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A141285 Largest part of the n-th partition of j in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j). 90
1, 2, 3, 2, 4, 3, 5, 2, 4, 3, 6, 3, 5, 4, 7, 2, 4, 3, 6, 5, 4, 8, 3, 5, 4, 7, 3, 6, 5, 9, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 6, 5, 9, 4, 8, 7, 6, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also largest part of the n-th region of the set of partitions of j, if 1<=n<=A000041(j). For the definition of "region of the set of partitions of j" see A206437.

Also triangle read by rows: T(j,k) is the largest part of the k-th region in the last section of the set of partitions of j.

For row n >= 2 the rows of triangle are also the branches of a tree which is a projection of a three-dimensional structure of the section model of partitions of A135010, version tree. The branches of even rows give A182730. The branches of odd rows give A182731. Note that each column contains parts of the same size. It appears that the structure of A135010 is a periodic table of integer partitions. See also A210979 and A210980.

Also column 1 of: A193870, A206437, A210941, A210942, A210943. - Omar E. Pol, Sep 01 2013

Also row lengths of A211009. - Omar E. Pol, Feb 06 2014

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..4000

Omar E. Pol, Illustration of initial terms

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

FORMULA

a(n) = A001511(A228354(n)). - Omar E. Pol, Aug 22 2013

EXAMPLE

Written as a triangle T(j,k) the sequence begins:

1;

2;

3;

2, 4;

3, 5;

2, 4, 3, 6;

3, 5, 4, 7;

2, 4, 3, 6, 5, 4, 8;

3, 5, 4, 7, 3, 6, 5, 9;

2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10;

3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11;

...

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

n  A000041                a(n)

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

1  = p(1)                   1

2  = p(2)                 2 .

3  = p(3)                   . 3

4                         2 .

5  = p(4)               4   .

6                           . 3

7  = p(5)                   .   5

8                         2 .

9                       4   .

10                    3     .

11 = p(6)           6       .

12                          . 3

13                          .   5

14                          .     4

15 = p(7)                   .       7

...

From Omar E. Pol, Aug 22 2013: (Start)

Illustration of initial terms (n = 1..11) in three ways: as the largest parts of the partitions of 6 (see A026792), also as the largest parts of the regions of the diagram, also as the diagonal of triangle. By definition of "region" the largest part of the n-th region is also the largest part of the n-th partition

(see below):

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

.                  Diagram         Triangle in which

Partitions       of regions       rows are partitions

of 6           and partitions   and columns are regions

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

.                _ _ _ _ _ _

6                _ _ _      |                         6

3+3              _ _ _|_    |                       3 3

4+2              _ _    |   |                     4   2

2+2+2            _ _|_ _|_  |                   2 2   2

5+1              _ _ _    | |                 5       1

3+2+1            _ _ _|_  | |               3 1       1

4+1+1            _ _    | | |             4   1       1

2+2+1+1          _ _|_  | | |           2 2   1       1

3+1+1+1          _ _  | | | |         3   1   1       1

2+1+1+1+1        _  | | | | |       2 1   1   1       1

1+1+1+1+1+1       | | | | | |     1 1 1   1   1       1

...

The equivalent sequence for compositions is A001511. Explanation: for the positive integer j the diagram of regions of the set of compositions of j has 2^(j-1) regions. The largest part of the n-th region is A001511(n). The number of parts is A006519(n). On the other hand the diagram of regions of the set of partitions of j has A000041(j) regions. The largest part of the n-th region is a(n) = A001511(A228354(n)). The number of parts is A194446(n). Both diagrams have j sections. The diagram for partitions can be interpreted as one of the three views of a three dimensional diagram of compositions in which the rows of partitions are in orthogonal direction to the rest. For the first five sections of the diagrams see below:

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

.          Diagram                           Diagram

.         of regions                        of regions

.      and compositions                   and partitions

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

.      j = 1 2 3 4 5                     j = 1 2 3 4 5

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

n  A001511                    A228354  a(n)

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

1    1     _| | | | | ............ 1    1    _| | | | |

2    2     _ _| | | | ............ 2    2    _ _| | | |

3    1     _|   | | |    ......... 4    3    _ _ _| | |

4    3     _ _ _| | | ../  ....... 6    2    _ _|   | |

5    1     _| |   | |    / ....... 8    4    _ _ _ _| |

6    2     _ _|   | | ../ /   .... 12   3    _ _ _|   |

7    1     _|     | |    /   /   . 16   5    _ _ _ _ _|

8    4     _ _ _ _| | ../   /   /

9    1     _| | |   |      /   /

10   2     _ _| |   |     /   /

11   1     _|   |   |    /   /

12   3     _ _ _|   | ../   /

13   1     _| |     |      /

14   2     _ _|     |     /

15   1     _|       |    /

16   5     _ _ _ _ _| ../

...

Also we can draw an infinite Dyck path in which the n-th odd-indexed line segment has a(n) up-steps and the n-th even-indexed line segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two successive valleys at height 0 is also the partition number A000041(n). See below:

.                                 5

.                                 /\                 3

.                   4            /  \           4    /\

.                   /\          /    \          /\  /

.         3        /  \     3  /      \        /  \/

.    2    /\   2  /    \    /\/        \   2  /

. 1  /\  /  \  /\/      \  /            \  /\/

. /\/  \/    \/          \/              \/

.

.(End)

CROSSREFS

Where records occur give A000041, n>=1. Column 1 is A158478. Row j has length A187219(j). Row sums give A138137. Right border gives A000027.

Cf. A000041, A135010, A182730, A182731, A182732, A182733, A182982, A182983, A182703, A193870, A194446, A194447, A194550, A206437, A210979, A210980, A211978, A220517, A225600, A225610.

Sequence in context: A257907 A139094 A159081 * A286531 A157893 A264116

Adjacent sequences:  A141282 A141283 A141284 * A141286 A141287 A141288

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Aug 01 2008

EXTENSIONS

Edited by Omar E. Pol, Nov 28 2010

Better definition and edited by Omar E. Pol, Oct 17 2013

STATUS

approved

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Last modified September 20 15:44 EDT 2017. Contains 292275 sequences.