A182995 -id:A182995 - cp's OEIS frontend

A138121 Triangle read by rows in which row n lists the partitions of n that do not contain 1 as a part in juxtaposed reverse-lexicographical order followed by A000041(n-1) 1's.

1, 2, 1, 3, 1, 1, 4, 2, 2, 1, 1, 1, 5, 3, 2, 1, 1, 1, 1, 1, 6, 3, 3, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 7, 4, 3, 5, 2, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 4, 4, 5, 3, 6, 2, 3, 3, 2, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 5, 4, 6, 3, 3, 3, 3, 7, 2, 4, 3, 2, 5, 2, 2, 3, 2, 2

Offset: 1

Omar E. Pol, Mar 21 2008

Mirror of triangle A135010.

			Triangle begins:
[1];
[2],[1];
[3],[1],[1];
[4],[2,2],[1],[1],[1];
[5],[3,2],[1],[1],[1],[1],[1];
[6],[3,3],[4,2],[2,2,2],[1],[1],[1],[1],[1],[1],[1];
[7],[4,3],[5,2],[3,2,2],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1];
...
The illustration of the three views of the section model of partitions (version "tree" with seven sections) shows the connection between several sequences.
---------------------------------------------------------
Partitions                A194805            Table 1.0
.  of 7       p(n)        A194551             A135010
---------------------------------------------------------
7              15                    7     7 . . . . . .
4+3                                4       4 . . . 3 . .
5+2                              5         5 . . . . 2 .
3+2+2                          3           3 . . 2 . 2 .
6+1            11    6       1             6 . . . . . 1
3+3+1                  3     1             3 . . 3 . . 1
4+2+1                    4   1             4 . . . 2 . 1
2+2+2+1                    2 1             2 . 2 . 2 . 1
5+1+1           7            1   5         5 . . . . 1 1
3+2+1+1                      1 3           3 . . 2 . 1 1
4+1+1+1         5        4   1             4 . . . 1 1 1
2+2+1+1+1                  2 1             2 . 2 . 1 1 1
3+1+1+1+1       3            1 3           3 . . 1 1 1 1
2+1+1+1+1+1     2          2 1             2 . 1 1 1 1 1
1+1+1+1+1+1+1   1            1             1 1 1 1 1 1 1
.               1                         ---------------
.               *<------- A000041 -------> 1 1 2 3 5 7 11
.                         A182712 ------->   1 0 2 1 4 3
.                         A182713 ------->     1 0 1 2 2
.                         A182714 ------->       1 0 1 1
.                                                  1 0 1
.                         A141285           A182703  1 0
.                    A182730   A182731                 1
---------------------------------------------------------
.                              A138137 --> 1 2 3 6 9 15..
---------------------------------------------------------
.       A182746 <--- 4 . 2 1 0 1 2 . 4 ---> A182747
---------------------------------------------------------
.
.       A182732 <--- 6 3 4 2 1 3 5 4 7 ---> A182733
.                    . . . . 1 . . . .
.                    . . . 2 1 . . . .
.                    . 3 . . 1 2 . . .
.      Table 2.0     . . 2 2 1 . . 3 .     Table 2.1
.                    . . . . 1 2 2 . .
.                            1 . . . .
.
.  A182982  A182742       A194803       A182983  A182743
.  A182992  A182994       A194804       A182993  A182995
---------------------------------------------------------
.
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms (n = 1..6). The table shows the six sections of the set of partitions of 6. Note that before the dissection the set of partitions was in the ordering mentioned in A026792. More generally, the six sections of the set of partitions of 6 also can be interpreted as the first six sections of the set of partitions of any integer >= 6.
Illustration of initial terms:
---------------------------------------
n  j     Diagram          Parts
---------------------------------------
.         _
1  1     |_|              1;
.         _ _
2  1     |_  |            2,
2  2       |_|            .  1;
.         _ _ _
3  1     |_ _  |          3,
3  2         | |          .  1,
3  3         |_|          .  .  1;
.         _ _ _ _
4  1     |_ _    |        4,
4  2     |_ _|_  |        2, 2,
4  3           | |        .  1,
4  4           | |        .  .  1,
4  5           |_|        .  .  .  1;
.         _ _ _ _ _
5  1     |_ _ _    |      5,
5  2     |_ _ _|_  |      3, 2,
5  3             | |      .  1,
5  4             | |      .  .  1,
5  5             | |      .  .  1,
5  6             | |      .  .  .  1,
5  7             |_|      .  .  .  .  1;
.         _ _ _ _ _ _
6  1     |_ _ _      |    6,
6  2     |_ _ _|_    |    3, 3,
6  3     |_ _    |   |    4, 2,
6  4     |_ _|_ _|_  |    2, 2, 2,
6  5               | |    .  1,
6  6               | |    .  .  1,
6  7               | |    .  .  1,
6  8               | |    .  .  .  1,
6  9               | |    .  .  .  1,
6  10              | |    .  .  .  .  1,
6  11              |_|    .  .  .  .  .  1;
...
(End)

