A211978 -id:A211978 - cp's OEIS frontend

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.

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.

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).

A225596 Sum of largest parts of all partitions of n plus n. Also, total number of parts in all partitions of n plus n.

Original entry on oeis.org

0, 2, 5, 9, 16, 25, 41, 61, 94, 137, 202, 286, 411, 569, 794, 1083, 1479, 1982, 2662, 3517, 4650, 6073, 7921, 10229, 13198, 16876, 21548, 27321, 34573, 43482, 54593, 68166, 84959, 105399, 130496, 160911, 198050, 242849, 297239, 362626, 441586, 536145

Offset: 0

Omar E. Pol, Aug 01 2013

nonn

a(n) is also the number of horizontal toothpicks (or the total length of all horizontal boundary segments) in the diagram of regions of the set of partitions of n, see example. A093694(n) is the number of vertical toothpicks in the diagram. See also A225610. For a minimalist version of the diagram see A211978. For the definition of "region" see A206437.

			For n = 7 the sum of largest parts of all partitions of 7 plus 7 is (7+4+5+3+6+3+4+2+5+3+4+2+3+2+1) + 7 = 54 + 7 = 61. On the other hand the number of toothpicks in horizontal direction in the diagram of regions of the set of partitions of 7 is equal to 61, so a(7) = 61.
.
.                  Diagram of regions       Horizontal
Partitions         and partitions of 7      toothpicks
of 7
.                     _ _ _ _ _ _ _
7                    |_ _ _ _      |             7
4+3                  |_ _ _ _|_    |             4
5+2                  |_ _ _    |   |             5
3+2+2                |_ _ _|_ _|_  |             3
6+1                  |_ _ _      | |             6
3+3+1                |_ _ _|_    | |             3
4+2+1                |_ _    |   | |             4
2+2+2+1              |_ _|_ _|_  | |             2
5+1+1                |_ _ _    | | |             5
3+2+1+1              |_ _ _|_  | | |             3
4+1+1+1              |_ _    | | | |             4
2+2+1+1+1            |_ _|_  | | | |             2
3+1+1+1+1            |_ _  | | | | |             3
2+1+1+1+1+1          |_  | | | | | |             2
1+1+1+1+1+1+1        |_|_|_|_|_|_|_|             1
.                                                7
.                                              _____
.                                       Total   61
.

Cf. A000041, A006128, A093694, A135010, A139250, A141285, A186114, A194446, A194447, A206437, A211978, A220517, A225600, A225610.

a(n) = A006128(n) + n = A225610(n) - A093694(n).

a(n) = n + sum_{k=1..A000041(n)} A141285(k), n >= 1.

A225598 Triangle read by rows: T(n,k) = sum of all parts of all regions of the set of partitions of n whose largest part is k.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 1, 5, 5, 9, 1, 5, 8, 9, 12, 1, 7, 11, 15, 12, 20, 1, 7, 14, 19, 19, 20, 25, 1, 9, 17, 29, 24, 33, 25, 38, 1, 9, 23, 33, 36, 42, 39, 38, 49, 1, 11, 26, 47, 46, 61, 49, 61, 49, 69, 1, 11, 32, 55, 63, 76, 70, 76, 76, 69, 87, 1, 13, 38, 73, 78, 110, 87, 111, 95, 108, 87, 123

Offset: 1

Omar E. Pol, Aug 02 2013

For the definition of region see A206437.

T(n,k) is also the sum of all parts that end in the k-th column of the diagram of regions of the set of partitions of n (see Example section).

			For n = 5 and k = 3 the set of partitions of 5 contains two regions whose largest part is 3, they are third region which contains three parts [3, 1, 1] and the sixth region which contains only one part [3]. Therefore the sum of all parts is 3 + 1 + 1 + 3 = 8, so T(5,3) = 8.
.
.    Diagram    Illustration of parts ending in column k:
.    for n=5      k=1   k=2     k=3       k=4        k=5
.   _ _ _ _ _                                  _ _ _ _ _
.  |_ _ _    |                _ _ _           |_ _ _ _ _|
.  |_ _ _|_  |               |_ _ _|  _ _ _ _       |_ _|
.  |_ _    | |          _ _          |_ _ _ _|        |_|
.  |_ _|_  | |         |_ _|  _ _ _      |_ _|        |_|
.  |_ _  | | |          _ _  |_ _ _|       |_|        |_|
.  |_  | | | |      _  |_ _|     |_|       |_|        |_|
.  |_|_|_|_|_|     |_|   |_|     |_|       |_|        |_|
.
k = 1 2 3 4 5
.
The 5th row lists:  1     5       8         9         12
.
Triangle begins:
1;
1,  3;
1,  3,  5;
1,  5,  5,  9;
1,  5,  8,  9, 12;
1,  7, 11, 15, 12,  20;
1,  7, 14, 19, 19,  20, 25;
1,  9, 17, 29, 24,  33, 25,  38;
1,  9, 23, 33, 36,  42, 39,  38, 49;
1, 11, 26, 47, 46,  61, 49,  61, 49,  69;
1, 11, 32, 55, 63,  76, 70,  76, 76,  69, 87;
1, 13, 38, 73, 78, 110, 87, 111, 95, 108, 87, 123;

