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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. 6
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, 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, 16, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

Row lengths is A000079.

Row sums give A001792(n-1).

LINKS

Table of n, a(n) for n=1..84.

EXAMPLE

For n = 5 the diagram of regions of the set of compositions of 5 has 2^(5-1) regions, see below:

------------------------------------------------------

.          A006519

.         as a tree

.         of number        Diagram

Region    of parts       of regions     Composition

------------------------------------------------------

.                         _ _ _ _ _

1      | 1          |    |_| | | | |    1, 1, 1, 1, 1

2      |   2        |    |_ _| | | |    2, 1, 1, 1

3      | 1          |    |_|   | | |    1, 2, 1, 1

4      |      4     |    |_ _ _| | |    3, 1, 1

5      | 1          |    |_| |   | |    1, 1, 2, 1

6      |   2        |    |_ _|   | |    2, 2, 1

7      | 1          |    |_|     | |    1, 3, 1

8      |        8   |    |_ _ _ _| |    4, 1

9      | 1          |    |_| | |   |    1, 1, 1, 2

10     |   2        |    |_ _| |   |    2, 1, 2

11     | 1          |    |_|   |   |    1, 2, 2

12     |      4     |    |_ _ _|   |    3, 2

13     | 1          |    |_| |     |    1, 1, 3

14     |   2        |    |_ _|     |    2, 3

15     | 1          |    |_|       |    1, 4

16     |         16 |    |_ _ _ _ _|    5

.

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.

Triangle begins:

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;

32,16,8,8,4,4,4,4,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;

...

CROSSREFS

Cf. A000079, A001511, A001792, A006519, A011782, A065120, A187818, A228525, A228369.

Sequence in context: A290935 A031424 A013942 * A088423 A006839 A268267

Adjacent sequences:  A187813 A187814 A187815 * A187817 A187818 A187819

KEYWORD

nonn,tabf,easy

AUTHOR

Omar E. Pol, Sep 10 2013

STATUS

approved

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Last modified July 9 13:27 EDT 2020. Contains 335543 sequences. (Running on oeis4.)