A138137 -id:A138137 - cp's OEIS frontend

A144121 Number of nonprime parts in the last section of the set of partitions of n.

1, 1, 2, 4, 5, 9, 12, 20, 26, 39, 51, 76, 96, 136, 175, 241, 304, 412, 517, 686, 859, 1117, 1392, 1794, 2217, 2818, 3478, 4373, 5363, 6694, 8168, 10113, 12295, 15105, 18289, 22347, 26932, 32712, 39302, 47481, 56825, 68347

Offset: 1

Omar E. Pol, Sep 11 2008

First differences of A144119.

Cf. A018252, A135010, A138121, A138137, A144117, A144118, A144119, A144120.

a(n) = A138137(n)-A144120(n) = A144119(n)-A144119(n-1).

A194796 Imbalance of the number of parts of all partitions of n.

Original entry on oeis.org

0, -1, 0, -3, 0, -8, 0, -17, 3, -31, 10, -58, 22, -101, 52, -167, 104, -278, 191, -451, 344, -711, 594, -1119, 983, -1730, 1606, -2635, 2555, -3990, 3978, -5972, 6118, -8835, 9269, -12986, 13835, -18917, 20454, -27320, 29900, -39204, 43268, -55846, 62112

Offset: 1

Omar E. Pol, Feb 01 2012

sign

Consider the three-dimensional structure of the shell model of partitions, version "tree" (see the illustration in A194795). 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. For more information see A135010.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A006128, A096541, A135010, A138121, A138135, A138137, A197595, A194797, A194809.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    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:= proc(n) option remember;
      (-1)^n*(b(n-1, n-1)[2]-b(n, n)[2])+`if`(n=1, 0, a(n-1))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Apr 04 2012

b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0 || i == 1, {1, 0}, 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]]}]]; a[n_] := a[n] = (-1)^n*(b[n-1, n-1][[2]] - b[n, n][[2]]) + If[n == 1, 0, a[n-1]]; Table [a[n], {n, 1, 60}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

vector(50, n, sum(k=1, n, (-1)^(k-1)*(numdiv(k)-1+sum(j=1, k-1, (numdiv(j)-1)*(numbpart(k-j)-numbpart(k-j-1)))))) \\ Altug Alkan, Nov 11 2015

a(n) = Sum_{k=1..n} (-1)^(k-1)*A138135(k).

More terms from Alois P. Heinz, Apr 04 2012

A206438 Triangle read by rows which lists the squares of the parts of A135010.

Original entry on oeis.org

1, 1, 4, 1, 1, 9, 1, 1, 1, 4, 4, 16, 1, 1, 1, 1, 1, 4, 9, 25, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 16, 9, 9, 36, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 9, 4, 25, 9, 16, 49, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 16, 4, 9, 9, 4, 36, 9, 25

Offset: 1

Omar E. Pol, Feb 08 2012

Volumes of the parts in the section model of partitions version "boxes" in which each part of size k has a volume = k^2. Row sums of this triangle give A206440 and partial sums of A206440 give A066183.

			Written as a triangle:
1;
1,4;
1,1,9;
1,1,1,4,4,16;
1,1,1,1,1,4,9,25;
1,1,1,1,1,1,1,4,4,4,4,16,9,9,36;
1,1,1,1,1,1,1,1,1,1,1,4,4,9,4,25,9,16,49;

Alois P. Heinz, Rows n = 1..23, flattened

Row n has length A138137(n).
Row sums give A206440.
Right border gives positives A000290.
Cf. A066183, A135010, A138121, A141285.

Table[Reverse@ConstantArray[{1}, PartitionsP[n - 1]] ~Join~ DeleteCases[Sort@PadRight[Reverse/@Cases[IntegerPartitions[n], x_ /; Last[x] != 1]], x_ /; x == 0, 2], {n, 1, 8}] ^2  // Flatten (* Robert Price, May 28 2020 *)

a(n) = A135010(n)^2.

A210951 Triangle read by rows: T(n,k) = number of parts in the k-th column of the shell model of partitions considering only the n-th shell and with its parts aligned to the right margin.

