A207779 -id:A207779 - cp's OEIS frontend

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.

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).

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.

A299473 a(n) = 3*p(n), where p(n) is the number of partitions of n.

Original entry on oeis.org

3, 3, 6, 9, 15, 21, 33, 45, 66, 90, 126, 168, 231, 303, 405, 528, 693, 891, 1155, 1470, 1881, 2376, 3006, 3765, 4725, 5874, 7308, 9030, 11154, 13695, 16812, 20526, 25047, 30429, 36930, 44649, 53931, 64911, 78045, 93555, 112014, 133749, 159522, 189783, 225525, 267402, 316674, 374262, 441819, 520575, 612678

Offset: 0

Omar E. Pol, Feb 10 2018

nonn

For n >= 1, a(n) is also the number of vertices in the minimalist diagram of partitions of n, in which A139582(n) is the number of line segments and A000041(n) is the number of open regions (see example).

			Construction of a minimalist version of a modular table of partitions in which a(n) is the number of vertices of the diagram after n-th stage (n = 1..6):
-----------------------------------------------------------------------------------
n.........:    1     2       3         4           5           6   (stage)
A000041(n):    1     2       3         5           7          11   (open regions)
A139582(n):    2     4       6        10          14          22   (line segments)
a(n)......:    3     6       9        15          21          33   (vertices)
-----------------------------------------------------------------------------------
r     p(n)
-----------------------------------------------------------------------------------
.
1 .... 1 .... _|   _| |   _| | |   _| | | |   _| | | | |   _| | | | | |
2 .... 2 ......... _ _|   _ _| |   _ _| | |   _ _| | | |   _ _| | | | |
3 .... 3 ................ _ _ _|   _ _ _| |   _ _ _| | |   _ _ _| | | |
4                                  _ _|   |   _ _|   | |   _ _|   | | |
5 .... 5 ......................... _ _ _ _|   _ _ _ _| |   _ _ _ _| | |
6                                             _ _ _|   |   _ _ _|   | |
7 .... 7 .................................... _ _ _ _ _|   _ _ _ _ _| |
8                                                          _ _|   |   |
9                                                          _ _ _ _|   |
10                                                         _ _ _|     |
11 .. 11 ................................................. _ _ _ _ _ _|
.
The r-th horizontal line segment has length A141285(r).
The r-th vertical line segment has length A194446(r).
An infinite diagram is a minimalist table of all partitions of all positive integers.

Shawn A. Broyles, Table of n, a(n) for n = 0..1000

k times partition numbers: A000041 (k=1), A139582 (k=2), this sequence (k=3), A299474 (k=4).

Cf. A135010, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299475.

a(n) = 3*A000041(n) = A000041(n) + A139582(n).

a(n) = A299475(n) - 1, n >= 1.

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.

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

A299473 a(n) = 3*p(n), where p(n) is the number of partitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Comments

Examples

Crossrefs