Column 1 is A000012. Column 2 are the numbers >= 3 of A109613. Row sums give A066186. Right border gives A046746. Second right border gives A046746.

Cf. A000041, A066186, A135010, A141285, A186114, A186412, A187219, A194446, A206437, A207779, A211978, A225597, A225600, A225610.

A299774 Irregular triangle read by rows in which row n lists the indices of the partitions into equal parts in the list of colexicographically ordered partitions of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 5, 1, 7, 1, 8, 10, 11, 1, 15, 1, 16, 21, 22, 1, 27, 30, 1, 31, 41, 42, 1, 56, 1, 57, 69, 73, 76, 77, 1, 101, 1, 102, 134, 135, 1, 160, 172, 176, 1, 177, 221, 230, 231, 1, 297, 1, 298, 353, 380, 384, 385, 1, 490, 1, 491, 604, 615, 626, 627, 1

Offset: 1

Omar E. Pol, Mar 29 2018

Note that n is one of the partitions of n into equal parts.
If n is even then row n ending in [p(n) - 1, p(n)], where p(n) = A000041(n).
T(n,k) > p(n - 1), if 1 < k <= A000005(n).
Removing the 1's then all terms of the sequence are in increasing order.
If n is even then row n starts with [1, p(n - 1) + 1]. - David A. Corneth and Omar E. Pol, Aug 26 2018

			Triangle begins:
  1;
  1,   2;
  1,   3;
  1,   4,   5;
  1,   7;
  1,   8,  10,  11;
  1,  15;
  1,  16,  21,  22;
  1,  27,  30;
  1,  31,  41,  42;
  1,  56;
  1,  57,  69,  73,  76,  77;
  1, 101;
  1, 102, 134, 135;
  1, 160, 172, 176;
  ...
For n = 6 the partitions of 6 into equal parts are [1, 1, 1, 1, 1, 1], [2, 2, 2], [3, 3] and [6]. Then we have that in the list of colexicographically ordered partitions of 6 these partitions are in the rows 1, 8, 10 and 11 respectively as shown below, so the 6th row of the triangle is [1, 8, 10, 11].
-------------------------------------------------------------
   p      Diagram        Partitions of 6
-------------------------------------------------------------
        _ _ _ _ _ _
   1   |_| | | | | |    [1, 1, 1, 1, 1, 1]  <--- equal parts
   2   |_ _| | | | |    [2, 1, 1, 1, 1]
   3   |_ _ _| | | |    [3, 1, 1, 1]
   4   |_ _|   | | |    [2, 2, 1, 1]
   5   |_ _ _ _| | |    [4, 1, 1]
   6   |_ _ _|   | |    [3, 2, 1]
   7   |_ _ _ _ _| |    [5, 1]
   8   |_ _|   |   |    [2, 2, 2]  <--- equal parts
   9   |_ _ _ _|   |    [4, 2]
  10   |_ _ _|     |    [3, 3]  <--- equal parts
  11   |_ _ _ _ _ _|    [6]  <--- equal parts
.