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 0, 0, 1, 5, 0, 0, 0, 1, 7, 0, 0, 0, 1, 3, 11, 0, 0, 0, 0, 1, 3, 15, 0, 0, 0, 0, 1, 3, 6, 22, 0, 0, 0, 0, 0, 1, 4, 7, 30, 0, 0, 0, 0, 0, 1, 3, 7, 11, 42, 0, 0, 0, 0, 0, 0, 1, 4, 9, 13, 56, 0, 0, 0, 0, 0, 0, 1, 3, 8, 15, 20, 77, 0, 0, 0

Offset: 1

Omar E. Pol, Apr 22 2012

			For n = 6 and k = 1..6 the 6th shell looks like this:
-------------------------
k: 1,  2,  3,  4,  5,  6
-------------------------
.                      6
.                  3 + 3
.                  4 + 2
.              2 + 2 + 2
.                      1
.                      1
.                      1
.                      1
.                      1
.                      1
.                      1
.
The total number of parts in columns 1-6 are
.  0,  0,  0,  1,  3, 11, the same as the 6th row of triangle.
Triangle begins:
1;
0, 2;
0, 0, 3;
0, 0, 1, 5;
0, 0, 0, 1, 7;
0, 0, 0, 1, 3, 11;
0, 0, 0, 0, 1, 3, 15;
0, 0, 0, 0, 1, 3, 6, 22;
0, 0, 0, 0, 0, 1, 4, 7, 30;
0, 0, 0, 0, 0, 1, 3, 7, 11, 42;
0, 0, 0, 0, 0, 0, 1, 4, 9, 13, 56;
0, 0, 0, 0, 0, 0, 1, 3, 8, 15, 20, 77;

Row sums give A138137. Column sums converge to A000070. Right border gives A000041, n >= 1.

Cf. A135010, A138121, A194714, A210945, A210950, A210952, A210953.

A210955 Triangle read by rows: T(n,k) = total number of parts <= k in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 5, 5, 6, 5, 6, 7, 7, 8, 7, 11, 13, 14, 14, 15, 11, 14, 16, 17, 18, 18, 19, 15, 23, 26, 29, 30, 31, 31, 32, 22, 29, 35, 37, 39, 40, 41, 41, 42, 30, 45, 51, 56, 59, 61, 62, 63, 63, 64, 42, 57, 67, 72, 76, 78, 80, 81, 82, 82, 83

Offset: 1

Omar E. Pol, May 01 2012

Row n lists the partial sums of row n of triangle A182703.

			1,
1,   2,
2,   2,  3,
3,   5,  5,  6,
5,   6,  7,  7,  8,
7,  11, 13, 14, 14, 15,
11, 14, 16, 17, 18, 18, 19,
15, 23, 26, 29, 30, 31, 31, 32,
22, 29, 35, 37, 39, 40, 41, 41, 42;

Column 1 is A000041. Right border gives A138137.

Cf. A135010, A138121, A182703, A206437, A206562, A207031, A207032, A207383, A208476, A210947, A210956.

T(n,k) = Sum_{j=1..k} A182703(n,j).

More terms from Alois P. Heinz, May 25 2013

A228109 Height after n-th step of an infinite staircase which is the lower part of a structure whose upper part is the infinite Dyck path of A228110.

Original entry on oeis.org

0, -1, 0, -1, 0, -1, 0, -1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 2, 1, 2, 1, 0, -1, 0, -1, 0, 1, 2, 1, 2, 3, 4, 3, 4, 3, 2, 1, 0, -1, 0, -1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 5, 6, 5, 4, 5, 4, 3, 2, 1, 0, -1

