cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A186412 Sum of all parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 3, 7, 4, 25, 2, 6, 3, 13, 5, 4, 38, 3, 7, 4, 14, 3, 9, 5, 49, 2, 6, 3, 13, 5, 4, 23, 4, 10, 6, 5, 69, 3, 7, 4, 14, 3, 9, 5, 27, 5, 4, 15, 7, 6, 87, 2, 6, 3, 13, 5, 4, 23, 4, 10, 6, 5, 39, 3, 9, 5, 19, 4, 12, 7, 6, 123
Offset: 1

Views

Author

Omar E. Pol, Aug 12 2011

Keywords

Comments

Also triangle read by rows: T(j,k) = sum of all parts in the k-th region of the last section of the set of partitions of j. See Example section. For more information see A135010. - Omar E. Pol, Nov 26 2011
For the definition of "region" see A206437. - Omar E. Pol, Aug 19 2013

Examples

			Contribution from Omar E. Pol, Nov 26 2011 (Start):
Written as a triangle:
1;
3;
5;
2,9;
3,12;
2,6,3,20;
3,7,4,25;
2,6,3,13,5,4,38;
3,7,4,14,3,9,5,49;
2,6,3,13,5,4,23,4,10,6,5,69;
3,7,4,14,3,9,5,27,5,4,15,7,6,87;
2,6,3,13,5,4,23,4,10,6,5,39,3,9,5,19,4,12,7,6,123;
(End)
From _Omar E. Pol_, Aug 18 2013: (Start)
Illustration of initial terms (first seven regions):
.                                             _ _ _ _ _
.                                     _ _ _  |_ _ _ _ _|
.                           _ _ _ _  |_ _ _|       |_ _|
.                     _ _  |_ _ _ _|                 |_|
.             _ _ _  |_ _|     |_ _|                 |_|
.       _ _  |_ _ _|             |_|                 |_|
.   _  |_ _|     |_|             |_|                 |_|
.  |_|   |_|     |_|             |_|                 |_|
.
.   1     3       5     2         9       3          12
.
(End)

Crossrefs

Row sums of triangle A186114 and of triangle A193870.
Row j has length A187219(j).
Row sums give A138879.
Right border gives A046746, j >= 1.
Records give A046746, j >= 1.
Partial sums give A182244.
Cf. A000041, A002865, A066186, A135010, A138121, A138879, A194436, A194437, A194438, A194439, A194446, A194447.

Programs

  • Mathematica
    lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
A186412 = {}; l = {};
For[j = 1, j <= 50, 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]];
  ];
A186412  (* Robert Price, Jul 25 2020 *)

Formula

a(A000041(n)) = A046746(n).

A211009 Triangle read by rows: T(n,k) = number of cells in the k-column of the n-th region of j in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j), 1<=k<=A141285(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 1, 2, 7, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 4, 11, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 15, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 7, 22
Offset: 1

Views

Author

Omar E. Pol, Oct 21 2012

Keywords

Comments

Also the finite sequence a(1)..a(r), where a(r) is a record in the sequence, is also a finite triangle read by rows: T(n,k) = number of cells in the k-column of the n-th region of the integer whose number of partitions is equal to a(r).
T(n,k) is also 1 plus the number of holes between T(n,k) and the previous member in the column k of triangle.
T(n,k) is also the height of the column mentioned in the definition, in a three-dimensional model of the set of partitions of j, in which the regions appear rotated 90 degrees and where the pivots are the largest part of every region (see A141285). For the definition of "region" see A206437. - Omar E. Pol, Feb 06 2014

Examples

			The irregular triangle begins:
1;
1, 2;
1, 1, 3;
1, 1;
1, 1, 2, 5;
1, 1, 1;
1, 1, 1, 2, 7;
1, 1;
1, 1, 2, 2;
1, 1, 1;
1, 1, 1, 2, 4, 11;
1, 1, 1;
1, 1, 1, 2, 2;
1, 1, 1, 1;
1, 1, 1, 1, 2, 4, 15;
1, 1;
1, 1, 2, 2;
1, 1, 1;
1, 1, 1, 2, 4, 4;
1, 1, 1, 1, 1;
1, 1, 1, 1;
1, 1, 1, 1, 2, 3, 7, 22;
...
From _Omar E. Pol_, Feb 06 2014: (Start)
Illustration of initial terms:
.    _
.   |_|
.    1
.      _
.    _|_|
.   |_ _|
.    1 2
.        _
.       |_|
.    _ _|_|
.   |_ _ _|
.    1 1 3
.    _ _
.   |_ _|
.    1 1
.          _
.         |_|
.         |_|
.        _|_|
.    _ _|_ _|
.   |_ _ _ _|
.    1 1 2 5
.
(End)

Links

Crossrefs

Records give positive terms of A000041. Row n has length A141285(n). Row sums give A186412.
Cf. A040051, A135010, A182244, A182703, A186114, A187219, A193870, A194446, A206437.

Extensions

Better definition from Omar E. Pol, Feb 06 2014

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

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Apr 23 2012

Keywords

Examples

			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)

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Formula

a(A000041(n)) = A006128(n), n >= 1.
a(A000041(n)) = A182727(A000041(n)). - Omar E. Pol, May 24 2012

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

Views

Author

Omar E. Pol, Apr 23 2012

Keywords

Comments

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.