Row n has length A000005(n).
Right border gives A000041, n >= 1.
Column 1 gives A000012.
Records give A317296.
Cf. A211992 (partitions in colexicographic order).
Cf. A027750, A135010, A141285, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A299474, A299475, A299773, A299775.

row(n) = {if(n == 1, return([1])); my(nd = numdiv(n), res = vector(nd)); res[1] = 1; res[nd] = numbpart(n); if(nd > 2, t = nd - 1; p = vecsort(partitions(n)); forstep(i = #p - 1, 2, -1, if(p[i][1] == p[i][#p[i]], res[t] = i; t--; if(t==1, return(res)))), return(res))} \\ David A. Corneth, Aug 17 2018

Terms a(46) and beyond from David A. Corneth, Aug 16 2018

A350333 Irregular triangle read by rows in which row n lists all elements of the arrangement of the correspondence divisor/part related to the partitions of n in the following order: row n lists the n-th row of A026792 followed by the n-th row of A338156.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 4, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 3, 2, 4, 1, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 2, 4, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 6, 3, 3, 4, 2, 2, 2, 2, 5, 1

Offset: 1

Omar E. Pol, Dec 25 2021

			Triangle begins:
[1], [1];
[2, 1, 1], [1, 2, 1];
[3, 2, 1, 1, 1, 1], [1, 3, 1, 2, 1, 1];
[4, 2, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1], [1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1];
...
Illustration of the first six rows of triangle in an infinite table:
.
|---|---------|-----|-------|---------|-----------|-------------|---------------|
| n |         |  1  |   2   |    3    |     4     |      5      |       6       |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
|   |         |     |       |         |           |             |  6            |
| P |         |     |       |         |           |             |  3 3          |
| A |         |     |       |         |           |             |  4 2          |
| R |         |     |       |         |           |             |  2 2 2        |
| T |         |     |       |         |           |  5          |  5 1          |
| I |         |     |       |         |           |  3 2        |  3 2 1        |
| T |         |     |       |         |  4        |  4 1        |  4 1 1        |
| I |         |     |       |         |  2 2      |  2 2 1      |  2 2 1 1      |
| O |         |     |       |  3      |  3 1      |  3 1 1      |  3 1 1 1      |
| N |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |  2 1 1 1 1    |
| S |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |  1 1 1 1 1 1  |
----|---------|-----|-------|---------|-----------|-------------|---------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |  1 2 3     6  |
|   | A027750 |     |  1    |  1 2    |  1   3    |  1 2   4    |  1       5    |
|   | A027750 |     |       |  1      |  1 2      |  1   3      |  1 2   4      |
|   | A027750 |     |       |  1      |  1 2      |  1   3      |  1 2   4      |
|   | A027750 |     |       |         |  1        |  1 2        |  1   3        |
| D | A027750 |     |       |         |  1        |  1 2        |  1   3        |
| I | A027750 |     |       |         |  1        |  1 2        |  1   3        |
| V | A027750 |     |       |         |           |  1          |  1 2          |
| I | A027750 |     |       |         |           |  1          |  1 2          |
| S | A027750 |     |       |         |           |  1          |  1 2          |
| O | A027750 |     |       |         |           |  1          |  1 2          |
| R | A027750 |     |       |         |           |  1          |  1 2          |
| S | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|   | A027750 |     |       |         |           |             |  1            |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
For n = 6 in the upper zone of the above table we can see the partitions of 6 in reverse-colexicographic order in accordance with the 6th row of A026792.
In the lower zone of the table we can see the terms from the 6th row of A338156, these are the divisors of the numbers from the 6th row of A176206.
Note that in the lower zone of the table every row gives A027750.
The total number of rows in the table is equal to A000070(6+1) = 30.
The remarkable fact is that the elements in the lower zone of the arrangement are the same as the elements in the upper zone but in other order.
For an explanation of the connection of the elements of the upper zone with the elements of the lower zone, that is the correspondence divisor/part, see A338156.
For n = 10 we can see a representation of the upper zone (the partitions) and of the lower zone (the divisors) with the two polycubes described in A221529 respectively: a prism of partitions and a tower whose terraces are the symmetric representation of sigma(m), for m = 1..10. Each polycube has A066186(10) = 420 cubic cells, hence the total number of cubic cells is equal to A220909(10) = 840, equaling the sum of the 10th row of this triangle.

Row sums give A220909.
Row lengths give A211978.
Cf. A350357 (analog for the last section of the set of partitions of n).
Cf. A000041, A000070, A006128, A026792, A027750, A066186, A135010, A176206, A221529, A237593, A336811, A336812, A338156, A340035.

A225597 Triangle read by rows: T(n,k) = total number of parts of all regions of the set of partitions of n whose largest part is k.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 3, 5, 1, 3, 4, 5, 7, 1, 4, 5, 7, 7, 11, 1, 4, 6, 8, 9, 11, 15, 1, 5, 7, 11, 10, 15, 15, 22, 1, 5, 9, 12, 13, 17, 19, 22, 30, 1, 6, 10, 16, 15, 22, 21, 29, 30, 42, 1, 6, 12, 18, 19, 25, 26, 32, 38, 42, 56, 1, 7, 14, 23, 22, 33, 29, 41, 42, 54, 56, 77

Offset: 1

Omar E. Pol, Aug 02 2013

For the definition of "region" see A206437.

