A225600 -id:A225600 - cp's OEIS frontend

A220517 First differences of A225600. Also A141285 and A194446 interleaved.

1, 1, 2, 2, 3, 3, 2, 1, 4, 5, 3, 1, 5, 7, 2, 1, 4, 2, 3, 1, 6, 11, 3, 1, 5, 2, 4, 1, 7, 15, 2, 1, 4, 2, 3, 1, 6, 4, 5, 1, 4, 1, 8, 22, 3, 1, 5, 2, 4, 1, 7, 4, 3, 1, 6, 2, 5, 1, 9, 30, 2, 1, 4, 2, 3, 1, 6, 4, 5, 1, 4, 1, 8, 7, 4, 1, 7, 2, 6, 1, 5, 1, 10, 42

Offset: 1

Omar E. Pol, Feb 07 2013

Number of toothpicks added at n-th stage to the toothpick structure (related to integer partitions) of A225600.

			Written as an irregular triangle in which row n has length 2*A187219(n) we can see that the right border gives A000041 and the previous term of the last term in row n is n.
1,1;
2,2;
3,3;
2,1,4,5;
3,1,5,7;
2,1,4,2,3,1,6,11;
3,1,5,2,4,1,7,15;
2,1,4,2,3,1,6,4,5,1,4,1,8,22;
3,1,5,2,4,1,7,4,3,1,6,2,5,1,9,30;
2,1,4,2,3,1,6,4,5,1,4,1,8,7,4,1,7,2,6,1,5,1,10,42;
.
Illustration of the first seven rows of triangle as a minimalist diagram of regions of the set of partitions of 7:
.      _ _ _ _ _ _ _
. 15   _ _ _ _      |
.      _ _ _ _|_    |
.      _ _ _    |   |
.      _ _ _|_ _|_  |
. 11   _ _ _      | |
.      _ _ _|_    | |
.      _ _    |   | |
.      _ _|_ _|_  | |
.  7   _ _ _    | | |
.      _ _ _|_  | | |
.  5   _ _    | | | |
.      _ _|_  | | | |
.  3   _ _  | | | | |
.  2   _  | | | | | |
.  1    | | | | | | |
.
.      1 2 3 4 5 6 7
.
Also using the elements of this diagram we can draw a 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 height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). See below:
.
7..................................
.                                 /\
5....................            /  \                /\
.                   /\          /    \          /\  /
3..........        /  \        /      \        /  \/
2.....    /\      /    \    /\/        \      /
1..  /\  /  \  /\/      \  /            \  /\/
0 /\/  \/    \/          \/              \/
. 0,2,  6,   12,         24,             40... = A211978
.  1, 4,   9,       19,           33... = A179862
.

Omar E. Pol, Visualization of regions in a minimalist diagram for A006128
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000041, A006128, A135010, A138137, A141285, A179862, A186114, A186412, A187219, A194446, A206437, A211978, A220517, A225600, A225610.

a(2n-1) = A141285(n); a(2n) = A194446(n), n >= 1

A141285 Largest part of the n-th partition of j in the list of colexicographically ordered partitions of j, if 1 <= n <= A000041(j).

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 2, 4, 3, 6, 3, 5, 4, 7, 2, 4, 3, 6, 5, 4, 8, 3, 5, 4, 7, 3, 6, 5, 9, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 6, 5, 9, 4, 8, 7, 6, 12

Offset: 1

Omar E. Pol, Aug 01 2008

Also largest part of the n-th region of the set of partitions of j, if 1 <= n <= A000041(j). For the definition of "region of the set of partitions of j" see A206437.
Also triangle read by rows: T(j,k) is the largest part of the k-th region in the last section of the set of partitions of j.
For row n >= 2 the rows of triangle are also the branches of a tree which is a projection of a three-dimensional structure of the section model of partitions of A135010, version tree. The branches of even rows give A182730. The branches of odd rows give A182731. Note that each column contains parts of the same size. It appears that the structure of A135010 is a periodic table of integer partitions. See also A210979 and A210980.
Also column 1 of: A193870, A206437, A210941, A210942, A210943. - Omar E. Pol, Sep 01 2013
Also row lengths of A211009. - Omar E. Pol, Feb 06 2014

