A187816 - OEIS

A187816

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

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:

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