

A141285


Largest part of the nth partition of j in the list of colexicographically ordered partitions of j, if 1 <= n <= A000041(j).


97



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
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OFFSET

1,2


COMMENTS

Also largest part of the nth 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 kth 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 threedimensional 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 nth region is also the largest part of the nth 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^(j1) regions. The largest part of the nth 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 nth 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 nth oddindexed line segment has a(n) upsteps and the nth evenindexed line segment has A194446(n) downsteps. Note that the height of the nth 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)


MATHEMATICA

Last/@DeleteCases[DeleteCases[Sort@PadRight[Reverse/@IntegerPartitions[13]], x_ /; x == 0, 2], {_}] (* updated Robert Price, May 15 2020 *)


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 * A342165 A286531 A351453
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