			Written as a triangle T(j,k) the sequence begins:
  1;
  2;
  3;
  2, 4;
  3, 5;
  2, 4, 3, 6;
  3, 5, 4, 7;
  2, 4, 3, 6, 5, 4, 8;
  3, 5, 4, 7, 3, 6, 5, 9;
  2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10;
  3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8,  7, 6, 11;
  ...
  ------------------------------------------
  n  A000041                a(n)
  ------------------------------------------
   1 = p(1)                   1
   2 = p(2)                 2 .
   3 = p(3)                   . 3
   4                        2 .
   5 = p(4)               4   .
   6                          . 3
   7 = p(5)                   .   5
   8                        2 .
   9                      4   .
  10                    3     .
  11 = p(6)           6       .
  12                          . 3
  13                          .   5
  14                          .     4
  15 = p(7)                   .       7
  ...
From _Omar E. Pol_, Aug 22 2013: (Start)
Illustration of initial terms (n = 1..11) in three ways: as the largest parts of the partitions of 6 (see A026792), also as the largest parts of the regions of the diagram, also as the diagonal of triangle. By definition of "region" the largest part of the n-th region is also the largest part of the n-th partition (see below):
  --------------------------------------------------------
  .                  Diagram         Triangle in which
  Partitions       of regions       rows are partitions
  of 6           and partitions   and columns are regions
  --------------------------------------------------------
  .                _ _ _ _ _ _
  6                _ _ _      |                         6
  3+3              _ _ _|_    |                       3 3
  4+2              _ _    |   |                     4   2
  2+2+2            _ _|_ _|_  |                   2 2   2
  5+1              _ _ _    | |                 5       1
  3+2+1            _ _ _|_  | |               3 1       1
  4+1+1            _ _    | | |             4   1       1
  2+2+1+1          _ _|_  | | |           2 2   1       1
  3+1+1+1          _ _  | | | |         3   1   1       1
  2+1+1+1+1        _  | | | | |       2 1   1   1       1
  1+1+1+1+1+1       | | | | | |     1 1 1   1   1       1
  ...
The equivalent sequence for compositions is A001511. Explanation: for the positive integer j the diagram of regions of the set of compositions of j has 2^(j-1) regions. The largest part of the n-th region is A001511(n). The number of parts is A006519(n). On the other hand the diagram of regions of the set of partitions of j has A000041(j) regions. The largest part of the n-th region is a(n) = A001511(A228354(n)). The number of parts is A194446(n). Both diagrams have j sections. The diagram for partitions can be interpreted as one of the three views of a three dimensional diagram of compositions in which the rows of partitions are in orthogonal direction to the rest. For the first five sections of the diagrams see below:
  --------------------------------------------------------
  .          Diagram                           Diagram
  .         of regions                        of regions
  .      and compositions                   and partitions
  ---------------------------------------------------------
  .      j = 1 2 3 4 5                     j = 1 2 3 4 5
  ---------------------------------------------------------
   n  A001511                    A228354  a(n)
  ---------------------------------------------------------
   1   1     _| | | | | ............ 1    1    _| | | | |
   2   2     _ _| | | | ............ 2    2    _ _| | | |
   3   1     _|   | | |    ......... 4    3    _ _ _| | |
   4   3     _ _ _| | | ../  ....... 6    2    _ _|   | |
   5   1     _| |   | |    / ....... 8    4    _ _ _ _| |
   6   2     _ _|   | | ../ /   .... 12   3    _ _ _|   |
   7   1     _|     | |    /   /   . 16   5    _ _ _ _ _|
   8   4     _ _ _ _| | ../   /   /
   9   1     _| | |   |      /   /
  10   2     _ _| |   |     /   /
  11   1     _|   |   |    /   /
  12   3     _ _ _|   | ../   /
  13   1     _| |     |      /
  14   2     _ _|     |     /
  15   1     _|       |    /
  16   5     _ _ _ _ _| ../
  ...
Also we can draw an infinite Dyck path in which the n-th odd-indexed line segment has a(n) up-steps and the n-th even-indexed line segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two successive valleys at height 0 is also the partition number A000041(n). See below:
.                                 5
.                                 /\                 3
.                   4            /  \           4    /\
.                   /\          /    \          /\  /
.         3        /  \     3  /      \        /  \/
.    2    /\   2  /    \    /\/        \   2  /
. 1  /\  /  \  /\/      \  /            \  /\/
. /\/  \/    \/          \/              \/
.
.(End)

Alois P. Heinz, Table of n, a(n) for n = 1..4000
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of the seven regions of 5

Where records occur give A000041, n>=1. Column 1 is A158478. Row j has length A187219(j). Row sums give A138137. Right border gives A000027.

Cf. A000041, A135010, A182730, A182731, A182732, A182733, A182982, A182983, A182703, A193870, A194446, A194447, A194550, A206437, A210979, A210980, A211978, A220517, A225600, A225610.

Last/@DeleteCases[DeleteCases[Sort@PadRight[Reverse/@IntegerPartitions[13]], x_ /; x == 0, 2], {}] (* updated _Robert Price, May 15 2020 *)

a(n) = A001511(A228354(n)). - Omar E. Pol, Aug 22 2013

Edited by Omar E. Pol, Nov 28 2010

Better definition and edited by Omar E. Pol, Oct 17 2013

A225620 Indices of partitions in the table of compositions of A228351.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 14, 15, 16, 20, 24, 26, 28, 30, 31, 32, 36, 40, 42, 48, 52, 56, 58, 60, 62, 63, 64, 72, 80, 84, 96, 100, 104, 106, 112, 116, 120, 122, 124, 126, 127, 128, 136, 144, 160, 164, 168, 170, 192, 200, 208, 212, 224, 228, 232, 234, 240, 244, 248, 250, 252, 254, 255

Offset: 1

Omar E. Pol, Aug 03 2013

Also triangle read by rows in which T(n,k) is the decimal representation of a binary number whose mirror represents the k-th partition of n according with the list of juxtaposed reverse-lexicographically ordered partitions of the positive integers (A026792).
In order to construct this sequence as a triangle we use the following rules:
- In the list of A026792 we replace each part of size j of the k-th partition of n by concatenation of j - 1 zeros and only one 1.
- Then replace this new set of parts by the concatenation of its parts.
- Then replace this string by its mirror version which is a binary number.
T(n,k) is the decimal value of this binary number, which represents the k-th partition of n (see example).
The partitions of n are represented by a subsequence with A000041(n) integers starting with 2^(n-1) and ending with 2^n - 1, n >= 1. The odd numbers of the sequence are in A000225.
First differs from A065609 at a(23).
Conjecture: this sequence is a sorted version of b(n) where b(2^k) = 2^k for k >= 0, b(n) = A080100(n)*(2*b(A053645(n)) + 1) otherwise. - Mikhail Kurkov, Oct 21 2023

