A194550 -id:A194550 - cp's OEIS frontend

A141285 Largest part of the n-th partition of j in the list of colexicographically ordered partitions of j, if 1 <= n <= A000041(j).

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

Offset: 1

Omar E. Pol, Aug 01 2008

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

			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)

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

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.

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

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

Edited by Omar E. Pol, Nov 28 2010

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

A194805 Number of parts that are visible in one of the three views of the section model of partitions version "tree" with n sections.

Original entry on oeis.org

0, 1, 2, 4, 7, 11, 17, 25, 36, 51, 71, 97, 132, 177, 235, 310, 406, 527, 681, 874, 1116, 1418, 1793, 2256, 2829, 3532, 4393, 5445, 6727, 8282, 10168, 12445, 15190, 18491, 22452, 27192, 32859, 39613, 47651, 57199, 68522, 81920, 97756, 116434, 138435

Offset: 0

Omar E. Pol, Jan 27 2012

nonn

The mentioned view of the section model looks like a tree (see example). Note that every column contains the same parts. For more information about the section model of partitions see A135010 and A194803.

Number of partitions of 2n-1 such that n-1 or n is a part, for n >=1. - Clark Kimberling, Mar 01 2014

			Illustration of one of the three views with seven sections:
.
.                   1
.                 2 1
.                   1 3
.                 2 1
.               4   1
.                   1 3
.                   1   5
.                 2 1
.               4   1
.             3     1
.           6       1
.                     3
.                       5
.                         4
.                           7
.
There are 25 parts that are visible, so a(7) = 25.
Using the formula we have a(7) = p(7) + p(7-1) - 1 = 15 + 11 - 1 = 25, where p(n) is the number of partitions of n.

Robert Price, Table of n, a(n) for n = 0..5000

Cf. A000041, A000065, A084376, A135010, A138121, A141285, A194550, A194803, A194804.

Table[Count[IntegerPartitions[2 n - 1],  p_ /; Or[MemberQ[p, n - 1], MemberQ[p, n]]], {n, 50}]  (* Clark Kimberling, Mar 01 2014 *)
Table[PartitionsP[n] + PartitionsP[n-1] - 1, {n, 0, 44}] (* Robert Price, May 12 2020 *)

a(n) = A084376(n) - 1.
a(n) = A000041(n) + A000041(n-1) - 1, if n >= 1.
a(n) = A000041(n) + A000065(n-1), if n >= 1.

A194551 a(n) is the n-th largest part that are visible in one of the three views of the version "Tree" of the section model of partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 6, 4, 7, 5, 4, 8, 3, 6, 5, 9, 4, 7, 6, 5, 10, 5, 4, 8, 7, 6, 11, 3, 6, 5, 9, 4, 8, 7, 6, 12, 4, 7, 6, 5, 10, 5, 9, 8, 7, 13, 5, 4, 8, 7, 6, 11, 6, 5, 10, 9, 8, 7, 14, 3, 6, 5, 9, 4, 8, 7, 6, 12, 7, 6, 11, 5, 10, 9, 8, 15

Offset: 1

Omar E. Pol, Nov 22 2011

nonn

It appears that if this is written as a triangle (see example) and n >= 3 then row n has the following property:
If n is congruent to 0 (mod 3) then row n converge to the sequence 3,6,5,9,4,8,7,6,12... in which the records are the numbers >= 3 that are congruent to 0 (mod 3).
If n is congruent to 1 (mod 3) then row n converge to the sequence 4,7,6,5,10,5,9,8,7,13... in which the records are the numbers >= 4 that are congruent to 1 (mod 3).
If n is congruent to 2 (mod 3) then row n converge to the sequence 5,4,8,7,6,11,6,5,10,9,8,7,14... in which the records are the numbers >= 5 that are congruent to 2 (mod 3).
For more information see A135010.

			Written as a triangle begins:
1;
2;
3;
4;
5;
3,6;
4,7;
5,4,8;
3,6,5,9;
4,7,6,5,10;
5,4,8,7,6,11;
3,6,5,9,4,8,7,6,12;
4,7,6,5,10,5,9,8,7,13;
5,4,8,7,6,11,6,5,10,9,8,7,14;
...
Row n has length A008483(n), if n >= 3.

Robert Price, Table of n, a(n) for n = 1..56954, 50 rows.

Cf. A008483, A135010, A138121, A141285, A182730, A182731, A182732, A182733, A194550.

Join[{1},Table[Drop[l = Last/@DeleteCases[Sort@PadRight[Reverse /@ Cases[IntegerPartitions[n], x_ /; Last[x] != 1]], x_ /; x == 0, 2], First@FirstPosition[l, n - 2, {0}]], {n, 2, 15}]] // Flatten  (* Robert Price, May 15 2020 *)

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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).

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A194805 Number of parts that are visible in one of the three views of the section model of partitions version "tree" with n sections.

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A194551 a(n) is the n-th largest part that are visible in one of the three views of the version "Tree" of the section model of partitions.

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