A139582 -id:A139582 - cp's OEIS frontend

A299475 a(n) is the number of vertices in the diagram of partitions of n (see example).

1, 4, 7, 10, 16, 22, 34, 46, 67, 91, 127, 169, 232, 304, 406, 529, 694, 892, 1156, 1471, 1882, 2377, 3007, 3766, 4726, 5875, 7309, 9031, 11155, 13696, 16813, 20527, 25048, 30430, 36931, 44650, 53932, 64912, 78046, 93556, 112015, 133750, 159523, 189784, 225526, 267403, 316675, 374263, 441820, 520576, 612679

Offset: 0

Omar E. Pol, Feb 11 2018

nonn

For n >= 1, A299474(n) is the number of edges and A000041(n) is the number of regions in the mentioned diagram (see example and Euler's formula).

			Construction 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)
a(n)......:   4     7      10        16          22            34     (vertices)
A299474(n):   4     8      12        20          28            44     (edges)
A000041(n):   1     2       3         5           7            11     (regions)
--------------------------------------------------------------------------------
r     p(n)
--------------------------------------------------------------------------------
.             _    _ _    _ _ _    _ _ _ _    _ _ _ _ _    _ _ _ _ _ _
1 .... 1 ....|_|  |_| |  |_| | |  |_| | | |  |_| | | | |  |_| | | | | |
2 .... 2 .........|_ _|  |_ _| |  |_ _| | |  |_ _| | | |  |_ _| | | | |
3 .... 3 ................|_ _ _|  |_ _ _| |  |_ _ _| | |  |_ _ _| | | |
4                                 |_ _|   |  |_ _|   | |  |_ _|   | | |
5 .... 5 .........................|_ _ _ _|  |_ _ _ _| |  |_ _ _ _| | |
6                                            |_ _ _|   |  |_ _ _|   | |
7 .... 7 ....................................|_ _ _ _ _|  |_ _ _ _ _| |
8                                                         |_ _|   |   |
9                                                         |_ _ _ _|   |
10                                                        |_ _ _|     |
11 .. 11 .................................................|_ _ _ _ _ _|
.
Apart from the axis x, the r-th horizontal line segment has length A141285(r), equaling the largest part of the r-th region of the diagram.
Apart from the axis y, the r-th vertical line segment has length A194446(r), equaling the number of parts in the r-th region of the diagram.
The total number of parts equals the sum of largest parts.
Note that every diagram contains all previous diagrams.
An infinite diagram is a table of all partitions of all positive integers.

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

Cf. A000041, A135010, A139582, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299474.

a(n) = if (n==0, 1, 1+3*numbpart(n)); \\ Michel Marcus, Jul 15 2018

a(0) = 1; a(n) = 1 + 3*A000041(n), n >= 1.

a(n) = A299474(n) - A000041(n) + 1, n >= 1 (Euler's formula).

A179862 An unrestricted partition statistic: sum of A179864 over row n.

Original entry on oeis.org

1, 4, 9, 19, 33, 59, 93, 150, 226, 342, 494, 721, 1011, 1425, 1960, 2695, 3633, 4903, 6506, 8633, 11312, 14796, 19157, 24773, 31744, 40608, 51578, 65372, 82341, 103522, 129428, 161505, 200589, 248614, 306869, 378051, 463987, 568387, 693989, 845754, 1027625

Offset: 1

Alford Arnold, Aug 02 2010

nonn

Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n minus the number of partitions of n. - Omar E. Pol, Jul 15 2013

Sum of the hook-lengths of the (1,1)-cells of the Ferrers diagrams over all partitions of n. Example: a(3) = 9 because in each of the partitions 3, 21, and 111 the (1,1)-cell has hook-length 3. Comment follows at once from the previous comment. - Emeric Deutsch, Dec 20 2015