Robert Price, Table of n, a(n) for n = 1..16851, 25 rows
Omar E. Pol, A section model of partitions (2D and 3D) [From _Omar E. Pol_, Sep 07 2008]
Robert Price, Mathematica Program to Generate Diagram

Row n has length A138137(n).
Rows sums give A138879.
Cf. A000041, A135010, A138879, A138880, A141285, A182703, A194812, A206437, A211009.

less[run1_, run2_] := (lg1 = run1 // Length; lg2 = run2 // Length; lg = Max[lg1, lg2]; r1 = If[lg1 == lg, run1, PadRight[run1, lg, 0]]; r2 = If[lg2 == lg, run2, PadRight[run2, lg, 0]]; Order[r1, r2] != -1); row[n_] := Join[Array[1 &, {PartitionsP[n - 1]}], Sort[Reverse /@ Select[IntegerPartitions[n], FreeQ[#, 1] &], less]] // Flatten // Reverse; Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Jan 15 2013 *)
Table[Reverse/@Reverse@DeleteCases[Sort@PadRight[Reverse/@Cases[IntegerPartitions[n], x_ /; Last[x]!=1]], x_ /; x==0, 2]~Join~ConstantArray[{1}, PartitionsP[n - 1]], {n, 1, 9}]  // Flatten (* Robert Price, May 11 2020 *)

A182994 Sum of all parts of the n-th subshell of the head of the last section of the set of partitions of any even integer >= 2n.

Original entry on oeis.org

2, 6, 16, 32, 64, 132, 224, 404, 704, 1156, 1880, 3060, 4748, 7396, 11346, 17054, 25454, 37706, 54980, 79756, 114702, 163394, 231288, 325408, 454138, 630542, 870504, 1194536, 1631196, 2216992, 2997542, 4036022, 5411108, 7223636

Offset: 1

Omar E. Pol, Feb 06 2011

nonn

The last section of the set of partitions of 2n contains n subshells.

Also first differences of A182736. - Omar E. Pol, Mar 03 2011

			a(5)=64 because the 5th subshell of the head of the last section of any even integer >= 10 looks like this:
(10 . . . . . . . . . )
( 5 . . . . 5 . . . . )
( 6 . . . . . 4 . . . )
( 7 . . . . . . 3 . . )
( 4 . . . 3 . . 3 . . )
.                (2 . )
.                (2 . )
.                (2 . )
.                (2 . )
.                (2 . )
.                (2 . )
.                (2 . )
There are 17 parts whose sum is 10+5+5+6+4+7+3+4+3+3+2+2+2+2+2+2+2 = 10*5 + 2*7 = 64, so a(5)=64.

Cf. A135010, A138880, A182736, A182742, A182982, A182992, A182995.

a(1) = 2. a(n) = A138880(2n) - A138880(2n-2), n >= 2.