T(n,k) is also the number of parts that end in the k-th column of the diagram of regions of the set of partitions of n (see Example section).

			For n = 5 and k = 3 the set of partitions of 5 contains two regions whose largest part is 3, they are third region which contains three parts [3, 1, 1] and the sixth region which contains only one part [3]. Therefore the total number of parts is 3 + 1 = 4, so T(5,3) = 4.
.
.    Diagram    Illustration of parts ending in column k:
.    for n=5      k=1   k=2     k=3       k=4        k=5
.   _ _ _ _ _                                  _ _ _ _ _
.  |_ _ _    |                _ _ _           |_ _ _ _ _|
.  |_ _ _|_  |               |_ _ _|  _ _ _ _       |_ _|
.  |_ _    | |          _ _          |_ _ _ _|        |_|
.  |_ _|_  | |         |_ _|  _ _ _      |_ _|        |_|
.  |_ _  | | |          _ _  |_ _ _|       |_|        |_|
.  |_  | | | |      _  |_ _|     |_|       |_|        |_|
.  |_|_|_|_|_|     |_|   |_|     |_|       |_|        |_|
.
k = 1 2 3 4 5
.
The 5th row lists:  1     3       4         5          7
.
Triangle begins:
1;
1,  2;
1,  2,  3;
1,  3,  3,  5;
1,  3,  4,  5,  7;
1,  4,  5,  7,  7, 11;
1,  4,  6,  8,  9, 11, 15;
1,  5,  7, 11, 10, 15, 15, 22;
1,  5,  9, 12, 13, 17, 19, 22, 30;
1,  6, 10, 16, 15, 22, 21, 29, 30, 42;
1,  6, 12, 18, 19, 25, 26, 32, 38, 42, 56;
1,  7, 14, 23, 22, 33, 29, 41, 42, 54, 56, 77;

Column 1 is A000012. Column 2 are the numbers => 2 of A008619. Row sums give A006128, n>=1. Right border gives A000041, n>=1. Second right border gives A000041, n>=1.

Cf. A006128, A133041, A135010, A138137, A139582, A141285, A182377, A186114, A186412, A187219, A193870, A194446, A206437, A207779, A211978, A220517, A225598, A225600, A225610.

A244968 Area between two valleys at height 0 under the infinite Dyck path related to partitions 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, multiplied by 2.

Original entry on oeis.org

1, 4, 9, 28, 54, 151

Offset: 1

Omar E. Pol, Nov 08 2014

			For k = 6, the diagram 1 represents the partitions of 6. 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
.      _ _ _ _ _ _                           _ _ _ _ _ _
. 11  |_ _ _      |       6                  _ _ _      |
. 10  |_ _ _|_    |       3+3                _ _ _|_    |
.  9  |_ _    |   |       4+2                _ _    |   |
.  8  |_ _|_ _|_  |       2+2+2              _ _|_ _|_  |
.  7  |_ _ _    | |       5+1                _ _ _    | |
.  6  |_ _ _|_  | |       3+2+1              _ _ _|_  | |
.  5  |_ _    | | |       4+1+1              _ _    | | |
.  4  |_ _|_  | | |       2+2+1+1            _ _|_  | | |
.  3  |_ _  | | | |       3+1+1+1            _ _  | | | |
.  2  |_  | | | | |       2+1+1+1+1          _  | | | | |
.  1  |_|_|_|_|_|_|       1+1+1+1+1+1         | | | | | |
.
Then we use the elements from the above diagram to draw an infinite Dyck path in which the j-th odd-indexed segment has A141285(j) up-steps and the j-th even-indexed segment has A194446(j) down-steps.
For the illustration of initial terms we use two opposite Dyck paths, as shown below:
11 ...........................................................
.                                                            /\
.                                                           /
.                                                          /
7 ..................................                      /
.                                  /\                    /
5 ....................            /  \                /\/
.                    /\          /    \          /\  /
3 ..........        /  \        /      \        /  \/
2 .....    /\      /    \    /\/        \      /
1 ..  /\  /  \  /\/      \  /            \  /\/
0  /\/  \/    \/          \/              \/
.  \/\  /\    /\          /\              /\
.     \/  \  /  \/\      /  \            /  \/\
.   1      \/      \    /    \/\        /      \
.      4            \  /        \      /        \  /\
.           9        \/          \    /          \/  \
.                                 \  /                \/\
.                    28            \/                    \
.                                                         \
.                                  54                      \
.                                                           \
.                                                            \/
.
The diagram is infinite. Note that the n-th largest peak between two valleys at height 0 is also the partition number A000041(n).
Calculations:
a(1) = 1.
a(2) = 2^2 = 4.
a(3) = 3^2 = 9.
a(4) = 2^2-1^2+5^2 = 4-1+25 = 28.
a(5) = 3^2-2^2+7^2 = 9-4+49 = 54.
a(6) = 2^2-1^2+5^2-3^2+6^2-5^2+11^2 = 4-1+25-9+36-25+121 = 151.