Offset: 0

Omar E. Pol, Aug 13 2013

sign

The master diagram of regions of the set of partitions of all positive integers is a total dissection of the first quadrant of the square grid in which the j-th horizontal line segments has length A141285(j) and the j-th vertical line segment has length A194446(j). For the definition of "region" see A206437. The first A000041(k) regions of the diagram represent the set of partitions of k in colexicographic order (see A211992). The length of the j-th horizontal line segment equals the largest part of the j-th partition of k and equals the largest part of the j-th region of the diagram. The length of the j-th vertical line segment (which is the line segment ending in row j) equals the number of parts in the j-th region.
For k = 5, the diagram 1 represents the partitions of 5. The diagram 2 shows separately the boundary segments southwest sides of the first seven regions of the diagram 1, see below:
.
j   Diagram 1              Diagram 2
     _  
7 |  _    |                                |  _
6 |  _|  |                          | _       |
5 |     | |                  |                   |
4 | |_  | |              |     |_                |
3 |   | | |        |             |               |
2 |  | | | |    |      |           |               |
1 |||_|||  |  |    |          |              |_
.
.    1 2 3 4 5
.
a(n) is the height after n-th step of an infinite staircase which is the lower part of a diagram of regions of the set of partitions of all positive integers. The upper part of the diagram is the infinite Dyck path mentioned in A228110. The diagram shows the shape of the successive regions of the set of partitions of all positive integers. The area of the n-th region is A186412(n).
For the height of the peaks and the valleys in the infinite Dyck path see A229946.

			Illustration of initial terms (n = 1..53):
5
4                                                      /
3                                 /\/\                /
2                                /    \            /\/
1                   /\/\      /\/      \        /\/
0          /\    /\/    \    /          \    /\/
-1 \/\/\/\/  \/\/        \/\/            \/\/
-2
The diagram shows the Dyck pack mentioned in A228110 together with the staircase illustrated above. The area of the n-th region is equal to A186412(n).
.
7...................................
.                                  /\
5.....................            /  \                /\
.                    /\          /    \          /\  / /
3...........        /  \        / /\/\ \        /  \/ /
2......    /\      /    \    /\/ /    \ \      /   /\/
1...  /\  /  \  /\/ /\/\ \  / /\/      \ \  /\/ /\/
0  /\/  \/ /\ \/ /\/    \ \/ /          \ \/ /\/
-1 \/\/\/\/  \/\/        \/\/            \/\/
.
Region:
.   1  2    3   4     5      6      7       8    9   10

Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A228110, A229946.

A228110 Height after n-th step of the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, n >= 0, k >= 1.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 1, 2, 3, 4, 5, 4, 3, 4, 5, 6, 5, 6, 7, 8, 9, 10, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 2, 3, 4, 5, 6, 7, 6, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 13, 14, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

Offset: 0

Omar E. Pol, Aug 10 2013

The master diagram of regions of the set of partitions of all positive integers is a total dissection of the first quadrant of the square grid in which the j-th horizontal line segments has length A141285(j) and the j-th vertical line segment has length A194446(j). For the definition of "region" see A206437. The first A000041(k) regions of the diagram represent the set of partitions of k in colexicographic order (see A211992). The length of the j-th horizontal line segment equals the largest part of the j-th partition of k and equals the largest part of the j-th region of the diagram. The length of the j-th vertical line segment (which is the line segment ending in row j) equals the number of parts in the j-th region.
For k = 7, the diagram 1 represents the partitions of 7. The diagram 2 is a minimalist version of the structure which does not contain the axes [X, Y].  See below:
.
.  j     Diagram 1         Partitions           Diagram 2
.        _   _                            _   _  
. 15  |  _       |      7                   _        |
. 14  |  _ |    |      4+3                  _ |    |
. 13  |  _    |   |      5+2                  _    |   |
. 12  |  _| |_  |      3+2+2                _| |_  |
. 11  |  _      | |      6+1                  _      | |
. 10  |  _|    | |      3+3+1               _ |    | |
.  9  |     |   | |      4+2+1                   |   | |
.  8  | |_ |  | |      2+2+2+1             |_ |  | |
.  7  |  _    | | |      5+1+1                _    | | |
.  6  |  _|  | | |      3+2+1+1             _ |  | | |
.  5  |     | | | |      4+1+1+1                 | | | |
.  4  | |_  | | | |      2+2+1+1+1           |_  | | | |
.  3  |   | | | | |      3+1+1+1+1             | | | | |
.  2  |  | | | | | |      2+1+1+1+1+1          | | | | | |
.  1  |||_|||_|_|      1+1+1+1+1+1+1       | | | | | | |
.
.      1 2 3 4 5 6 7
.
The second diagram has the property that if the number of regions is also the number of partitions of k so the sum of the lengths of all horizontal line segment equals the sum of the lengths of all vertical line segments and equals A006128(k), for k >= 1.
Also the diagram has the property that it can be transformed in a Dyck path (see example).
The sequence gives the height of the infinite Dyck path after n-th step.
The absolute values of the first differences give A000012.
For the height of the peaks and the valleys in the infinite Dyck path see A229946.
Q: Is this infinite Dyck path a fractal?

