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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). 99
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 Links 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 section of the set of partitions 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

Robert Price, Table of n, a(n) for n = 1..321, first 75 regions.

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

Robert Price, Mathematica program to draw diagram up to n=28

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 section of the set of partitions 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];

MATHEMATICA

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0, 2];

reg = {}; l = {};

For[j = 1, j <= 22, j++,

  mx = Max@lex[j][[j]]; AppendTo[l, mx];

  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];

  AppendTo[reg, Take[Reverse[First /@ lex[mx]], j - i]];

  ];

Flatten@reg  (* Robert Price, Apr 21 2020, revised Jul 24 2020 *)

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, 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,look

AUTHOR

Omar E. Pol, Feb 14 2012

EXTENSIONS

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

Minor edits by Omar E. Pol, Apr 23 2020

STATUS

approved

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Last modified November 28 05:30 EST 2021. Contains 349401 sequences. (Running on oeis4.)