Examples

			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.

Crossrefs

Other versions: A207380, A210970, A210979, A210980, A210991.
Cf. A135010, A138121, A141285, A194446, A182181, A182727, A186114, A206437.

Formula

a(n) = A182244(n) + A182727(n) + A182181(n), n >= 1.
a(A000041(n)) = 2*A006128(n) + A066186(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

Views

Author

Omar E. Pol, Jan 25 2011

Keywords

Comments

Question: Is there some connection with fractals?

Examples

			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;

Links

Crossrefs

Partial sums of A141285. Row j has length A187219(j). Right border gives A006128.
Cf. A135010, A141285, A182728, A182729, A186114, A206437, A182181, A182244, A210990.

Formula

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

Extensions

New name from Omar E. Pol, Apr 26 2012

A182276 Sum of all parts minus the total number of parts of the shell model of partitions with n regions.

Original entry on oeis.org

0, 1, 3, 4, 8, 10, 15, 16, 20, 22, 31, 33, 38, 41, 51, 52, 56, 58, 67, 71, 74, 90, 92, 97, 100, 110, 112, 119, 123, 142, 143, 147, 149, 158, 162, 165, 181, 184, 192, 197, 201, 228, 230, 235, 238, 248, 250, 257, 261, 280, 284, 287, 299, 305, 310, 341
Offset: 1

Views

Author

Omar E. Pol, Apr 23 2012

Keywords

Comments

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

Examples

			Written has a triangle:
0,
1,
3,
4,    8;
10,  15;
16,  20, 22, 31;
33,  38, 41, 51;
52,  56, 58, 67, 71, 74, 90;
92,  97,100,110,112,119,123,142;
143,147,149,158,162,165,181,184,192,197,201,228;
230,235,238,248,250,257,261,280,284,287,299,305,310,341;

Crossrefs

Row j has length A187219(j). Right border gives A196087.
Cf. A000041, A066186, A006128, A135010, A138121, A182181, A182244, A186114, A206437, A207035.

Programs

  • Mathematica
    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[]]];
  t = Take[Reverse[First /@ lex[mx]], j - i];
  AppendTo[reg, Total@t - Length@t]
  ];
Accumulate@reg  (* Robert Price, Jul 25 2020 *)

Formula

a(n) = A182244(n) - A182181(n).
a(A000041(n)) = A196087(n).

A211004 Number of distinct regions in the set of partitions of n.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51
Offset: 1

Views

Author

Omar E. Pol, Oct 22 2012

Keywords

Comments

The number of regions in the set of partitions of n equals the number of partitions of n. The sequence counts only the distinct regions. For the definition of "regions of the set of partitions of n" (or more simply "regions of n") see A206437.
Is this the same as A001840 for all positive integers? If not, where is the first place these sequences differ?

Examples

			For n = 6 the 11 regions of 6 are [1], [2,1], [3,1,1], [2], [4,2,1,1,1], [3], [5,2,1,1,1,1,1], [2], [4,2], [3], [6,3,2,2,1,1,1,1,1,1,1]. These number are the first A006128(6) terms of triangle A206437 in which the first A000041(6) rows are the 11 regions of 6. We can see that the 8th region is equal to the 4th region: [2] = [2]. Also the 10th region is equal to the 6th region: [3] = [3]. There are two repeated regions, therefore a(6) = A000041(6) - 2 = 11 - 2 = 9.

Links

Crossrefs

Cf. A000041, A006128, A135010, A141285, A182244, A186114, A186412, A187219, A193870, A194446, A194447, A206437, A211009.

A220487 Partial sums of triangle A206437.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 11, 15, 17, 18, 19, 20, 23, 28, 30, 31, 32, 33, 34, 35, 37, 41, 43, 46, 52, 55, 57, 59, 60, 61, 62, 63, 64, 65, 66, 69, 74, 76, 80, 87, 90, 92, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 107, 111, 113, 116, 122, 125, 127, 129, 134, 138
Offset: 1

Views

Author

Omar E. Pol, Jan 18 2013

Keywords

Examples

			When written as an irregular triangle in which row j has length A194446(j) then the right border gives A182244. Also the records of row lengths give the partition numbers (A000041) of the positive integers as shown below:
1;
3, 4;
7, 8, 9;
11;
15,17,18,19,20;
23;
28,30,31,32,33,34,35;
37;
41,43;
46;
52,55,57,59,60,61,62,63,64,65,66;
69;
74,76;
80;
87,90,92,94,95,96,97,98,99,100,101,102,103,104,105;
...
Also when written as an irregular triangle in which row j has length A138137(j) then the right border gives A066186 as shown below:
1;
3, 4;
7, 8, 9;
11,15,17,18,19,20;
23,28,30,31,32,33,34,35;
37,41,43,46,52,55,57,59,60,61,62,63,64,65,66;
69,74,76,80,87,90,92,94,95,96,97,98,99,100,101,102,103,104,105;
...

Crossrefs

Cf. A000041, A006128, A066186, A135010, A138137, A138879, A141285, A182181, A182244, A186412, A194446, A206437.

Formula

a(A182181(n)) = A182244(n), n >= 1.
a(A006128(n)) = A066186(n), n >= 1.
Showing 1-8 of 8 results.

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