			T(6,8) = 58 because 58 in base 2 is 111010 whose mirror is 010111 which is the concatenation of 01, 01, 1, 1, whose number of digits are 2, 2, 1, 1, which are also the 8th partition of 6.
Illustration of initial terms:
The sequence represents a table of partitions (see below):
--------------------------------------------------------
.            Binary                        Partitions
n  k  T(n,k) number  Mirror   Diagram       (A026792)
.                                          1 2 3 4 5 6
--------------------------------------------------------
.                             _
1  1     1       1    1        |           1,
.                             _ _
1  1     2      10    01      _  |           2,
2  2     3      11    11       | |         1,1,
.                             _ _ _
3  1     4     100    001     _ _  |           3,
3  2     6     110    011     _  | |         2,1,
3  3     7     111    111      | | |       1,1,1,
.                             _ _ _ _
4  1     8    1000    0001    _ _    |           4,
4  2    10    1010    0101    _ _|_  |         2,2,
4  3    12    1100    0011    _ _  | |         3,1,
4  4    14    1110    0111    _  | | |       2,1,1,
4  5    15    1111    1111     | | | |     1,1,1,1,
.                             _ _ _ _ _
5  1    16   10000    00001   _ _ _    |           5,
5  2    20   10100    00101   _ _ _|_  |         3,2,
5  3    24   11000    00011   _ _    | |         4,1,
5  4    26   11010    01011   _ _|_  | |       2,2,1,
5  5    28   11100    00111   _ _  | | |       3,1,1,
5  6    30   11110    01111   _  | | | |     2,1,1,1,
5  7    31   11111    11111    | | | | |   1,1,1,1,1,
.                             _ _ _ _ _ _
6  1    32  100000    000001  _ _ _      |           6
6  2    36  100100    001001  _ _ _|_    |         3,3,
6  3    40  101000    000101  _ _    |   |         4,2,
6  4    42  101010    010101  _ _|_ _|_  |       2,2,2,
6  5    48  110000    000011  _ _ _    | |         5,1,
6  6    52  110100    001011  _ _ _|_  | |       3,2,1,
6  7    56  111000    000111  _ _    | | |       4,1,1,
6  8    58  111010    010111  _ _|_  | | |     2,2,1,1,
6  9    60  111100    001111  _ _  | | | |     3,1,1,1,
6  10   62  111110    011111  _  | | | | |   2,1,1,1,1,
6  11   63  111111    111111   | | | | | | 1,1,1,1,1,1,
.
Triangle begins:
  1;
  2,   3;
  4,   6,  7;
  8,  10, 12, 14, 15;
  16, 20, 24, 26, 28, 30, 31;
  32, 36, 40, 42, 48, 52, 56, 58, 60, 62, 63;
  ...
From _Gus Wiseman_, Apr 01 2020: (Start)
Using the encoding of A066099, this sequence ranks all finite nonempty multisets, as follows.
   1: {1}
   2: {2}
   3: {1,1}
   4: {3}
   6: {1,2}
   7: {1,1,1}
   8: {4}
  10: {2,2}
  12: {1,3}
  14: {1,1,2}
  15: {1,1,1,1}
  16: {5}
  20: {2,3}
  24: {1,4}
  26: {1,2,2}
  28: {1,1,3}
  30: {1,1,1,2}
  31: {1,1,1,1,1}
(End)

Column 1 is A000079. Row n has length A000041(n). Right border gives A000225.
Cf. A026792, A065609, A135010, A141285, A186114, A194446, A194546, A206437, A207779, A211978, A225600, A225610, A228351.
The case covering an initial interval is A333379 or A333380.
All of the following pertain to compositions in the order of A066099.
- The weakly increasing version is this sequence.
- The weakly decreasing version is A114994.
- The strictly increasing version is A333255.
- The strictly decreasing version is A333256.
- The unequal version is A233564.
- The equal version is A272919.
- The case covering an initial interval is A333217.
- Initial intervals are ranked by A164894.
- Reversed initial intervals are ranked by A246534.
Cf. A000120, A029931, A048793, A070939, A124766, A233249, A333219, A333220.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],LessEqual@@stc[#]&] (* Gus Wiseman, Apr 01 2020 *)

Conjecture: a(A000070(m) - k) = 2^m - A228354(k) for m > 0, 0 < k <= A000041(m). - Mikhail Kurkov, Oct 20 2023

A187219 Number of partitions of n that do not contain parts less than the smallest part of the partitions of n-1.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137, 165, 210, 253, 320, 383, 478, 574, 708, 847, 1039, 1238, 1507, 1794, 2167, 2573, 3094, 3660, 4378, 5170, 6153, 7245, 8591, 10087, 11914, 13959, 16424, 19196, 22519, 26252, 30701, 35717

Offset: 1

Omar E. Pol, Dec 09 2011

nonn

Essentially the same as A002865, but here a(1) = 1 not 0.
Also number of regions in the last section of the set of partitions of n.
Also number of partitions of n+k that are formed by k+1 sections, k >= 0 (Cf. A194799). - Omar E. Pol, Jan 30 2012
For the definition of region see A206437. - Omar E. Pol, Aug 13 2013
Partial sums give A000041, n >= 1. - Omar E. Pol, Sep 04 2013
Also the number of partitions of n with no parts greater than the number of ones. - Spencer Miller, Jan 28 2023

			From _Omar E. Pol_, Aug 13 2013: (Start)
Illustration of initial terms as number of regions:
.                                           _ _ _ _ _ _
.                                          |_ _ _      |
.                                          |_ _ _|_    |
.                                          |_ _    |   |
.                               _ _ _ _ _  |_ _|_ _|_  |
.                              |_ _ _    |           | |
.                     _ _ _ _  |_ _ _|_  |           | |
.                    |_ _    |         | |           | |
.             _ _ _  |_ _|_  |         | |           | |
.       _ _  |_ _  |       | |         | |           | |
.   _  |_  |     | |       | |         | |           | |
.  |_|   |_|     |_|       |_|         |_|           |_|
.
.   1     1       1         2           2             4
.
(End)

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
Omar E. Pol, Illustration of the seven regions of 5

Cf. A000041, A002865, A083751, A135010, A186114, A193870, A206437, A225600, A225610.