			Illustration of initial terms (n = 1..59):
.
11 ...........................................................
.                                                            /
.                                                           /
.                                                          /
7 ..................................                      /
.                                  /\                    /
5 ....................            /  \                /\/
.                    /\          /    \          /\  /
3 ..........        /  \        /      \        /  \/
2 .....    /\      /    \    /\/        \      /
1 ..  /\  /  \  /\/      \  /            \  /\/
.  /\/  \/    \/          \/              \/
.
Note that the j-th largest peak between two valleys at height 0 is also the partition number A000041(j).
Written as an irregular triangle in which row k has length 2*A138137(k), the sequence begins:
0,1;
0,1,2,1;
0,1,2,3,2,1;
0,1,2,1,2,3,4,5,4,3,2,1;
0,1,2,3,2,3,4,5,6,7,6,5,4,3,2,1;
0,1,2,1,2,3,4,5,4,3,4,5,6,5,6,7,8,9,10,11,10,9,8,7,6,5,4,3,2,1;
0,1,2,3,2,3,4,5,6,7,6,5,6,7,8,9,8,9,10,11,12,13,14,15,14,13,12,11,10,9,8,7,6,5,4,3,2,1;
...

Omar E. Pol, Visualization of regions in a diagram for A006128

Column 1 is A000004. Both column 2 and the right border are in A000012. Both columns 3 and 5 are in A007395.

Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A229946.

A229946 Height of the peaks and the valleys in the Dyck path whose j-th ascending line segment has A141285(j) steps and whose j-th descending line segment has A194446(j) steps.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 2, 1, 5, 0, 3, 2, 7, 0, 2, 1, 5, 3, 6, 5, 11, 0, 3, 2, 7, 5, 9, 8, 15, 0, 2, 1, 5, 3, 6, 5, 11, 7, 12, 11, 15, 14, 22, 0, 3, 2, 7, 5, 9, 8, 15, 11, 14, 13, 19, 17, 22, 21, 30, 0, 2, 1, 5, 3, 6, 5, 11, 7, 12, 11, 15, 14, 22, 15, 19, 18, 25, 23, 29, 28, 33, 32, 42, 0