			From _Omar E. Pol_, Jul 15 2013: (Start)
Illustration of initial terms using 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). a(n) is the x-coordinate of the mentioned largest peak. Note that this Dyck path is infinite.
.
7..................................
.                                 /\
5....................            /  \                /\
.                   /\          /    \          /\  /
3..........        /  \        /      \        /  \/
2.....    /\      /    \    /\/        \      /
1..  /\  /  \  /\/      \  /            \  /\/
0 /\/  \/    \/          \/              \/
. 0,2,  6,   12,         24,             40... = A211978
.  1, 4,   9,       19,           33... = this sequence (End)

Cf. A179864.

a(n) = Sum_{k=1..A000041(n)} A179864(n,k).
a(n) = A211978(n) - A000041(n). - Omar E. Pol, Jul 15 2013
a(n) = A225600(A139582(n)-1), n>= 1. - Omar E. Pol, Jul 25 2013

More terms from Omar E. Pol, Jul 15 2013

A304789 Number T(n,k) of partitions of 2n whose Ferrers-Young diagram allows exactly k different domino tilings; triangle T(n,k), n>=0, 0<=k<=A304790(n), read by rows.

Original entry on oeis.org

0, 1, 0, 2, 0, 4, 1, 1, 6, 2, 2, 2, 10, 3, 4, 1, 2, 6, 14, 4, 6, 4, 4, 0, 2, 2, 12, 22, 5, 8, 7, 6, 2, 4, 4, 0, 0, 4, 1, 2, 25, 30, 6, 10, 12, 10, 4, 6, 6, 0, 2, 8, 2, 4, 0, 2, 0, 0, 4, 2, 0, 2, 46, 44, 7, 12, 17, 14, 8, 8, 8, 0, 4, 12, 5, 6, 0, 8, 2, 0, 8, 4, 0, 4, 0, 0, 0, 2, 2, 0, 0, 4, 1, 2, 0, 0, 2, 0, 1

Offset: 0

Alois P. Heinz, May 18 2018

			T(2,2) = 1: 22.
T(3,0) = 1: 321.
T(3,1) = 6: 111111, 21111, 3111, 411, 51, 6.
T(3,2) = 2: 2211, 42.
T(3,3) = 2: 222, 33.
T(8,36) = 1: 4444.
Triangle T(n,k) begins:
   0,  1;
   0,  2;
   0,  4, 1;
   1,  6, 2,  2;
   2, 10, 3,  4,  1,  2;
   6, 14, 4,  6,  4,  4, 0, 2, 2;
  12, 22, 5,  8,  7,  6, 2, 4, 4, 0, 0, 4, 1, 2;
  25, 30, 6, 10, 12, 10, 4, 6, 6, 0, 2, 8, 2, 4, 0, 2, 0, 0, 4, 2, 0, 2;

Alois P. Heinz, Rows n = 0..20, flattened
Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Domino
Wikipedia, Domino tiling
Wikipedia, Ferrers diagram
Wikipedia, Mutilated chessboard problem
Wikipedia, Partition (number theory)
Wikipedia, Young tableau, Diagrams
Index entries for sequences related to dominoes

Columns k=0-1 give: A304710, A139582(n) = 2*A000041(n) for n>0.
Row sums give A058696(n) or A000041(2n).
Cf. A000290, A000712, A001105, A048574, A052837, A304662, A304790.

h:= proc(l, f) option remember; local k; if min(l[])>0 then
     `if`(nops(f)=0, 1, h(map(u-> u-1, l[1..f[1]]), subsop(1=[][], f)))
    else for k from nops(l) while l[k]>0 by -1 do od;
        `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f), 0)+
        `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)
      fi
    end:
g:= l-> x^`if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
          `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
b:= (n, i, l)-> `if`(n=0 or i=1, g([l[], 1$n]), b(n, i-1, l)
                  +b(n-i, min(n-i, i), [l[], i])):
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(2*n$2, [])):
seq(T(n), n=0..11);

Sum_{k>0} k * T(n,k) = A304662(n).
T(n,A304790(n)) = 1 for n in { A001105 }.
Sum_{k>=0} T(n,k) = A058696(n) = A000041(2n).
Sum_{k>=1} T(n,k) = A000712(n).
Sum_{k>=2} T(n,k) = A048574(n) = A052837(n).