Join[{1}, Drop[CoefficientList[Series[1 / Product[(1 - x^k)^1, {k, 2, 50}], {x, 0, 50}], x], 2]] (* Vincenzo Librandi, Feb 15 2018 *)
A187219[nmax_]:=Join[{1},Differences[PartitionsP[Range[nmax]]]];
A187219[100] (* Paolo Xausa, Feb 17 2023 *)

a(n) = A083751(n) + 1. - Omar E. Pol, Mar 04 2012

a(n) = A002865(n), if n >= 2. - Omar E. Pol, Aug 13 2013

Better definition from Omar E. Pol, Sep 04 2013

A194446 Number of parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 2, 1, 15, 1, 2, 1, 4, 1, 1, 22, 1, 2, 1, 4, 1, 2, 1, 30, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 42, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 3, 1, 1, 56, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 1, 77, 1, 2, 1

Offset: 1

Omar E. Pol, Nov 26 2011

For the definition of "region" of the set of partitions of j, see A206437.
a(n) is also the number of positive integers in the n-th row of triangle A186114. a(n) is also the number of positive integers in the n-th row of triangle A193870.
Also triangle read by rows: T(j,k) = number of parts in the k-th region of the last section of the set of partitions of j. See example. For more information see A135010.
a(n) is also the length of the n-th vertical line segment in the minimalist diagram of regions and partitions. The length of the n-th horizontal line segment is A141285(n). See also A194447. - Omar E. Pol, Mar 04 2012
From Omar E. Pol, Aug 19 2013: (Start)
In order to construct this sequence with a cellular automaton we use the following rules: We start in the first quadrant of the square grid with no toothpicks. At stage n we place A141285(n) toothpicks of length 1 connected by their endpoints in horizontal direction starting from the point (0, n). Then we place toothpicks of length 1 connected by their endpoints in vertical direction starting from the exposed toothpick endpoint downward up to touch the structure or up to touch the x-axis. a(n) is the number of toothpicks in vertical direction added at n-th stage (see example section and A139250, A225600, A225610).
a(n) is also the length of the n-th descendent line segment in an infinite Dyck path in which the length of the n-th ascendent line segment is A141285(n). See Example section. For more information see A211978, A220517, A225600.
(End)
The equivalent sequence for compositions is A006519. - Omar E. Pol, Aug 22 2013

			Written as an irregular triangle the sequence begins:
  1;
  2;
  3;
  1, 5;
  1, 7;
  1, 2, 1, 11;
  1, 2, 1, 15;
  1, 2, 1,  4, 1, 1, 22;
  1, 2, 1,  4, 1, 2,  1, 30;
  1, 2, 1,  4, 1, 1,  7,  1, 2, 1, 1, 42;
  1, 2, 1,  4, 1, 2,  1,  8, 1, 1, 3,  1, 1, 56;
  1, 2, 1,  4, 1, 1,  7,  1, 2, 1, 1, 12, 1,  2, 1, 4, 1, 2, 1, 1, 77;
  ...
From _Omar E. Pol_, Aug 18 2013: (Start)
Illustration of initial terms (first seven regions):
.                                             _ _ _ _ _
.                                     _ _ _  |_ _ _ _ _|
.                           _ _ _ _  |_ _ _|       |_ _|
.                     _ _  |_ _ _ _|                 |_|
.             _ _ _  |_ _|     |_ _|                 |_|
.       _ _  |_ _ _|             |_|                 |_|
.   _  |_ _|     |_|             |_|                 |_|
.  |_|   |_|     |_|             |_|                 |_|
.
.   1     2       3     1         5       1           7
.
The next figure shows a minimalist diagram of the first seven regions. The n-th horizontal line segment has length A141285(n). a(n) is the length of the n-th vertical line segment, which is the vertical line segment ending in row n (see also A225610).
.      _ _ _ _ _
.  7   _ _ _    |
.  6   _ _ _|_  |
.  5   _ _    | |
.  4   _ _|_  | |
.  3   _ _  | | |
.  2   _  | | | |
.  1    | | | | |
.
.      1 2 3 4 5
.
Illustration of initial terms from an infinite Dyck path in which the length of the n-th ascendent line segment is A141285(n). a(n) is the length of the n-th descendent line segment.
.                                    /\
.                                   /  \
.                      /\          /    \
.                     /  \        /      \
.            /\      /    \    /\/        \
.       /\  /  \  /\/      \  / 1          \
.    /\/  \/    \/ 1        \/              \
.     1   2     3           5               7
.
(End)

Robert Price, Table of n, a(n) for n = 1..5603
Omar E. Pol, Illustration of the seven regions of 5

Row j has length A187219(j). Right border gives A000041, j >= 1. Records give A000041, j >= 1. Row sums give A138137.

Cf. A002865, A006128, A135010, A138121, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194447.

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
A194446 = {}; l = {};
For[j = 1, j <= 30, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[A194446, j - i];
  ];
A194446   (* Robert Price, Jul 25 2020 *)

a(n) = A141285(n) - A194447(n). - Omar E. Pol, Mar 04 2012

A194447 Rank of the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

0, 0, 0, 1, -1, 2, -2, 1, 2, 2, -5, 2, 3, 3, -8, 1, 2, 2, 2, 4, 3, -14, 2, 3, 3, 3, 2, 4, 4, -21, 1, 2, 2, 2, 4, 3, 1, 3, 5, 5, 4, -32, 2, 3, 3, 3, 2, 4, 4, 1, 4, 3, 5, 6, 5, -45, 1, 2, 2, 2, 4, 3, 1, 3, 5, 5, 4, -2, 2, 4, 4, 5, 3, 6, 6, 5, -65

Offset: 1

Omar E. Pol, Dec 04 2011

Here the rank of a "region" is defined to be the largest part minus the number of parts (the same idea as the Dyson's rank of a partition).
Also triangle read by rows: T(j,k) = rank of the k-th region of the last section of the set of partitions of j.
The sum of every row is equal to zero.
Note that in some rows there are several negative terms. - Omar E. Pol, Oct 27 2012
For the definition of "region" see A206437. See also A225600 and A225610. - Omar E. Pol, Aug 12 2013

