A210990 -id:A210990 - cp's OEIS frontend

A182181 Total number of parts in the section model of partitions of A135010 with n regions.

1, 3, 6, 7, 12, 13, 20, 21, 23, 24, 35, 36, 38, 39, 54, 55, 57, 58, 62, 63, 64, 86, 87, 89, 90, 94, 95, 97, 98, 128, 129, 131, 132, 136, 137, 138, 145, 146, 148, 149, 150, 192, 193, 195, 196, 200, 201, 203, 204, 212, 213, 214, 217, 218, 219, 275

Offset: 1

Omar E. Pol, Apr 23 2012

			The first four regions of the section model of partitions are [1],[2, 1],[3, 1, 1],[2]. We can see that there are seven parts so a(4) = 7.
Written as a triangle begins:
    1;
    3;
    6;
    7,  12;
   13,  20;
   21,  23,  24,  35;
   36,  38,  39,  54;
   55,  57,  58,  62,  63,  64,  86;
   87,  89,  90,  94,  95,  97,  98, 128;
  129, 131, 132, 136, 137, 138, 145, 146, 148, 149, 150, 192;
  193, 195, 196, 200, 201, 203, 204, 212, 213, 214, 217, 218, 219, 275;
  ...
From _Omar E. Pol_, Oct 20 2014: (Start)
Illustration of initial terms:
.                                                _ _ _ _ _
.                                      _ _ _    |_ _ _    |
.                            _ _ _ _  |_ _ _|_  |_ _ _|_  |
.                    _ _    |_ _    | |_ _    | |_ _    | |
.            _ _ _  |_ _|_  |_ _|_  | |_ _|_  | |_ _|_  | |
.      _ _  |_ _  | |_ _  | |_ _  | | |_ _  | | |_ _  | | |
.  _  |_  | |_  | | |_  | | |_  | | | |_  | | | |_  | | | |
. |_| |_|_| |_|_|_| |_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_|_|
.
.  1    3      6       7        12        13         20
.
.                                          _ _ _ _ _ _
.                             _ _ _       |_ _ _      |
.                _ _ _ _     |_ _ _|_     |_ _ _|_    |
.   _ _         |_ _    |    |_ _    |    |_ _    |   |
.  |_ _|_ _ _   |_ _|_ _|_   |_ _|_ _|_   |_ _|_ _|_  |
.  |_ _ _    |  |_ _ _    |  |_ _ _    |  |_ _ _    | |
.  |_ _ _|_  |  |_ _ _|_  |  |_ _ _|_  |  |_ _ _|_  | |
.  |_ _    | |  |_ _    | |  |_ _    | |  |_ _    | | |
.  |_ _|_  | |  |_ _|_  | |  |_ _|_  | |  |_ _|_  | | |
.  |_ _  | | |  |_ _  | | |  |_ _  | | |  |_ _  | | | |
.  |_  | | | |  |_  | | | |  |_  | | | |  |_  | | | | |
.  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|_|
.
.       21           23           24            35
(End)

Robert Price, Table of n, a(n) for n = 1..204226, rows 1-50.
Omar E. Pol, Illustration of the seven regions of 5

Partial sums of A194446.
Row j has length A187219(j).
Right border gives A006128.
For the definition of "region" see A206437.
Cf. A000041, A135010, A138121, A141285, A182244, A182276, A182727, A186114, A210990.

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
reg = {}; l = {};
For[j = 1, j <= 56, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[reg, j - i];
  ];
Accumulate@reg  (* Robert Price, Apr 22 2020, revised Jul 25 2020 *)

a(A000041(n)) = A006128(n), n >= 1.

a(A000041(n)) = A182727(A000041(n)). - Omar E. Pol, May 24 2012

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).

A182244 Sum of all parts of the shell model of partitions of A135010 with n regions.

Original entry on oeis.org

1, 4, 9, 11, 20, 23, 35, 37, 43, 46, 66, 69, 76, 80, 105, 107, 113, 116, 129, 134, 138, 176, 179, 186, 190, 204, 207, 216, 221, 270, 272, 278, 281, 294, 299, 303, 326, 330, 340, 346, 351, 420, 423, 430, 434, 448, 451, 460, 465, 492, 497, 501, 516, 523, 529, 616

Offset: 1

Omar E. Pol, Apr 23 2012

			The first four regions of the shell model of partitions are [1],[2, 1],[3, 1, 1],[2], so a(4) = (1)+(2+1)+(3+1+1)+(2) = 11.
Written as a triangle begins:
1;
4;
9;
11,  20;
23,  35;
37,  43, 46, 66;
69,  76, 80,105;
107,113,116,129,134,138,176;
179,186,190,204,207,216,221,270;
272,278,281,294,299,303,326,330,340,346,351,420;
423,430,434,448,451,460,465,492,497,501,516,523,529,616;
...
From _Omar E. Pol_, Aug 08 2013: (Start)
Illustration of initial terms:
.                                                _ _ _ _ _
.                                      _ _ _    |_ _ _    |
.                            _ _ _ _  |_ _ _|_  |_ _ _|_  |
.                    _ _    |_ _    | |_ _    | |_ _    | |
.            _ _ _  |_ _|_  |_ _|_  | |_ _|_  | |_ _|_  | |
.      _ _  |_ _  | |_ _  | |_ _  | | |_ _  | | |_ _  | | |
.  _  |_  | |_  | | |_  | | |_  | | | |_  | | | |_  | | | |
. |_| |_|_| |_|_|_| |_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_|_|
.
.  1    4      9       11       20        23        35
.
.                                          _ _ _ _ _ _
.                             _ _ _       |_ _ _      |
.                _ _ _ _     |_ _ _|_     |_ _ _|_    |
.   _ _         |_ _    |    |_ _    |    |_ _    |   |
.  |_ _|_ _ _   |_ _|_ _|_   |_ _|_ _|_   |_ _|_ _|_  |
.  |_ _ _    |  |_ _ _    |  |_ _ _    |  |_ _ _    | |
.  |_ _ _|_  |  |_ _ _|_  |  |_ _ _|_  |  |_ _ _|_  | |
.  |_ _    | |  |_ _    | |  |_ _    | |  |_ _    | | |
.  |_ _|_  | |  |_ _|_  | |  |_ _|_  | |  |_ _|_  | | |
.  |_ _  | | |  |_ _  | | |  |_ _  | | |  |_ _  | | | |
.  |_  | | | |  |_  | | | |  |_  | | | |  |_  | | | | |
.  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|_|
.
.       37           43           46           66
(End)