Cf. A000041, A135010, A141285, A193870, A194446, A194447, A206437, A211009, A211978, A220517, A225600, A225610, A228109, A228110, A228350, A230440, A233968.

A299775 Irregular triangle read by rows in which row n lists the indices of the partitions into consecutive parts in the list of colexicographically ordered partitions of n.

Original entry on oeis.org

1, 2, 2, 3, 5, 6, 7, 6, 11, 14, 15, 22, 25, 29, 30, 25, 42, 55, 56

Offset: 1

Omar E. Pol, Mar 29 2018

If n > 1 and n is odd then row n ending in [p(n) - 1, p(n)], where p(n) is A000041(n).

			Triangle begins:
   1;
   2;
   2,  3;
   5;
   6,  7;
   6, 11;
  14, 15;
  22;
  25, 29, 30;
  25, 42;
  55, 56;
...
For n = 9 the partitions of 9 into consecutive parts are [4, 3, 2], [5, 4] and [9]. Then we have that in the list of colexicographically ordered partitions of 9 these partitions are in the rows 25, 29 and 30 respectively as shown below, so the 9th row of the triangle is [25, 29, 30].
--------------------------------------------------------
   p         Diagram          Partitions of 9
--------------------------------------------------------
        1 2 3 4 5 6 7 8 9
        _ _ _ _ _ _ _ _ _
   1   |_| | | | | | | | |   [1, 1, 1, 1, 1, 1, 1, 1, 1]
   2   |_ _| | | | | | | |   [2, 1, 1, 1, 1, 1, 1, 1]
   3   |_ _ _| | | | | | |   [3, 1, 1, 1, 1, 1, 1]
   4   |_ _|   | | | | | |   [2, 2, 1, 1, 1, 1, 1]
   5   |_ _ _ _| | | | | |   [4, 1, 1, 1, 1, 1]
   6   |_ _ _|   | | | | |   [3, 2, 1, 1, 1, 1]
   7   |_ _ _ _ _| | | | |   [5, 1, 1, 1, 1]
   8   |_ _|   |   | | | |   [2, 2, 2, 1, 1, 1]
   9   |_ _ _ _|   | | | |   [4, 2, 1, 1, 1]
  10   |_ _ _|     | | | |   [3, 3, 1, 1, 1]
  11   |_ _ _ _ _ _| | | |   [6, 1, 1, 1]
  12   |_ _ _|   |   | | |   [3, 2, 2, 1, 1]
  13   |_ _ _ _ _|   | | |   [5, 2, 1, 1]
  14   |_ _ _ _|     | | |   [4, 3, 1, 1]
  15   |_ _ _ _ _ _ _| | |   [7, 1, 1]
  16   |_ _|   |   |   | |   [2, 2, 2, 2, 1]
  17   |_ _ _ _|   |   | |   [4, 2, 2, 1]
  18   |_ _ _|     |   | |   [3, 3, 2, 1]
  19   |_ _ _ _ _ _|   | |   [6, 2, 1]
  20   |_ _ _ _ _|     | |   [5, 3, 1]
  21   |_ _ _ _|       | |   [4, 4, 1]
  22   |_ _ _ _ _ _ _ _| |   [8, 1]
  23   |_ _ _|   |   |   |   [3, 2, 2, 2]
  24   |_ _ _ _ _|   |   |   [5, 2, 2]
  25   |_ _ _ _|     |   |   [4, 3, 2]   <--- Consecutive parts
  26   |_ _ _ _ _ _ _|   |   [7, 2]
  27   |_ _ _|     |     |   [3, 3, 3]
  28   |_ _ _ _ _ _|     |   [6, 3]
  29   |_ _ _ _ _|       |   [5, 4]   <--- Consecutive parts
  30   |_ _ _ _ _ _ _ _ _|   [9]   <--- Consecutive parts
.