			In the triangle T(j,k) for j = 6 the number of regions in the last section of the set of partitions of 6 is equal to 4. The first region given by [2] has rank 2-1 = 1. The second region given by [4,2] has rank 4-2 = 2. The third region given by [3] has rank 3-1 = 2. The fourth region given by [6,3,2,2,1,1,1,1,1,1,1] has rank 6-11 = -5 (see below):
From _Omar E. Pol_, Aug 12 2013: (Start)
---------------------------------------------------------
.    Regions       Illustration of ranks of the regions
---------------------------------------------------------
.    For J=6        k=1     k=2      k=3        k=4
.  _ _ _ _ _ _                              _ _ _ _ _ _
. |_ _ _      |                     _ _ _   .          |
. |_ _ _|_    |           _ _ _ _   * * .|    .        |
. |_ _    |   |     _ _   * * .  |              .      |
. |_ _|_ _|_  |     * .|        .|                .    |
.           | |                                     .  |
.           | |                                       .|
.           | |                                       *|
.           | |                                       *|
.           | |                                       *|
.           | |                                       *|
.           |_|                                       *|
.
So row 6 lists:     1       2         2              -5
(End)
Written as a triangle begins:
0;
0;
0;
1,-1;
2,-2;
1,2,2,-5;
2,3,3,-8;
1,2,2,2,4,3,-14;
2,3,3,3,2,4,4,-21;
1,2,2,2,4,3,1,3,5,5,4,-32;
2,3,3,3,2,4,4,1,4,3,5,6,5,-45;
1,2,2,2,4,3,1,3,5,5,4,-2,2,4,4,5,3,6,6,5,-65;
2,3,3,3,2,4,4,1,4,3,5,6,5,-3,3,5,5,4,5,4,7,7,6,-88;

Row j has length A187219(j). The absolute value of the last term of row j is A000094(j+1). Row sums give A000004.

Cf. A000041, A002865, A135010, A138121, A138137, A138879, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A206437.

a(n) = A141285(n) - A194446(n). - Omar E. Pol, Dec 05 2011

A211978 Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n.

Original entry on oeis.org

0, 2, 6, 12, 24, 40, 70, 108, 172, 256, 384, 550, 798, 1112, 1560, 2136, 2926, 3930, 5288, 6996, 9260, 12104, 15798, 20412, 26348, 33702, 43044, 54588, 69090, 86906, 109126, 136270, 169854, 210732, 260924, 321752, 396028, 485624, 594402, 725174, 883092, 1072208

Offset: 0

Omar E. Pol, Jan 03 2013

nonn

Also twice A006128, because the total number of parts in all partitions of n equals the sum of largest parts of all partitions of n. For a proof without words see the illustration of initial terms. Note that the sum of the lengths of all horizontal segments equals the sum of largest parts of all partitions of n. On the other hand, the sum of the lengths of all vertical segments equals the total number of parts of all partition of n. Therefore the sum of lengths of all horizontal segments equals the sum of lengths of all vertical segments.
a(n) is also the sum of the semiperimeters of the Ferrers boards of the partitions of n. Example: a(2)=6; indeed, the Ferrers boards of the partitions [2] and [1,1] of 2 are 2x1 rectangles; the sum of their semiperimeters is 3 + 3 = 6. - Emeric Deutsch, Oct 07 2016
a(n) is also the sum of the semiperimeters of the regions of the set of partitions of n. See the first illustration in the Example section. For more information see A278355. - Omar E. Pol, Nov 23 2016

			Illustration of initial terms as a minimalist diagram of regions of the set of partitions of n, for n = 1..6:
.                                         _ _ _ _ _ _
.                                         _ _ _      |
.                                         _ _ _|_    |
.                                         _ _    |   |
.                             _ _ _ _ _   _ _|_ _|_  |
.                             _ _ _    |  _ _ _    | |
.                   _ _ _ _   _ _ _|_  |  _ _ _|_  | |
.                   _ _    |  _ _    | |  _ _    | | |
.           _ _ _   _ _|_  |  _ _|_  | |  _ _|_  | | |
.     _ _   _ _  |  _ _  | |  _ _  | | |  _ _  | | | |
. _   _  |  _  | |  _  | | |  _  | | | |  _  | | | | |
.  |   | |   | | |   | | | |   | | | | |   | | | | | |
.
. 2    6     12        24         40          70
.
Also using the elements from the 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) as shown below:
.
11...........................................................
.                                                           /\
.                                                          /  \
.                                                         /    \
7..................................                      /      \
.                                 /\                    /        \
5....................            /  \                /\/          \
.                   /\          /    \          /\  /              \
3..........        /  \        /      \        /  \/                \
2.....    /\      /    \    /\/        \      /                      \
1..  /\  /  \  /\/      \  /            \  /\/                        \
0 /\/  \/    \/          \/              \/                            \
. 0,2,  6,   12,         24,             40,                          70...
.

Michael De Vlieger, Table of n, a(n) for n = 0..3000
Toufik Mansour, Armend Sh. Shabani, Enumerations on bargraphs, Discrete Math. Lett. (2019) Vol. 2, 65-94.
Omar E. Pol, Visualization of regions in a diagram for A006128
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139582 is twice A000041. A220909 is twice A066186.

Cf. A006128, A135010, A141285, A186114, A193870, A187219, A194446, A194447, A206437, A211026, A220517, A225600, A278355.