Offset: 0

Omar E. Pol, Nov 03 2013

Also 0 together the alternating sums of A220517.
The master diagram of regions of the set of partitions of all positive integers is a total dissection of the first quadrant of the square grid in which the j-th horizontal line segments has length A141285(j) and the j-th vertical line segment has length A194446(j). For the definition of "region" see A206437. The first A000041(k) regions of the diagram represent the set of partitions of k in colexicographic order (see A211992). The length of the j-th horizontal line segment equals the largest part of the j-th partition of k and equals the largest part of the j-th region of the diagram. The length of the j-th vertical line segment (which is the line segment ending in row j) equals the number of parts in the j-th region.
For k = 7, the diagram 1 represents the partitions of 7. The diagram 2 is a minimalist version of the structure which does not contain the axes [X, Y].  See below:
.
.  j     Diagram 1         Partitions           Diagram 2
.        _   _                            _   _  
. 15  |  _       |      7                   _        |
. 14  |  _ |    |      4+3                  _ |    |
. 13  |  _    |   |      5+2                  _    |   |
. 12  |  _| |_  |      3+2+2                _| |_  |
. 11  |  _      | |      6+1                  _      | |
. 10  |  _|    | |      3+3+1               _ |    | |
.  9  |     |   | |      4+2+1                   |   | |
.  8  | |_ |  | |      2+2+2+1             |_ |  | |
.  7  |  _    | | |      5+1+1                _    | | |
.  6  |  _|  | | |      3+2+1+1             _ |  | | |
.  5  |     | | | |      4+1+1+1                 | | | |
.  4  | |_  | | | |      2+2+1+1+1           |_  | | | |
.  3  |   | | | | |      3+1+1+1+1             | | | | |
.  2  |  | | | | | |      2+1+1+1+1+1          | | | | | |
.  1  |||_|||_|_|      1+1+1+1+1+1+1       | | | | | | |
.
.      1 2 3 4 5 6 7
.
The second diagram has the property that if the number of regions is also the number of partitions of k so the sum of the lengths of all horizontal line segment equals the sum of the lengths of all vertical line segments and equals A006128(k), for k >= 1.
Also the diagram has the property that it can be transformed in a Dyck path (see example).
The height of the peaks and the valleys of the infinite Dyck path give this sequence.
Q: Is this Dyck path a fractal?

			Illustration of initial terms (n = 0..21):
.                                                             11
.                                                             /
.                                                            /
.                                                           /
.                                   7                      /
.                                   /\                 6  /
.                     5            /  \           5    /\/
.                     /\          /    \          /\  / 5
.           3        /  \     3  /      \        /  \/
.      2    /\   2  /    \    /\/        \   2  /   3
.   1  /\  /  \  /\/      \  / 2          \  /\/
.   /\/  \/    \/ 1        \/              \/ 1
.  0 0   0     0           0               0
.
Note that the k-th largest peak between two valleys at height 0 is also A000041(k) and the next term is always 0.
.
Written as an irregular triangle in which row k has length 2*A187219(k), k >= 1, the sequence begins:
0,1;
0,2;
0,3;
0,2,1,5;
0,3,2,7;
0,2,1,5,3,6,5,11;
0,3,2,7,5,9,8,15;
0,2,1,5,3,6,5,11,7,12,11,15,14,22;
0,3,2,7,5,9,8,15,11,14,13,19,17,22,21,30;
0,2,1,5,3,6,5,11,7,12,11,15,14,22,15,19,18,25,23,29,28,33,32,42;
...

Omar E. Pol, Visualization of regions in a diagram for A006128

Column 1 is A000004. Right border gives A000041 for the positive integers.

Cf. A006128, A135010, A138137, A139582, A141285, A186412, A187219, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610.

a(0) = 0; a(n) = a(n-1) + (-1)^(n-1)*A220517(n), n >= 1.

A230440 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of partitions of n that do not contain 1 as a part in colexicographic order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 18 2013

The n-th row of triangle lists the parts of the n-th section of the set of partitions of any integer >= n. For the definition of "section" see A135010.

			Illustration of initial terms (row = 1..6). The table shows the six sections of the set of partitions of 6 in three ways. Note that before the dissection, the set of partitions was in colexicographic order, see A211992. More generally, in a master model, 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.
---------------------------------------------------------
n  j     Diagram          Parts              Parts
---------------------------------------------------------
.         _
1  1     |_|              1;                 1;
.           _
2  1      _| |              1,                 1,
2  2     |_ _|              2;               2;
.             _
3  1         | |              1,                 1,
3  2      _ _| |              1,               1,
3  3     |_ _ _|              3;             3;
.               _
4  1           | |              1,                 1,
4  2           | |              1,               1,
4  3      _ _ _| |              1,             1,
4  4     |_ _|   |            2,2,           2,2,
4  5     |_ _ _ _|              4;           4;
.                 _
5  1             | |              1,                 1,
5  2             | |              1,               1,
5  3             | |              1,             1,
5  4             | |              1,             1,
5  5      _ _ _ _| |              1,           1,
5  6     |_ _ _|   |            3,2,         3,2,
5  7     |_ _ _ _ _|              5;         5;
.                   _
6  1               | |              1,                 1,
6  2               | |              1,               1,
6  3               | |              1,             1,
6  4               | |              1,             1,
6  5               | |              1,           1,
6  6               | |              1,           1,
6  7      _ _ _ _ _| |              1,         1,
6  8     |_ _|   |   |          2,2,2,       2,2,2,
6  9     |_ _ _ _|   |            4,2,       4,2,
6  10    |_ _ _|     |            3,3,       3,3,
6  11    |_ _ _ _ _ _|              6;       6;
...
Triangle begins:
[1];
[1],[2];
[1],[1],[3];
[1],[1],[1],[2,2],[4];
[1],[1],[1],[1],[1],[3,2],[5];
[1],[1],[1],[1],[1],[1],[1],[2,2,2],[4,2],[3,3],[6];
...