A211026 Number of segments needed to draw (on the infinite square grid) a diagram of regions and partitions of n.

Original entry on oeis.org

4, 6, 8, 12, 16, 24, 32, 46, 62, 86, 114, 156, 204, 272, 354, 464, 596, 772, 982, 1256, 1586, 2006, 2512, 3152, 3918, 4874, 6022, 7438, 9132, 11210, 13686, 16700, 20288, 24622, 29768, 35956, 43276, 52032, 62372, 74678, 89168, 106350

Offset: 1

Omar E. Pol, Oct 29 2012

nonn

On the infinite square grid the diagram of regions of the set of partitions of n is represented by a rectangle with base = n and height = A000041(n). The rectangle contains n shells. Each shell contains regions. Each row of a region is a part. Each part of size k contains k cells. The number of regions equals the number of partitions of n (see illustrations in the links section). For a minimalist version see A139582. For the definition of "region of n" see A206437.

Omar E. Pol, Illustration of initial terms of A211026 and of A066186
Omar E. Pol, Illustration of the seven regions of 5

Cf. A000041, A052810, A135010, A139582, A141285, A186412, A186114, A187219, A193870, A194446, A194447, A206437, A211009

a(n) = 2*A000041(n) + 2 = 2*A052810(n) = A139582(n) + 2.

a(18) corrected by Georg Fischer, Apr 11 2024

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.

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).

A262445 Number of exact 3-colored partitions such that no adjacent parts have the same color.

Original entry on oeis.org

0, 0, 0, 6, 24, 72, 186, 438, 990, 2142, 4560, 9492, 19620, 40068, 81534, 164892, 332808, 669528, 1345554, 2699448, 5412636, 10843038, 21714972, 43467342, 86995428, 174069306, 348265164, 696694692, 1393652298, 2787646380, 5575837836, 11152384044, 22305891948, 44613248352, 89228806704, 178460625402, 356925987924

Offset: 0

Ran Pan, Sep 23 2015

nonn

a(1) = a(2) = 0 because we need to use exactly three colors, which means the number of parts should be greater than two.

All terms are multiples of 6.

			a(3)=6 because there are three partitions of 3 and there are no ways to color [3] or [2,1] but there are six ways to color [1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Ran Pan, A note on enumerating colored integer partitions, arXiv:1509.06107 [math.CO], 2015.
Ran Pan, Exercise S, Project P.

Cf. A000041, A262444, A139582, A262495, A320545.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
a:= n-> `if`(n=0, 0, b(n$2, 2)/2*3-6*b(n$2, 1)+3):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 23 2015

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]]; a[n_] := If[n == 0, 0, b[n, n, 2]/2*3 - 6*b[n, n, 1] + 3]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)

G.f.: 3/2*Product_{k>=1} (1/(1-2*x^k)) - 6*Product_{k>=1} (1/(1-x^k)) + 3/(1-x) + 3/2.
a(n) = A262444(n) - 6*A000041(n) + 3, for n >= 1.
a(n) = 6 * A262495(n,3). - Alois P. Heinz, Sep 24 2015

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.

cp's OEIS Frontend

A299475 a(n) is the number of vertices in the diagram of partitions of n (see example).

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A179862 An unrestricted partition statistic: sum of A179864 over row n.

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A304789 Number T(n,k) of partitions of 2n whose Ferrers-Young diagram allows exactly k different domino tilings; triangle T(n,k), n>=0, 0<=k<=A304790(n), read by rows.

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A211026 Number of segments needed to draw (on the infinite square grid) a diagram of regions and 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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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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A262445 Number of exact 3-colored partitions such that no adjacent parts have the same color.

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