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%I #29 Sep 22 2013 16:23:00
%S 1,1,1,3,1,1,7,3,1,1,16,7,3,1,1,36,16,7,3,1,1,80,36,16,7,3,1,1,176,80,
%T 36,16,7,3,1,1,384,176,80,36,16,7,3,1,1,832,384,176,80,36,16,7,3,1,1,
%U 1792,832,384,176,80,36,16,7,3,1,1,3840,1792,832,384,176,80,36,16,7,3,1,1
%N Triangle read by rows: T(n,k) = total number of occurrences of parts k in the n-th section of the set of compositions (ordered partitions) of any integer >= n.
%C Here, define "n-th section of the set of compositions of any integer >= n" to be the set formed by all parts that occur as a result of taking all compositions (ordered partitions) of n and then remove all parts of the compositions of n-1, if n >= 1. Hence the n-th section of the set of compositions of any integer >= n is also the last section of the set of compositions of n. Note that by definition the ordering of compositions is not relevant. For the visualization of the sections here we use a dissection of the diagram of compositions of n in colexicographic order, see example.
%C The equivalent sequence for partitions is A182703.
%C Row n lists the first n terms of A045891 in decreasing order.
%F T(n,k) = A045891(n-k), n >= 1, 1<=k<=n.
%e Illustration (using the colexicograpical order of compositions A228525) of the four sections of the set of compositions of 4, also the first four sections of the set of compositions of any integer >= 4:
%e .
%e . 1 2 3 4
%e . _ _ _ _
%e . |_| _| | | | | |
%e . |_ _| _ _| | | |
%e . |_| | | |
%e . |_ _ _| _ _ _| |
%e . |_| | |
%e . |_ _| |
%e . |_| |
%e . |_ _ _ _|
%e .
%e For n = 4 and k = 2, T(4,2) = 3 because there are 3 parts of size 2 in all compositions of 4, see below:
%e --------------------------------------------------------
%e . The last section Number of
%e . Composition of 4 of the set of parts of
%e . compositions of 4 size k
%e . -------------------- -------------------
%e . Diagram Diagram k = 1 2 3 4
%e . ------------------------------------------------------
%e . _ _ _ _ _
%e . 1+1+1+1 |_| | | | 1 | | 1 0 0 0
%e . 2+1+1 |_ _| | | 1 | | 1 0 0 0
%e . 1+2+1 |_| | | 1 | | 1 0 0 0
%e . 3+1 |_ _ _| | 1 _ _ _| | 1 0 0 0
%e . 1+1+2 |_| | | 1+1+2 |_| | | 2 1 0 0
%e . 2+2 |_ _| | 2+2 |_ _| | 0 2 0 0
%e . 1+3 |_| | 1+3 |_| | 1 0 1 0
%e . 4 |_ _ _ _| 4 |_ _ _ _| 0 0 0 1
%e . ---------
%e . Column sums give row 4: 7,3,1,1
%e .
%e Triangle begins:
%e 1;
%e 1, 1;
%e 3, 1, 1;
%e 7, 3, 1, 1;
%e 16, 7, 3, 1, 1;
%e 36, 16, 7, 3, 1, 1;
%e 80, 36, 16, 7, 3, 1, 1;
%e 176, 80, 36, 16, 7, 3, 1, 1;
%e 384, 176, 80, 36, 16, 7, 3, 1, 1;
%e 832, 384, 176, 80, 36, 16, 7, 3, 1, 1;
%e 1792, 832, 384, 176, 80, 36, 16, 7, 3, 1, 1;
%e 3840, 1792, 832, 384,176, 80, 36, 16, 7, 3, 1, 1;
%e 8192, 3840,1792, 832,384,176, 80, 36, 16, 7, 3, 1, 1;
%e ...
%Y Row sums give A045623. Every column gives A045891.
%Y Cf. A011782, A182703, A221876, A228350, A228366, A228367, A228370, A228526, A228527.
%K nonn,tabl
%O 1,4
%A _Omar E. Pol_, Aug 26 2013
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