Positive terms of A228716.
Row n has length A138137(n).
Row sums give A138879.
Right border gives A000027.
Cf. A000041, A135010, A138121, A141285, A182703, A187219, A193870, A194446, A206437, A207031, A207034, A207383, A207379, A211009.

A233968 Number of steps between two valleys at height 0 in the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, k >= 1.

Original entry on oeis.org

2, 4, 6, 12, 16, 30, 38, 64, 84, 128, 166, 248, 314, 448, 576, 790, 1004, 1358, 1708, 2264, 2844, 3694, 4614, 5936, 7354, 9342, 11544, 14502, 17816, 22220, 27144, 33584, 40878, 50192, 60828, 74276, 89596, 108778, 130772, 157918, 189116, 227374

Offset: 1

Omar E. Pol, Jan 14 2014

nonn

Also first differences of A211978.

			Illustration of initial terms as a dissection of a minimalist diagram of regions of the set of partitions of n, for n = 1..6:
.                                         _ _ _ _ _ _
.                                         _ _ _      |
.                                         _ _ _|_    |
.                                         _ _    |   |
.                             _ _ _ _ _      |   |   |
.                             _ _ _    |             |
.                   _ _ _ _        |   |             |
.                   _ _    |           |             |
.           _ _ _      |   |           |             |
.     _ _        |         |           |             |
. _      |       |         |           |             |
.  |     |       |         |           |             |
.
. 2    4      6       12          16          30
.
Also using the elements from the above diagram we can draw an infinite Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the n-th largest peak between two valleys at height 0 is also the partition number A000041(n).
7..................................
.                                 /\
5....................            /  \                /\
.                   /\          /    \          /\  /
3..........        /  \        /      \        /  \/
2.....    /\      /    \    /\/        \      /
1..  /\  /  \  /\/      \  /            \  /\/
0 /\/  \/    \/          \/              \/
.  2, 4,   6,       12,           16,...
.

Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A228109, A228110, A229946.

a(n) = 2*(A006128(n) - A006128(n-1)) = 2*A138137(n).

cp's OEIS Frontend

A144121 Number of nonprime parts in the last section of the set of partitions of n.

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A194796 Imbalance of the number of parts of all partitions of n.

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A206438 Triangle read by rows which lists the squares of the parts of A135010.

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A210951 Triangle read by rows: T(n,k) = number of parts in the k-th column of the shell model of partitions considering only the n-th shell and with its parts aligned to the right margin.

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A210955 Triangle read by rows: T(n,k) = total number of parts <= k in the last section of the set of partitions of n.

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A228109 Height after n-th step of an infinite staircase which is the lower part of a structure whose upper part is the infinite Dyck path of A228110.

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A228110 Height after n-th step of the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, n >= 0, k >= 1.

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A229946 Height of the peaks and the valleys in the Dyck path whose j-th ascending line segment has A141285(j) steps and whose j-th descending line segment has A194446(j) steps.

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A230440 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of partitions of n that do not contain 1 as a part in colexicographic order.

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A233968 Number of steps between two valleys at height 0 in the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, k >= 1.

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