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

Partial sums of A186412. Row j has length A187219(j). Right border gives A066186.

Cf. A000041, A135010, A138121, A141285, A182244, A182181, A182276, A186114, A206437, A210990.

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
A186412 = {}; l = {};
For[j = 1, j <= 56, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[A186412, Total@Take[Reverse[First /@ lex[mx]], j - i]];
  ];
Accumulate@A186412  (* Robert Price, Jul 25 2020 *)

a(A000041(k)) = A066186(k), k >= 1.

A210979 Total area of the shadows of the three views of the version "Tree" of the shell model of partitions with n shells.

Original entry on oeis.org

0, 3, 8, 15, 27, 42, 69, 102, 155, 225, 327, 458, 652, 894, 1232, 1669, 2257, 2999, 3996, 5242, 6877, 8928, 11564, 14845, 19045, 24223, 30756, 38815, 48877, 61195, 76496, 95124, 118067, 145930, 179991, 221160, 271268, 331538, 404463, 491948, 597253

Offset: 0

Omar E. Pol, Apr 28 2012

nonn

The physical model shows each part of a partition as an object, for example; a cube of side 1 which is labeled with the size of the part. Note that on the branches of the tree each column contains parts of the same size, as a periodic structure. For the large version of this model see A210980.

			For n = 7 the three views of the shell model of partitions version "tree" with seven shells looks like this:
.
.         A194805(7) = 25        A006128(7) = 54
.
.                        7       7
.                      4         4 3
.                    5           5 2
.                  3             3 2 2
.        6       1               6 1
.          3     1               3 3 1
.            4   1               4 2 1
.              2 1               2 2 2 1
.                1   5           5 1 1
.                1 3             3 2 1 1
.            4   1               4 1 1 1
.              2 1               2 2 1 1 1
.                1 3             3 1 1 1 1
.              2 1               2 1 1 1 1 1
.                1               1 1 1 1 1 1 1
-------------------------------------------------
.
.        6 3 4 2 1 3 5 4 7
.          3 2 2 1 2 2 3
.              2 1 2
.                1
.                1
.                1
.                1
.
.         A194803(7) = 23
.
The areas of the shadows of the three views are A006128(7) = 54, A194803(7) = 23 and A194805(7) = 25, therefore the total area of the three shadows is 54+23+25 = 102, so a(7) = 102.

Other versions: A207380, A210970, A210980, A210990, A210991.

Cf. A006128, A135010, A138121, A182703, A194803, A194805.

a(n) = A006128(n) + A194803(n) + A194805(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).

A182727 Sum of largest parts of the shell model of partitions with n regions.

Original entry on oeis.org

1, 3, 6, 8, 12, 15, 20, 22, 26, 29, 35, 38, 43, 47, 54, 56, 60, 63, 69, 74, 78, 86, 89, 94, 98, 105, 108, 114, 119, 128, 130, 134, 137, 143, 148, 152, 160, 164, 171, 177, 182, 192, 195, 200, 204, 211, 214, 220, 225, 234, 239, 243, 251, 258, 264, 275, 277, 281

Offset: 1

Omar E. Pol, Jan 25 2011

Question: Is there some connection with fractals?

			For n = 6 the largest parts of the first six regions of the shell model of partitions are 1, 2, 3, 2, 4, 3, so a(6) = 1+2+3+2+4+3 = 15.
Written as a triangle begins:
1;
3;
6;
8,   12;
15,  20;
22,  26, 29, 35;
38,  43, 47, 54;
56,  60, 63, 69, 74, 78, 86;
89,  94, 98,105,108,114,119,128;
130,134,137,143,148,152,160,164,171,177,182,192;
195,200,204,211,214,220,225,234,239,243,251,258,264,275;

Alois P. Heinz, Table of n, a(n) for n = 1..4000
Omar E. Pol, Illustration of the seven regions of 5

Partial sums of A141285. Row j has length A187219(j). Right border gives A006128.

Cf. A135010, A141285, A182728, A182729, A186114, A206437, A182181, A182244, A210990.

a(A000041(n)) = A182181(A000041(n)) = A006128(n). - Omar E. Pol, May 24 2012

New name from Omar E. Pol, Apr 26 2012

cp's OEIS Frontend

A182181 Total number of parts in the section model of partitions of A135010 with n regions.

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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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A182244 Sum of all parts of the shell model of partitions of A135010 with n regions.

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A210979 Total area of the shadows of the three views of the version "Tree" of the shell model of partitions with n shells.

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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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A182727 Sum of largest parts of the shell model of partitions with n regions.

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