Q := sum(x^j/(1-x^j), j = 1 .. i): R := product(1-x^j, j = 1 .. i): g := sum(x^i*(1+i+Q)/R, i = 1 .. 100): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 41); # Emeric Deutsch, Oct 07 2016

Array[2 Sum[DivisorSigma[0, m] PartitionsP[# - m], {m, #}] &, 42, 0] (* Michael De Vlieger, Mar 20 2020 *)

a(n) = 2*A006128(n).
a(n) = A225600(2*A000041(n)) = A225600(A139582(n)), n >= 1.
a(n) = (Sum_{m=1..p(n)} A194446(m)) + (Sum_{m=1..p(n)} A141285(m)) = 2*Sum_{m=1..p(n)} A194446(m) = 2*Sum_{m=1..p(n)} A141285(m), where p(n) = A000041(n), n >= 1.
The trivariate g.f. G(t,s,x) of the partitions of a nonnegative integer relative to weight (marked by x), number of parts (marked by t), and largest part (marked by s) is G(t,s,x) = Sum_{i>=1} t*s^i*x^i/product_{j=1..i} (1-tx^j). Setting s = t, we obtain the bivariate g.f. of the partitions relative to weight (marked by x) and semiperimeter of the Ferrers board (marked by t). The g.f. of a(n) is g(x) = Sum_{i>=1} ((x^i*(1 + i + Q(x))/R(x)), where Q(x) = sum_{j=1..i} (x^j/(1 - x^j)) and R(x) = product_{j=1..i}(1-x^j). g(x) has been obtained by setting t = 1 in dG(t,t,x))/dt. - Emeric Deutsch, Oct 07 2016

A225610 Total number of parts in all partitions of n plus the sum of largest parts in all partitions of n plus the number of partitions of n plus n.

Original entry on oeis.org

1, 4, 10, 18, 33, 52, 87, 130, 202, 295, 436, 617, 887, 1226, 1709, 2327, 3173, 4244, 5691, 7505, 9907, 12917, 16822, 21690, 27947, 35685, 45506, 57625, 72836, 91500, 114760, 143143, 178235, 220908, 273268, 336670, 414041, 507298, 620455, 756398, 920470

Offset: 0

Omar E. Pol, Jul 29 2013

nonn

a(n) is also the total number of toothpicks in a toothpick structure which represents a diagram of regions of the set of partitions of n, n >= 1. The number of horizontal toothpicks is A225596(n). The number of vertical toothpicks is A093694(n). The difference between vertical toothpicks and horizontal toothpicks is A000041(n) - n = A000094(n+1). The total area (or total number of cells) of the diagram is A066186(n). The number of parts in the k-th region is A194446(k). The area (or number of cells) of the k-th region is A186412(k). For the definition of "region" see A206437. For a minimalist version of the diagram (which can be transformed into a Dyck path) see A211978. See also A225600.

			For n = 7 the total number of parts in all partitions of 7 plus the sum of largest parts in all partitions of 7 plus the number of partitions of 7 plus 7 is equal to A006128(7) + A006128(7) + A000041(7) + 7 = 54 + 54 + 15 + 7 = 130. On the other hand the number of toothpicks in the diagram of regions of the set of partitions of 7 is equal to 130, so a(7) = 130.
.                               Diagram of regions
Partitions of 7                 and partitions of 7
.                                   _ _ _ _ _ _ _
7                               15 |_ _ _ _      |
4 + 3                              |_ _ _ _|_    |
5 + 2                              |_ _ _    |   |
3 + 2 + 2                          |_ _ _|_ _|_  |
6 + 1                           11 |_ _ _      | |
3 + 3 + 1                          |_ _ _|_    | |
4 + 2 + 1                          |_ _    |   | |
2 + 2 + 2 + 1                      |_ _|_ _|_  | |
5 + 1 + 1                        7 |_ _ _    | | |
3 + 2 + 1 + 1                      |_ _ _|_  | | |
4 + 1 + 1 + 1                    5 |_ _    | | | |
2 + 2 + 1 + 1 + 1                  |_ _|_  | | | |
3 + 1 + 1 + 1 + 1                3 |_ _  | | | | |
2 + 1 + 1 + 1 + 1 + 1            2 |_  | | | | | |
1 + 1 + 1 + 1 + 1 + 1 + 1        1 |_|_|_|_|_|_|_|
.
.                                   1 2 3 4 5 6 7
.
Illustration of initial terms as the number of toothpicks in a diagram of regions of the set of partitions of n, for n = 1..6:
.                                         _ _ _ _ _ _
.                                        |_ _ _      |
.                                        |_ _ _|_    |
.                                        |_ _    |   |
.                             _ _ _ _ _  |_ _|_ _|_  |
.                            |_ _ _    | |_ _ _    | |
.                   _ _ _ _  |_ _ _|_  | |_ _ _|_  | |
.                  |_ _    | |_ _    | | |_ _    | | |
.           _ _ _  |_ _|_  | |_ _|_  | | |_ _|_  | | |
.     _ _  |_ _  | |_ _  | | |_ _  | | | |_ _  | | | |
. _  |_  | |_  | | |_  | | | |_  | | | | |_  | | | | |
.|_| |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
.
. 4    10     18       33         52          87

Omar E. Pol, Illustration of the seven regions of 5
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000041, A000094, A006128, A066186, A093694, A133041, A135010, A138137, A139250, A139582, A141285, A182377, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225596, A225600.

a(n) = 2*A006128(n) + A000041(n) + n = A211978(n) + A133041(n) = A093694(n) + A006128(n) + n = A093694(n) + A225596(n).

A228370 Toothpick sequence from a diagram of compositions of the positive integers (see Comments lines for definition).