More terms from Omar E. Pol, Mar 03 2011

A182993 Number of parts of the n-th subshell of the head of the last section of the set of partitions of any odd integer >= 2n+1.

Original entry on oeis.org

1, 2, 5, 12, 21, 39, 73, 118, 198, 326, 510, 797, 1234, 1854, 2778, 4122, 6014, 8717, 12550, 17849, 25252, 35486, 49447, 68540, 94480, 129378, 176339, 239165, 322676, 433487, 579907, 772318, 1024691, 1354445, 1783504

Offset: 1

Omar E. Pol, Feb 06 2011

nonn

The last section of the set of partitions of 2n+1 contains n subshells.

Also first differences of A182735. - Omar E. Pol, Mar 03 2011

			a(5)=21 because the 5th subshell of the head of the last section of any odd integer >= 11 looks like this:
(11 . . . . . . . . . . )
( 6 . . . . . 5 . . . . )
( 7 . . . . . . 4 . . . )
( 8 . . . . . . . 3 . . )
( 4 . . . 4 . . . 3 . . )
( 5 . . . . 3 . . 3 . . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
The subshell has 21 parts, so a(5)=21.

Cf. A135010, A138135, A182735, A182743, A182983, A182992, A182995.

a(n) = A138135(2n+1) - A138135(2n-1).

More terms from Omar E. Pol, Mar 03 2011

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.

A194797 Imbalance of the sum of parts of all partitions of n.

Original entry on oeis.org

0, -2, 1, -7, 3, -21, 7, -49, 23, -97, 57, -195, 117, -359, 256, -624, 498, -1086, 909, -1831, 1634, -2986, 2833, -4847, 4728, -7700, 7798, -12026, 12537, -18633, 19745, -28479, 30723, -42955, 47100, -64284, 71136, -95228, 106402, -139718, 157327, -203495

Offset: 1

Omar E. Pol, Jan 31 2012

sign

Consider the three-dimensional structure of the shell model of partitions, version "tree" (see example). Note that only the parts > 1 produce the imbalance. The 1's are located in the central columns therefore they do not produce the imbalance. Note that every column contains exactly the same parts. For more information see A135010.

			For n = 6 the illustration of the three views of the shell model with 6 shells shows an imbalance (see below):
------------------------------------------------------
Partitions                Tree             Table 1.0
of 6.                    A194805            A135010
------------------------------------------------------
6                   6                     6 . . . . .
3+3                   3                   3 . . 3 . .
4+2                     4                 4 . . . 2 .
2+2+2                     2               2 . 2 . 2 .
5+1                         1   5         5 . . . . 1
3+2+1                       1 3           3 . . 2 . 1
4+1+1                   4   1             4 . . . 1 1
2+2+1+1                   2 1             2 . 2 . 1 1
3+1+1+1                     1 3           3 . . 1 1 1
2+1+1+1+1                 2 1             2 . 1 1 1 1
1+1+1+1+1+1                 1             1 1 1 1 1 1
------------------------------------------------------
.
.                   6 3 4 2 1 3 5
.     Table 2.0     . . . . 1 . .     Table 2.1
.      A182982      . . . 2 1 . .      A182983
.                   . 3 . . 1 2 .
.                   . . 2 2 1 . .
.                   . . . . 1
------------------------------------------------------
The sum of all parts > 1 on the left hand side is 34 and the sum of all parts > 1 on the right hand side is 13, so a(6) = -34 + 13 = -21. On the other hand for n = 6 the first n terms of A138880 are 0, 2, 3, 8, 10, 24 thus a(6) = 0-2+3-8+10-24 = -21.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000041, A002865, A135010, A138121, A138880, A141285, A182710, A182742, A182743, A182746, A182747, A182982, A182983, A182994, A182995, A194796, A194805.

