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A206437 Triangle read by rows: T(j,k) is the k-th part of the j-th region of the set of partitions of n, if 1<=j<=A000041(n). 94
1, 2, 1, 3, 1, 1, 2, 4, 2, 1, 1, 1, 3, 5, 2, 1, 1, 1, 1, 1, 2, 4, 2, 3, 6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 3, 5, 2, 4, 7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 8, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Here the j-th "region" of the set of partitions of n (or more simply the j-th "region" of n) is defined to be the first h elements of the sequence formed by the smallest parts in nonincreasing order of the partitions of the largest part of the j-th partition of n, with the list of partitions in lexicographic ordering, where h = j - i, and i is the index of the previous partition of n whose largest part is greater than the largest part of the j-th partition of n, or i = 0 if such previous largest part does not exist. The largest part of the j-th region of n is A141285(j) and the number of parts is h = A194446(j).

Some properties of the regions of n:

- The number of regions of n equals the number of partitions of n (see A000041).

- The set of regions of n contain the sets of regions of all positive integers previous to n.

- The first j regions of n are also first j regions of all integers greater than n.

- The sums of all largest parts of all regions of n equals the total number of parts of all regions of n. See A006128(n).

- If T(j,1) is a record in the sequence then the leading diagonals of triangle formed by the first j rows give the partitions of n (see example).

- The rank of a region is the largest part minus the number of parts (see A194447).

- The sum of all ranks of the regions of n is equal to zero.

How to make a diagram of the regions and partitions of n: in the first quadrant of the square grid we draw a horizontal line {[0, 0],[n, 0]} of length n. Then we draw a vertical line {[0, 0],[0, p(n)]} of length p(n) where p(n) is the number of partitions of n. Then, for j = 1..p(n), we draw a horizontal line {[0, j],[g, j] where g = A141285(j) is the largest part of the j-th partition of n, with the list of partitions in lexicographic order. Then, for n = 1 .. p(n), we draw a vertical line from the point [g,j] up to intercept the next segment in a lower row with respect to the axis "y". So we have a number of closed regions. Then we divide each region of n in horizontal rectangles with shorter sides = 1. We can see that in the original rectangle of area n*p(n) each row contains a set of rectangles whose areas are equal to the parts of one of the partitions of n. Then each region of n is labeled according to the position of its largest part on axis "y". Note that each region of n is similar to a mirror version of the Young diagram of one of the partitions of s, where s is the sum of all parts of the region. See the illustrations of the seven regions of 5 in the link section.

Note that if row j of triangle contains parts of size 1 then the parts of row j are the smallest parts of all partitions of T(j,1), (see A046746), and also T(j,1) is a record in the sequence and also j is the number of partitions of T(j,1), (see A000041). Otherwise, if row j does not contain parts of size 1 then the parts of row j are the emergent parts of the next record in the sequence (see A183152). Row j is also the partition of A186412(j).

Also triangle read by rows in which row r lists the parts of the last shell of r, ordered by regions, such that the previous parts to the part of size r are the emergent parts of the partitions of r (see A138152) and the rest are the smallest parts of the partitions of r (see example). - Omar E. Pol, Apr 28 2012

LINKS

Table of n, a(n) for n=1..86.

Omar E. Pol, Illustration of the seven regions of 5

Omar E. Pol, Illustration of initial terms, regions = 1..77 (2D view)

Omar E. Pol, Illustration of initial terms, regions = 1..30 (3D view)

Omar E. Pol, Visualization of regions in a diagram for A006128

EXAMPLE

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

Region     Triangle

j          of parts

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

1          1;

2          2,1;

3          3,1,1;

4          2;

5          4,2,1,1,1;

6          3;

7          5,2,1,1,1,1,1;

8          2;

9          4,2;

10         3;

11         6,3,2,2,1,1,1,1,1,1,1;

12         3;

13         5,2;

14         4;

15         7,3,2,2,1,1,1,1,1,1,1,1,1,1,1;

.

The rotated triangle shows each row as a partition:

.

.                           7

.                         4   3

.                       5       2

.                     3   2       2

.                   6               1

.                 3   3               1

.               4       2               1

.             2   2       2               1

.           5               1               1

.         3   2               1               1

.       4       1               1               1

.     2   2       1               1               1

.   3       1       1               1               1

. 2   1       1       1               1               1

1   1   1       1       1               1               1

.

Alternative interpretation of this sequence:

Triangle read by rows in which row r lists the parts of the last shell of r ordered by regions (see comments):

[1];

[2,1];

[3,1,1];

[2],[4,2,1,1,1];

[3],[5,2,1,1,1,1,1];

[2],[4,2],[3],[6,3,2,2,1,1,1,1,1,1,1];

[3],[5,2],[4],[7,3,2,2,1,1,1,1,1,1,1,1,1,1,1];

CROSSREFS

Positive integers in A193870. Column 1 is A141285. Row j has length A194446(j). Row sums give A186412. Records are A000027.

Cf. A000041, A046746, A135010, A138121, A182699, A182703, A182709, A183152, A186114, A187219, A193870, A194436-A194439, A194447, A194448, A196025, A198381.

Sequence in context: A281426 A070099 A126760 * A296085 A007740 A117811

Adjacent sequences:  A206434 A206435 A206436 * A206438 A206439 A206440

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Feb 14 2012

EXTENSIONS

Further edited by Omar E. Pol, Mar 31 2012, Jan 27 2013

STATUS

approved

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Last modified April 26 04:11 EDT 2019. Contains 322469 sequences. (Running on oeis4.)