Row n has length A001227(n).
Right border gives A000041, n >= 1.
Cf. A211992 (partitions in colexicographic order).
Cf. A299765 (partitions into consecutive parts).
For tables of partitions into consecutive parts see also A286000 and A286001.
Cf. A000041, A135010, A141285, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A299474, A299475, A299773, A299774.

A225599 Triangle read by rows: T(n,k) = sum of all parts that start in the k-th column of the diagram of regions of the set of partitions of n.

Original entry on oeis.org

1, 3, 1, 6, 1, 2, 12, 1, 4, 3, 20, 1, 4, 5, 5, 35, 1, 6, 8, 9, 7, 54, 1, 6, 10, 12, 11, 11, 86, 1, 8, 13, 20, 14, 19, 15, 128, 1, 8, 18, 23, 22, 25, 23, 22, 192, 1, 10, 21, 34, 30, 37, 29, 36, 30, 275, 1, 10, 26, 41, 41, 48, 41, 45, 46, 42, 399, 1, 12, 32, 56, 53, 72, 52, 67, 58, 66, 56

Offset: 1

Omar E. Pol, Aug 02 2013

For the construction of the diagram see A225600.

			For n = 5 and k = 3 the diagram of regions of the set of partitions of 5 contains three parts that start in the third column: two parts of size 1 and one part of size 2, therefore the sum of all parts that start in column 3 is 1 + 1 + 2 = 4, so T(5,3) = 4.
.
.                       Illustration of the parts
.    Diagram             that start in column k:
.    for n=5       k=1          k=2  k=3    k=4    k=5
.   _ _ _ _ _       _ _ _ _ _
.  |_ _ _    |     |_ _ _ _ _|               _ _
.  |_ _ _|_  |     |_ _ _|_                 |_ _|   _
.  |_ _    | |     |_ _ _ _|          _ _          |_|
.  |_ _|_  | |     |_ _|_            |_ _|   _     |_|
.  |_ _  | | |     |_ _ _|            _     |_|    |_|
.  |_  | | | |     |_ _|         _   |_|    |_|    |_|
.  |_|_|_|_|_|     |_|          |_|  |_|    |_|    |_|
.
k = 1 2 3 4 5
.
The 5th row lists:  20           1    4      5      5
.
Triangle begins:
1;
3,   1;
6,   1,  2;
12,  1,  4,  3;
20,  1,  4,  5,  5;
35,  1,  6,  8,  9,  7;
54,  1,  6, 10, 12, 11, 11;
86,  1,  8, 13, 20, 14, 19, 15;
128, 1,  8, 18, 23, 22, 25, 23, 22;
192, 1, 10, 21, 34, 30, 37, 29, 36, 30;
275, 1, 10, 26, 41, 41, 48, 41, 45, 46, 42;
399, 1, 12, 32, 56, 53, 72, 52, 67, 58, 66, 56;

Column 1-2: A006128, A000012. Row sums give A066186. Right border gives A000041.

Cf. A135010, A141285, A176572, A186114, A186412, A187219, A193870, A194446, A206437, A211978, A225598, A225600, A225610.

cp's OEIS Frontend

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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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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A225596 Sum of largest parts of all partitions of n plus n. Also, total number of parts in all partitions of n plus n.

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A225598 Triangle read by rows: T(n,k) = sum of all parts of all regions of the set of partitions of n whose largest part is k.

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A299774 Irregular triangle read by rows in which row n lists the indices of the partitions into equal parts in the list of colexicographically ordered partitions of n.

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A350333 Irregular triangle read by rows in which row n lists all elements of the arrangement of the correspondence divisor/part related to the partitions of n in the following order: row n lists the n-th row of A026792 followed by the n-th row of A338156.

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A225597 Triangle read by rows: T(n,k) = total number of parts of all regions of the set of partitions of n whose largest part is k.

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A244968 Area between two valleys at height 0 under the infinite Dyck path related to partitions 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, multiplied by 2.

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A299775 Irregular triangle read by rows in which row n lists the indices of the partitions into consecutive parts in the list of colexicographically ordered partitions of n.

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A225599 Triangle read by rows: T(n,k) = sum of all parts that start in the k-th column of the diagram of regions of the set of partitions of n.

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