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 8, 11, 15, 16, 17, 19, 21, 22, 23, 27, 35, 36, 37, 39, 41, 42, 43, 46, 50, 51, 52, 54, 56, 57, 58, 63, 79, 80, 81, 83, 85, 86, 87, 90, 94, 95, 96, 98, 100, 101, 102, 106, 114, 115, 116, 118, 120, 121, 122, 125, 129, 130, 131, 133, 135, 136, 137, 143, 175

Offset: 0

Omar E. Pol, Aug 21 2013

nonn

In order to construct this sequence we use the following rules:
We start in the first quadrant of the square grid with no toothpicks, so a(0) = 0.
If n is odd then at stage n we place the smallest possible number of toothpicks of length 1 connected by their endpoints in horizontal direction starting from the grid point (0, (n+1)/2) such that the x-coordinate of the exposed endpoint of the last toothpick is not equal to the x-coordinate of any outer corner of the structure.
If n is even then at stage n we place toothpicks of length 1 connected by their endpoints in vertical direction, starting from the exposed toothpick endpoint, downward up to touch the structure or up to touch the x-axis.
Note that the number of toothpick of added at stage (n+1)/2 in horizontal direction is also A001511(n) and the number of toothpicks added at stage n/2 in vertical direction is also A006519(n).
The sequence gives the number of toothpicks after n stages. A228371 (the first differences) gives the number of toothpicks added at the n-th stage.
After 2^k stages a new section of the structure is completed, so the structure can be interpreted as a diagram of the 2^(k-1) compositions of k in colexicographic order, if k >= 1 (see A228525). The infinite diagram can be interpreted as a table of compositions of the positive integers.
The equivalent sequence for partitions is A225600.

			For n = 32 the diagram represents the 16 compositions of 5. The structure has 79 toothpicks, so a(32) = 79. Note that the k-th horizontal line segment has length A001511(k) equals the largest part of the k-th region, and the k-th vertical line segment has length A006519(k) equals the number of parts of the k-th region.
----------------------------------------------------------
.                                    Triangle
Compositions                  of compositions (rows)
of 5          Diagram          and regions (columns)
----------------------------------------------------------
.            _ _ _ _ _
5            _        |                                 5
1+4          _|_      |                               1 4
2+3          _  |     |                             2   3
1+1+3        _|_|_    |                           1 1   3
3+2          _    |   |                         3       2
1+2+2        _|_  |   |                       1 2       2
2+1+2        _  | |   |                     2   1       2
1+1+1+2      _|_|_|_  |                   1 1   1       2
4+1          _      | |                 4               1
1+3+1        _|_    | |               1 3               1
2+2+1        _  |   | |             2   2               1
1+1+2+1      _|_|_  | |           1 1   2               1
3+1+1        _    | | |         3       1               1
1+2+1+1      _|_  | | |       1 2       1               1
2+1+1+1      _  | | | |     2   1       1               1
1+1+1+1+1     | | | | |   1 1   1       1               1
.
Illustration of initial terms (n = 1..16):
.
.                                   _        _
.                   _ _    _ _      _ _      _|_
.       _     _     _      _  |     _  |     _  |
.              |     |      | |      | |      | |
.
.       1      2     4      6        7        8
.
.
.                                            _ _
.                        _         _         _
.     _ _ _    _ _ _     _ _ _     _|_ _     _|_ _
.     _        _    |    _    |    _    |    _    |
.     _|_      _|_  |    _|_  |    _|_  |    _|_  |
.     _  |     _  | |    _  | |    _  | |    _  | |
.      | |      | | |     | | |     | | |     | | |
.
.       11       15        16        17        19
.
.
.                                _ _ _ _    _ _ _ _
.             _        _         _          _      |
.    _ _      _ _      _|_       _|_        _|_    |
.    _  |     _  |     _  |      _  |       _  |   |
.    _|_|_    _|_|_    _|_|_     _|_|_      _|_|_  |
.    _    |   _    |   _    |    _    |     _    | |
.    _|_  |   _|_  |   _|_  |    _|_  |     _|_  | |
.    _  | |   _  | |   _  | |    _  | |     _  | | |
.     | | |    | | |    | | |     | | |      | | | |
.
.      21       22       23        27          35
.

Cf. A001511, A005187, A006519, A006520, A011782, A139250, A187816, A187818, A206437, A225600, A228350, A228351, A228366, A228367, A228368, A228371, A228525, A228526.

def A228370(n): return sum(((m:=(i>>1)+1)&-m).bit_length() if i&1 else (m:=i>>1)&-m for i in range(1,n+1)) # Chai Wah Wu, Jul 14 2022

a(n) = sum_{k=1..n} A228371(k), n >= 1.
a(2n-1) = A005187(n) + A006520(n+1) - A006519(n), n >= 1.
a(2n) = A005187(n) + A006520(n+1), n >= 1.

A278355 a(n) = sum of the perimeters of the Ferrers boards of the partitions of n. Also, sum of the perimeters of the diagrams of the regions of the set of partitions of n.

Original entry on oeis.org

0, 4, 12, 24, 48, 80, 140, 216, 344, 512, 768, 1100, 1596, 2224, 3120, 4272, 5852, 7860, 10576, 13992, 18520, 24208, 31596, 40824, 52696, 67404, 86088, 109176, 138180, 173812, 218252, 272540, 339708, 421464, 521848, 643504, 792056, 971248, 1188804, 1450348, 1766184, 2144416, 2599164, 3141748, 3791248, 4563780

Offset: 0

Omar E. Pol, Nov 19 2016

nonn

a(n) is also 4 times the total number of parts in all partitions of n.
Hence a(n) is also 4 times the sum of largest parts of all partitions of n.
Hence a(n) is also twice the total number of parts in all partitions of n plus twice the sum of largest parts of all partitions of n.
a(n) is also the sum of the perimeters of the first n polygons constructed with the Dyck path (and its mirror) that arises from the minimalist diagram of the regions of the set of partitions of n. The n-th odd-indexed segment of the diagram has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. The k-th polygon of the diagram is associated to the k-th section of the set of partitions of n, with 1<=k<=n. See the bottom of Example section. For the definition of "section" see A135010. For the definition of "region" see A206437.

