A210979 -id:A210979 - 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

A207380 Total area of the shadows of the three views of a three-dimensional version of the shell model of partitions with n shells.

Original entry on oeis.org

0, 3, 10, 21, 42, 70, 122, 187, 298, 443, 667, 957, 1401, 1960, 2775, 3828, 5295, 7167, 9745, 12998, 17380, 22915, 30196, 39347, 51274, 66126, 85209, 108942, 139055, 176273, 223148, 280733, 352623, 440646, 549597, 682411, 845852, 1044084, 1286512, 1579582

Offset: 0

Omar E. Pol, Feb 17 2012

nonn

In this model each part of a partition can be represented by a cuboid of size 1 x 1 x L, where L is the size of the part. One of the views is a rectangle formed by ones whose area is n*A000041(n) = A066186(n). Each element of the first view is equal to the volume of a horizontal column parallel to the axis x. The second view is the n-th slice illustrated in A026792 which has A000041(n) levels and its area is A006128(n) equals the total number of parts of all partitions of n and equals the sum of largest parts of all partitions of n. Each zone contains a partition of n. Each element of the second view is equal to the volume of a horizontal column parallel to the axis y. The third view is a triangle because it is also the n-th slice of the tetrahedron of A209655. The area of triangle is A000217(n). Each element of the third view is equal to the volume of a vertical column parallel to the axis z. The sum of elements of each view is A066186(n) equals the area of the first view. For more information about the shell model of partitions see A135010 and A182703.

			For n = 5 the three views of the three-dimensional shell model of partitions with 5 shells look like this:
.
.   A066186(5) = 35     A006128(5) = 20
.
.         1 1 1 1 1     5
.         1 1 1 1 1     3 2
.         1 1 1 1 1     4 1
.         1 1 1 1 1     2 2 1
.         1 1 1 1 1     3 1 1
.         1 1 1 1 1     2 1 1 1
.         1 1 1 1 1     1 1 1 1 1
.
.
.         7 6 4 2 1
.           1 2 3 2
.             1 1 2
.               1 1
.                 1
.
.   A000217(5) = 15
.
The areas of the shadows of the three views are A066186(5) = 35, A006128(5) = 20 and A000217(5) = 15, therefore the total area of the three shadows is 35+20+15 = 70, so a(5) = 70.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Other versions: A210970, A210979, A210980, A210990, A210991.

Cf. A000041, A000217, A006128, A026792, A066186, A135010, A138121, A141285, A182703, A182715, A206437, A209655.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+g[1]]
      fi
    end:
a:= n-> n*b(n, n)[1] +b(n, n)[2] +n*(n+1)/2:
seq (a(n), n=0..50);  # Alois P. Heinz, Mar 22 2012

b[n_, i_] := b[n, i] = Module[{f, g}, If [n == 0 || i == 1, {1, n}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; Join[f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]] ]]]; a[n_] := n*b[n, n][[1]] + b[n, n][[2]] + n*(n+1)/2; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jun 18 2015, after Alois P. Heinz *)

a(n) = n * A000041(n) + A000217(n) + A006128(n) = A066186(n) + A000217(n) + A006128(n).

More terms from Alois P. Heinz, Mar 22 2012

A210970 Total area of the shadows of the three views of a three-dimensional version of the shell model of partitions with n shells.

Original entry on oeis.org

0, 3, 9, 18, 34, 55, 91, 136, 208, 301, 439, 616, 876, 1203, 1665, 2256, 3062, 4083, 5459, 7186, 9470, 12335, 16051, 20688, 26648, 34027, 43395, 54966, 69496, 87341, 109591, 136766, 170382, 211293, 261519, 322382, 396694, 486327, 595143, 725954, 883912

Offset: 0

Omar E. Pol, Apr 22 2012

nonn

For more information see A135010 and A182703.

			For n = 6 the illustration of the three views of a three-dimensional version of the shell model of partitions with 6 shells looks like this:
.
.   A006128(6) = 35     A006128(6) = 35
.
.                 6     6
.               3 3     3 3
.               4 2     4 2
.             2 2 2     2 2 2
.               5 1     5 1
.             3 2 1     3 2 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
.
.
.       1 2 5 9 12 6  \
.         1 1 3 5 6    \
.           1 1 2 4     \ 6th slice of
.             1 1 2     / tetrahedron A210961
.               1 1    /
.                 1   /
.
.      A000217(6) = 21
.
The areas of the shadows of the three views are A006128(6) = 35, A006128(6) = 35 and A000217(6) = 21, therefore the total area of the three shadows is 35+35+21 = 91, so a(6) = 91.

Other versions: A207380, A210979, A210980, A210990, A210991.

Cf. A000217, A006128, A066186, A135010, A138121, A182703, A206437, A210952, A210961.

a(n) = 2*A006128(n) + A000217(n).

A210990 Total area of the shadows of the three views of the shell model of partitions with n regions.

