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%I #23 Mar 13 2015 22:54:08
%S 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 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 16,8,8,8,8,4,4,4,4,4,4,4,4,2,2,2,2,2
%N Triangle read by rows in which row n lists the first 2^(n-1) terms of A006519 in nonincreasing order, n >= 1.
%C 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 Row lengths is A000079.
%C Row sums give A001792(n-1).
%e For n = 5 the diagram of regions of the set of compositions of 5 has 2^(5-1) regions, see below:
%e ------------------------------------------------------
%e . A006519
%e . as a tree
%e . of number Diagram
%e Region of parts of regions Composition
%e ------------------------------------------------------
%e . _ _ _ _ _
%e 1 | 1 | |_| | | | | 1, 1, 1, 1, 1
%e 2 | 2 | |_ _| | | | 2, 1, 1, 1
%e 3 | 1 | |_| | | | 1, 2, 1, 1
%e 4 | 4 | |_ _ _| | | 3, 1, 1
%e 5 | 1 | |_| | | | 1, 1, 2, 1
%e 6 | 2 | |_ _| | | 2, 2, 1
%e 7 | 1 | |_| | | 1, 3, 1
%e 8 | 8 | |_ _ _ _| | 4, 1
%e 9 | 1 | |_| | | | 1, 1, 1, 2
%e 10 | 2 | |_ _| | | 2, 1, 2
%e 11 | 1 | |_| | | 1, 2, 2
%e 12 | 4 | |_ _ _| | 3, 2
%e 13 | 1 | |_| | | 1, 1, 3
%e 14 | 2 | |_ _| | 2, 3
%e 15 | 1 | |_| | 1, 4
%e 16 | 16 | |_ _ _ _ _| 5
%e .
%e 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 Triangle begins:
%e 1;
%e 2,1;
%e 4,2,1,1;
%e 8,4,2,2,1,1,1,1;
%e 16,8,4,4,2,2,2,2,1,1,1,1,1,1,1,1;
%e 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 ...
%Y Cf. A000079, A001511, A001792, A006519, A011782, A065120, A187818, A228525, A228369.
%K nonn,tabf,easy
%O 1,2
%A _Omar E. Pol_, Sep 10 2013
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