with(combinat):
a:= proc(n) option remember;
      n *(-1)^n *(numbpart(n-1)-numbpart(n)) +a(n-1)
    end: a(0):=0:
seq(a(n), n=1..50); # Alois P. Heinz, Apr 04 2012

a[n_] := Sum[(-1)^(k-1)*k*(PartitionsP[k] - PartitionsP[k-1]), {k, 1, n}]; Array[a, 50] (* Jean-François Alcover, Dec 09 2016 *)

a(n) = Sum_{k=1..n} (-1)^(k-1)*k*(p(k)-p(k-1)), where p(k) is the number of partitions of k.
a(n) = b(1)-b(2)+b(3)-b(4)+b(5)-b(6)...+-b(n), where b(n) = A138880(n).
a(n) ~ -(-1)^n * Pi * sqrt(2) * exp(Pi*sqrt(2*n/3)) / (48*sqrt(n)). - Vaclav Kotesovec, Oct 09 2018

A194804 Sum 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, 4, 8, 15, 23, 40, 59, 92, 137, 202, 285, 418, 577, 802, 1106, 1511, 2019, 2724, 3598, 4755, 6226, 8107, 10462, 13523, 17280, 22029, 27953, 35350, 44416, 55763, 69579, 86634, 107459, 132914, 163768, 201475, 246841, 301822, 368033, 447790, 543206

Offset: 0

Omar E. Pol, Jan 27 2012

nonn

For the number of parts see A194803. For more information about the shell model see A135010 and A194805.

			Illustration of one of the three views with seven shells:
.
.        A182732 <- 6 3 4 2 1 3 5 4 7 -> A182733
.                   . . . . 1 . . . .
.                   . . . 2 1 . . . .
.      Table 2.0    . 3 . . 1 2 . . .    Table 2.1
.                   . . 2 2 1 . . 3 .
.                   . . . . 1 2 2 . .
.                           1 . . . .
.  A182742  A182982                   A182743  A182983
.  A182992  A182994                   A182993  A182995
.
The sum of parts that are visible is 1+1+1+1+1+1+1+2+2+2+2+2+2+2+3+3+3+3+4+4+5+6+7 = 59, so a(7) = 59. Using the formula we have a(7) = 7+24+28 = 59.

Cf. A002865, A066186, A135010, A138121, A138880, A182732, A182733, A182742, A182743, A182982, A182983, A182992-A182995, A194803, A194805.

a(n) = n + A138880(n-1) + A138880(n), if n >= 2.

A194450 Vertex number of a rectangular spiral which contains exactly between its edges the successive shells of the partitions of the positive integers.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 17, 21, 28, 33, 44, 50, 65, 72, 94, 102, 132, 141, 183, 193, 249, 260, 337, 349, 450, 463, 598, 612, 788, 803, 1034, 1050, 1347, 1364, 1749, 1767, 2257, 2276, 2903, 2923, 3715, 3736, 4738, 4760, 6015, 6038, 7613, 7637, 9595

Offset: 0

Omar E. Pol, Nov 01 2011

nonn

First differences give A194451, the length of the edges of the spiral. For more information see A135010 and A138121.

Cf. A000041, A000217, A006128, A026905, A066186, A135010, A138121, A138879, A138880, A182703, A182994, A182995, A194451.

a(2n-1) = A026905(n) + A000217(n) - n, if n >= 1.

a(2n) = A026905(n) + A000217(n), if n >= 1.

cp's OEIS Frontend

A138121 Triangle read by rows in which row n lists the partitions of n that do not contain 1 as a part in juxtaposed reverse-lexicographical order followed by A000041(n-1) 1's.

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A182994 Sum of all parts of the n-th subshell of the head of the last section of the set of partitions of any even integer >= 2n.

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A182993 Number of parts of the n-th subshell of the head of the last section of the set of partitions of any odd integer >= 2n+1.

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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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A194797 Imbalance of the sum of parts of all partitions of n.

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A194804 Sum 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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A194450 Vertex number of a rectangular spiral which contains exactly between its edges the successive shells of the partitions of the positive integers.

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