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A187816 Triangle read by rows in which row n lists the first 2^(n-1) terms of A006519 in nonincreasing order, n >= 1.

%I A187816 #23 Mar 13 2015 22:54:08
%S A187816 1,2,1,4,2,1,1,8,4,2,2,1,1,1,1,16,8,4,4,2,2,2,2,1,1,1,1,1,1,1,1,32,16,
%T A187816 8,8,4,4,4,4,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,64,32,16,
%U A187816 16,8,8,8,8,4,4,4,4,4,4,4,4,2,2,2,2,2
%N A187816 Triangle read by rows in which row n lists the first 2^(n-1) terms of A006519 in nonincreasing order, n >= 1.
%C A187816 T(n,k) is also the number of parts in the k-th largest region of the diagram of regions of the set of compositions of n, n >= 1, k >= 1, see example.
%C A187816 Row lengths is A000079.
%C A187816 Row sums give A001792(n-1).
%e A187816 For n = 5 the diagram of regions of the set of compositions of 5 has 2^(5-1) regions, see below:
%e A187816 ------------------------------------------------------
%e A187816 .          A006519
%e A187816 .         as a tree
%e A187816 .         of number        Diagram
%e A187816 Region    of parts       of regions     Composition
%e A187816 ------------------------------------------------------
%e A187816 .                         _ _ _ _ _
%e A187816 1      | 1          |    |_| | | | |    1, 1, 1, 1, 1
%e A187816 2      |   2        |    |_ _| | | |    2, 1, 1, 1
%e A187816 3      | 1          |    |_|   | | |    1, 2, 1, 1
%e A187816 4      |      4     |    |_ _ _| | |    3, 1, 1
%e A187816 5      | 1          |    |_| |   | |    1, 1, 2, 1
%e A187816 6      |   2        |    |_ _|   | |    2, 2, 1
%e A187816 7      | 1          |    |_|     | |    1, 3, 1
%e A187816 8      |        8   |    |_ _ _ _| |    4, 1
%e A187816 9      | 1          |    |_| | |   |    1, 1, 1, 2
%e A187816 10     |   2        |    |_ _| |   |    2, 1, 2
%e A187816 11     | 1          |    |_|   |   |    1, 2, 2
%e A187816 12     |      4     |    |_ _ _|   |    3, 2
%e A187816 13     | 1          |    |_| |     |    1, 1, 3
%e A187816 14     |   2        |    |_ _|     |    2, 3
%e A187816 15     | 1          |    |_|       |    1, 4
%e A187816 16     |         16 |    |_ _ _ _ _|    5
%e A187816 .
%e A187816 The first largest region in the diagram is the 16th region which contains 16 parts, so T(5,1) = 16. The second largest region is the 8th region which contains 8 parts, so T(5,2) = 8. The third and the fourth largest regions are both the 4th region and the 12th region, each contains 4 parts, so T(5,3) = 4 and T(5,4) = 4. And so on. The sequence of the number of parts of the k-th largest region of the diagram is [16, 8, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1], the same as the 5th row of triangle, as shown below.
%e A187816 Triangle begins:
%e A187816 1;
%e A187816 2,1;
%e A187816 4,2,1,1;
%e A187816 8,4,2,2,1,1,1,1;
%e A187816 16,8,4,4,2,2,2,2,1,1,1,1,1,1,1,1;
%e A187816 32,16,8,8,4,4,4,4,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
%e A187816 ...
%Y A187816 Cf. A000079, A001511, A001792, A006519, A011782, A065120, A187818, A228525, A228369.
%K A187816 nonn,tabf,easy
%O A187816 1,2
%A A187816 _Omar E. Pol_, Sep 10 2013