1,1,2,2,3,4];
%e A228349 [1],[1,2],[1],[1,1,2,3],[1],[1,2],[1],[1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5];
%t A228349 Table[Map[Length@ TakeWhile[IntegerDigits[#, 2], # == 1 &] &, Range[2^(# - 1), 2^# - 1]] &@ IntegerExponent[2 n, 2], {n, 32}] // Flatten (* _Michael De Vlieger_, May 23 2017 *)
%Y A228349 Main triangle: Right border gives A001511. Row j has length A006519(j). Row sums give A038712.
%Y A228349 Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A090996, A186114, A187816, A187818, A206437, A220482, A228347, A228348, A228350, A228351, A228366, A228367, A228370, A228371, A228525, A228526.
%K A228349 nonn,tabf
%O A228349 1,3
%A A228349 _Omar E. Pol_, Aug 26 2013