Original entry on oeis.org

0, 3, 10, 21, 26, 44, 51, 75, 80, 92, 99, 136, 143, 157, 166, 213, 218, 230, 237, 260, 271, 280, 348, 355, 369, 378, 403, 410, 427, 438, 526, 531, 543, 550, 573, 584, 593, 631, 640, 659, 672, 683, 804, 811, 825, 834, 859, 866, 883, 894, 938, 949, 958

Offset: 0

Omar E. Pol, Apr 23 2012

nonn

Each part is represented by a cuboid of sides 1 X 1 X k where k is the size of the part. For the definition of "regions of n" see A206437.

			For n = 11 the three views of the shell model of partitions with 11 regions look like this:
.
.   A182181(11) = 35            A182244(11) = 66
.
.   6                             * * * * * 6
.   3 3                      P    * * 3 * * 3
.   2   4                    a    * * * 4 * 2
.   2   2 2                  r    * 2 * 2 * 2
.   1       5                t    * * * * 5 1
.   1       2 3              i    * * 3 * 2 1
.   1       1   4            t    * * * 4 1 1
.   1       1   2 2          i    * 2 * 2 1 1
.   1       1   1   3        o    * * 3 1 1 1
.   1       1   1   1 2      n    * 2 1 1 1 1
.   1       1   1   1 1 1    s    1 1 1 1 1 1
. <------- Regions ------         ------------> N
.                            L
.                            a    1
.                            r    * 2
.                            g    * * 3
.                            e    * 2
.                            s    * * * 4
.                            t    * * 3
.                                 * * * * 5
.                            p    * 2
.                            a    * * * 4
.                            r    * * 3
.                            t    * * * * * 6
.                            s
.                               A182727(11) = 35
.
The areas of the shadows of the three views are A182244(11) = 66, A182181(11) = 35 and A182727(11) = 35, therefore the total area of the three shadows is 66+35+35 = 136, so a(11) = 136.

Other versions: A207380, A210970, A210979, A210980, A210991.

Cf. A135010, A138121, A141285, A194446, A182181, A182727, A186114, A206437.

a(n) = A182244(n) + A182727(n) + A182181(n), n >= 1.

a(A000041(n)) = 2*A006128(n) + A066186(n).

A210980 Total area of the shadows of the three views of the shell model of partitions, version "Tree", with n shells.

Original entry on oeis.org

0, 3, 10, 21, 42, 69, 123, 189, 304, 458, 693, 998, 1474, 2067, 2927, 4056, 5613, 7595, 10335, 13782, 18411, 24276, 31944, 41583, 54152, 69762, 89758, 114668, 146181, 185083, 234051, 294126, 368992, 460669, 573906, 711865, 881506, 1087023, 1338043

Offset: 0

Omar E. Pol, Apr 21 2012

nonn

Each part is represented by a cuboid 1 X 1 X L where L is the size of the part.

			For n = 7 the shadows of the three views of the shell model of partitions version "tree" with seven shells looks like this:
.                                        |  Partitions
.    A194805(7) = 25    A066186(7) = 105 |  of 7
.                                        |
.                   1    * * * * * * 1   |  7
.                 2      * * * 1 * * 2   |  4+3
.               2        * * * * 1 * 2   |  5+2
.             3          * * 1 * 2 * 3   |  3+2+2
.   1       2            * * * * * 1 2   |  6+1
.     2     3            * * 1 * * 2 3   |  3+3+1
.       2   3            * * * 1 * 2 3   |  4+2+1
.         3 4            * 1 * 2 * 3 4   |  2+2+2+1
.           3   1        * * * * 1 2 3   |  5+1+1
.           4 2          * * 1 * 2 3 4   |  3+2+1+1
.       1   4            * * * 1 2 3 4   |  4+1+1+1
.         2 5            * 1 * 2 3 4 5   |  2+2+1+1+1
.           5 1          * * 1 2 3 4 5   |  3+1+1+1+1
.         1 6            * 1 2 3 4 5 6   |  2+1+1+1+1+1
.           7            1 2 3 4 5 6 7   |  1+1+1+1+1+1+1
.   ----------------------------------   |
.                                        |
.   * * * * 1 * * * *                    |
.   * * * 1 2 * * * *                    |
.   * 1 * * 2 1 * * *                    |
.   * * 1 2 2 * * 1 *                    |
.   * * * * 2 2 1 * *                    |
.   1 2 2 3 2 * * * *                    |
.           2 3 2 2 1                    |
.                                        |
.    A194804(7) = 59                     |
.
Note that, as a variant, in this case each part is labeled with its position in the partition.
The areas of the shadows of the three views are A066186(7) = 105, A194804(7) = 59 and A194805(7) = 25, therefore the total area of the three shadows is 105+59+25 = 189, so a(7) = 189.

Other versions: A207380, A210970, A210979, A210990, A210991.

Cf. A000041, A066186, A135010, A138121, A141285, A182703, A194804, A194805, A206437.

a(n) = A066186(n) + A194804(n) + A194805(n), n >= 1.

A210991 Total area of the shadows of the three views of the shell model of partitions with n regions.