			For n = 5 consider the partitions of 5 in colexicographic order (as shown in the 5th row of the triangle A211992) and its associated diagram of regions as shown below:
.                                Regions            Minimalist
.         Partitions of 5        diagram             version
.                               _ _ _ _ _
.         1, 1, 1, 1, 1        |_| | | | |          _| | | | |
.         2, 1, 1, 1           |_ _| | | |          _ _| | | |
.         3, 1, 1              |_ _ _| | |          _ _ _| | |
.         2, 2, 1              |_ _|   | |          _ _|   | |
.         4, 1                 |_ _ _ _| |          _ _ _ _| |
.         3, 2                 |_ _ _|   |          _ _|     |
.         5                    |_ _ _ _ _|          _ _ _ _ _|
.
Then consider the following table which contains the Ferrers boards of the partitions of 5 and the diagram of every region of the set of partitions of 5:
-------------------------------------------------------------------------
| Partitions  |             |       |   Regions   |             |       |
|     of 5    |   Ferrers   | Peri- |     of 5    |   Region    | Peri- |
|(See A211992)|    board    | meter |(see A220482)|   diagram   | meter |
-------------------------------------------------------------------------
|                  _                |                 _                 |
|      1          |_|               |       1        |_|            4   |
|      1          |_|               |                   _               |
|      1          |_|               |       1         _|_|              |
|      1          |_|               |       2        |_|_|          8   |
|      1          |_|          12   |                     _             |
|                  _ _              |       1            |_|            |
|      2          |_|_|             |       1         _ _|_|            |
|      1          |_|               |       3        |_|_|_|       12   |
|      1          |_|               |                 _ _               |
|      1          |_|          12   |       2        |_|_|          6   |
|                  _ _ _            |                       _           |
|      3          |_|_|_|           |       1              |_|          |
|      1          |_|               |       1              |_|          |
|      1          |_|          12   |       1             _|_|          |
|                  _ _              |       2         _ _|_|_|          |
|      2          |_|_|             |       4        |_|_|_|_|     18   |
|      2          |_|_|             |                 _ _ _             |
|      1          |_|          10   |       3        |_|_|_|        8   |
|                  _ _ _ _          |                         _         |
|      4          |_|_|_|_|         |       1                |_|        |
|      1          |_|          12   |       1                |_|        |
|                  _ _ _            |       1                |_|        |
|      3          |_|_|_|           |       1                |_|        |
|      2          |_|_|        10   |       1               _|_|        |
|                  _ _ _ _ _        |       2         _ _ _|_|_|        |
|      6          |_|_|_|_|_|  12   |       5        |_|_|_|_|_|   24   |
|                                   |                                   |
-------------------------------------------------------------------------
|   Sum of perimeters:         80         <-- equals -->           80   |
-------------------------------------------------------------------------
The sum of the perimeters of the Ferrers boards is 12 + 12 + 12 + 10 + 12 + 10 + 12 = 80, so a(5) = 80.
On the other hand, the sum of the perimeters of the diagrams of regions is 4 + 8 + 12 + 6 + 18 + 8 + 24 = 80, equaling the sum of the perimeters of the Ferrers boards.
.
Illustration of first six polygons of an infinite diagram constructed with the boundary segments of the minimalist diagram of regions and its mirror (note that the diagram looks like reflections on a mountain lake):
11............................................................
.                                                            /\
.                                                           /  \
.                                                          /    \
7...................................                      /      \
.                                  /\                    /        \
5.....................            /  \                /\/          \
.                    /\          /    \          /\  /              \
3...........        /  \        /      \        /  \/                \
2.......   /\      /    \    /\/        \      /                      \
1...  /\  /  \  /\/      \  /            \  /\/                        \
0  /\/  \/    \/          \/              \/                            \
.  \/\  /\    /\          /\              /\                            /
.     \/  \  /  \/\      /  \            /  \/\                        /
.          \/      \    /    \/\        /      \                      /
.                   \  /        \      /        \  /\                /
.                    \/          \    /          \/  \              /
.                                 \  /                \/\          /
.                                  \/                    \        /
.                                                         \      /
.                                                          \    /
.                                                           \  /
.                                                            \/
n:
. 0 1  2   3          4             5                         6
Perimeter of the n-th polygon:
. 0 4  8  12         24            32                        60
a(n) is the sum of the perimeters of the first n polygons:
. 0 4 12  24         48            80                       140
.
For n = 5, the sum of the perimeters of the first five polygons is 4 + 8 + 12 + 24 + 32 = 80, so a(5) = 80.
For n = 6, the sum of the perimeters of the first six polygons is 4 + 8 + 12 + 24 + 32 + 60 = 140, so a(6) = 140.
For another version of the above diagram see A228109.

Cf. A000041, A006128, A135010, A138137, A139582, A141285, A194446, A211992, A220482, A225600, A211978, A233968, A244968.

a(n) = 4*A006128(n) = 2*A211978(n).
a(n) = 2*A225600(2*A000041(n)) = 2*A225600(A139582(n)), n >= 1.
a(n) = 2*((Sum_{m=1..p(n)} A194446(m)) + (Sum_{m=1..p(n)} A141285(m))) = 4*Sum_{m=1..p(n)} A194446(m) = 4*Sum_{m=1..p(n)} A141285(m), where p(n) = A000041(n), n >= 1.

cp's OEIS Frontend

A220517 First differences of A225600. Also A141285 and A194446 interleaved.

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A141285 Largest part of the n-th partition of j in the list of colexicographically ordered partitions of j, if 1 <= n <= A000041(j).

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A225620 Indices of partitions in the table of compositions of A228351.

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A187219 Number of partitions of n that do not contain parts less than the smallest part of the partitions of n-1.

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A194446 Number of parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

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A194447 Rank of the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

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A211978 Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n.

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A225610 Total number of parts in all partitions of n plus the sum of largest parts in all partitions of n plus the number of partitions of n plus n.

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