Original entry on oeis.org

0, 3, 9, 18, 21, 35, 39, 58, 61, 67, 71, 99, 103, 110, 115, 152, 155, 161, 165, 175, 181, 186, 238, 242, 249, 254, 265, 269, 277, 283, 352, 355, 361, 365, 375, 381, 386, 401, 406, 415, 422, 428, 522, 526, 533, 538, 549, 553, 561, 567, 584, 590, 595, 606

Offset: 0

Omar E. Pol, Apr 30 2012

nonn

It appears that if n is a partition number A000041 then the rotated structure with n regions shows each row as a partition of k such that A000041(k) = n (see example).

For the definition of "regions of n" see A206437.

			For n = 11 the three views of the shell model of partitions with 11 regions look like this:
.
.     A182181(11) = 35           A210692(11) = 29
.
.   1                                       1
.   1                                       1
.   1                                       1
.   1                                       1
.   1       1                             1 1
.   1       1                             1 1
.   1       1   1                       1 1 1
.   2       1   1                       1 1 2
.   2       1   1   1                 1 1 1 2
.   3   2   2   2   1 1             1 1 2 2 3
.   6 3 4 2 5 3 4 2 3 2 1         1 2 3 4 5 6
. <------- Regions ------         ------------> N
.                            L
.                            a    1
.                            r    * 2
.                            g    * * 3
.                            e    * 2
.                            s    * * * 4
.                            t    * * 3
.                                 * * * * 5
.                            p    * 2
.                            a    * * * 4
.                            r    * * 3
.                            t    * * * * * 6
.                            s
.
.                                A182727(11) = 35
.
The areas of the shadows of the three views are A182181(11) = 35, A182727(11) = 35 and A210692(11) = 29, therefore the total area of the three shadows is 35+35+29 = 99, so a(11) = 99.
Since n = 11 is a partition number A000041 we can see that the rotated structure with 11 regions shows each row as a partition of 6 because A000041(6) = 11. See below:
.
.                      6
.                    3   3
.                  4       2
.                2   2       2
.              5               1
.            3   2               1
.          4       1               1
.        2   2       1               1
.      3       1       1               1
.    2   1       1       1               1
.  1   1   1       1       1               1
.

Other versions: A207380, A210970, A210979, A210980, A210990.

Cf. A000041, A026905, A135010, A138121, A141285, A182703, A194446, A182181, A182727, A186114, A206437, A210692.

a(n) = A182181(n) + A182727(n) + A210692(n).

a(A000041(n)) = 2*A006128(n) + A026905(n).

A194803 Number of parts that are visible in one of the three views of the shell model of partitions version "Tree" with n shells.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 17, 23, 33, 46, 64, 86, 121, 161, 217, 291, 388, 507, 671, 870, 1131, 1458, 1872, 2383, 3042, 3840, 4841, 6076, 7605, 9460, 11765, 14544, 17950, 22073, 27077, 33092, 40395, 49113, 59611, 72162, 87185, 105035, 126366

Offset: 0

Omar E. Pol, Jan 27 2012

nonn

The physical model shows each part represented by an object, for example using a cube or a cuboid. In this case the small version of the model shows each part as a cube of side 1 which is labeled with the size of the part. On the same way the large version of the model shows each part as a cuboid of sides 1 x 1 x L where L is the size of the part. The cuboid is labeled with the level of the part. For the sum of parts see A194804. For more information about the shell model see A135010 and A194805.

			Illustration of one of the three views with seven shells:
1) Small version:
.
Level
1        A182732 <- 6 3 4 2 1 3 5 4 7 -> A182733
2                     3 2 2 1 2 2 3
3                         2 1 2
4                           1
5      Table 2.0            1            Table 2.1
6                           1
7                           1
.
.  A182742  A182982                   A182743  A182983
.  A182992  A182994                   A182993  A182995
.
2) Large version:
.
.                   . . . . 1 . . . .
.                   . . . 1 2 . . . .
.                   . 1 . . 2 1 . . .
.                   . . 1 2 2 . . 1 .
.                   . . . . 2 2 1 . .
.                   1 2 2 3 2 . . . .
.                           2 3 2 2 1
.
The large version shows the parts labeled with the level of the part where "the level of a part" is its position in the partition. In both versions there are 23 parts that are visible, so a(7) = 23. Also using the formula we have a(7) = 7+8+8 = 23.

Cf. A006128, A096541, A138135, A135010, A138121, A141285, A182732, A182733, A182742, A182743, A182982, A182983, A182992-A182995, A194804, A194805, A210979.

a(n) = n + A138135(n-1) + A138135(n), if n >= 2.

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).

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A207380 Total area of the shadows of the three views of a three-dimensional version of the shell model of partitions with n shells.

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A210970 Total area of the shadows of the three views of a three-dimensional version of the shell model of partitions with n shells.

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A210990 Total area of the shadows of the three views of the shell model of partitions with n regions.

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A210980 Total area of the shadows of the three views of the shell model of partitions, version "Tree", with n shells.

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A210991 Total area of the shadows of the three views of the shell model of partitions with n regions.

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A194803 Number of parts that are visible in one of the three views of the shell model of partitions version "Tree" with n shells.

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