A182703 -id:A182703 - cp's OEIS frontend

A211990 Triangle read by rows: T(n,k) = total number of regions in the last k shells of n.

1, 1, 2, 1, 2, 3, 2, 3, 4, 5, 2, 4, 5, 6, 7, 4, 6, 8, 9, 10, 11, 4, 8, 10, 12, 13, 14, 15, 7, 11, 15, 17, 19, 20, 21, 22, 8, 15, 19, 23, 25, 27, 28, 29, 30, 12, 20, 27, 31, 35, 37, 39, 40, 41, 42, 14, 26, 34, 41, 45, 49, 51, 53, 54, 55, 56, 21, 35, 47

Offset: 1

Omar E. Pol, Apr 27 2012

For the definition of "region of n" see A206437. For the definition of "last section of n" see A135010.

Apparently differs from A027300 at the right border.

			For n = 5 and k = 2 we have that the 4th shell of 5 contains two regions: [2] and [4,2,1,1,1]. Then we can see that the 5th shell of 5 contains two regions: [3] and [5,2,1,1,1,1,1]. Therefore the total number of regions in the last two shells of 5 is T(5,2) = 2+2 = 4 (see illustration in the link section).
Triangle begins:
1;
1,   2;
1,   2,  3;
2,   3,  4,  5;
2,   4,  5,  6,  7;
4,   6,  8,  9, 10, 11;
4,   8, 10, 12, 13, 14, 15;
7,  11, 15, 17, 19, 20, 21, 22;
8,  15, 19, 23, 25, 27, 28, 29, 30;
12, 20, 27, 31, 35, 37, 39, 40, 41, 42;
14, 26, 34, 41, 45, 49, 51, 53, 54, 55, 56;
21, 35, 47, 55, 62, 66, 70, 72, 74, 75, 76, 77;

Omar E. Pol, Illustration of the seven regions of 5

Mirror of triangle A211980. Column 1 is A187219. Right border gives A000041, n >= 1.

Cf. A027300, A135010, A138121, A182703, A186114, A206437, A212010, A212011.

T(n,k) = A000041(n) - A000041(n-k), if 1

T(n,k) = A000041(n), if k = n.

T(n,k) = Sum_{j=1..k} A187219(n-j+1,1).

A228524 Triangle read by rows: T(n,k) = total number of occurrences of parts k in the n-th section of the set of compositions (ordered partitions) of any integer >= n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 7, 3, 1, 1, 16, 7, 3, 1, 1, 36, 16, 7, 3, 1, 1, 80, 36, 16, 7, 3, 1, 1, 176, 80, 36, 16, 7, 3, 1, 1, 384, 176, 80, 36, 16, 7, 3, 1, 1, 832, 384, 176, 80, 36, 16, 7, 3, 1, 1, 1792, 832, 384, 176, 80, 36, 16, 7, 3, 1, 1, 3840, 1792, 832, 384, 176, 80, 36, 16, 7, 3, 1, 1

Offset: 1

Omar E. Pol, Aug 26 2013

Here, define "n-th section of the set of compositions of any integer >= n" to be the set formed by all parts that occur as a result of taking all compositions (ordered partitions) of n and then remove all parts of the compositions of n-1, if n >= 1. Hence the n-th section of the set of compositions of any integer >= n is also the last section of the set of compositions of n. Note that by definition the ordering of compositions is not relevant. For the visualization of the sections here we use a dissection of the diagram of compositions of n in colexicographic order, see example.
The equivalent sequence for partitions is A182703.
Row n lists the first n terms of A045891 in decreasing order.

			Illustration (using the colexicograpical order of compositions A228525) of the four sections of the set of compositions of 4, also the first four sections of the set of compositions of any integer >= 4:
.
.            1      2        3          4
.            _      _        _          _
.           |_|   _| |      | |        | |
.                |_ _|   _ _| |        | |
.                       |_|   |        | |
.                       |_ _ _|   _ _ _| |
.                                |_| |   |
.                                |_ _|   |
.                                |_|     |
.                                |_ _ _ _|
.
For n = 4 and k = 2, T(4,2) = 3 because there are 3 parts of size 2 in all compositions of 4, see below:
--------------------------------------------------------
.                         The last section    Number of
.   Composition of 4        of the set of      parts of
.                         compositions of 4     size k
. --------------------   -------------------
.            Diagram             Diagram    k = 1 2 3 4
. ------------------------------------------------------
.            _ _ _ _                    _
.  1+1+1+1  |_| | | |         1        | |      1 0 0 0
.    2+1+1  |_ _| | |         1        | |      1 0 0 0
.    1+2+1  |_|   | |         1        | |      1 0 0 0
.      3+1  |_ _ _| |         1   _ _ _| |      1 0 0 0
.    1+1+2  |_| |   |     1+1+2  |_| |   |      2 1 0 0
.      2+2  |_ _|   |       2+2  |_ _|   |      0 2 0 0
.      1+3  |_|     |       1+3  |_|     |      1 0 1 0
.        4  |_ _ _ _|         4  |_ _ _ _|      0 0 0 1
.                                              ---------
.                      Column sums give row 4:  7,3,1,1
.
Triangle begins:
1;
1,       1;
3,       1,   1;
7,       3,   1,   1;
16,      7,   3,   1,  1;
36,     16,   7,   3,  1,  1;
80,     36,  16,   7,  3,  1,   1;
176,    80,  36,  16,  7,  3,   1,  1;
384,   176,  80,  36, 16,  7,   3,  1,  1;
832,   384, 176,  80, 36, 16,   7,  3,  1,  1;
1792,  832, 384, 176, 80, 36,  16,  7,  3,  1, 1;
3840, 1792, 832, 384,176, 80,  36, 16,  7,  3, 1, 1;
8192, 3840,1792, 832,384,176,  80, 36, 16,  7, 3, 1, 1;
...

Row sums give A045623. Every column gives A045891.

Cf. A011782, A182703, A221876, A228350, A228366, A228367, A228370, A228526, A228527.

T(n,k) = A045891(n-k), n >= 1, 1<=k<=n.

A341049 Irregular triangle read by rows T(n,k) in which row n lists the terms of n-th row of A336811 in nondecreasing order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 04 2021

All divisors of all terms of n-th row are also all parts of the last section of the set of partitions of n.
All divisors of all terms of the first n rows are also all parts of all partitions of n. In other words: all divisors of the first A000070(n-1) terms of the sequence are also all parts of all partitions of n.
For further information about the correspondence divisor/part see A338156 and A336812.

			Triangle begins:
1;
2;
1, 3;
1, 2, 4;
1, 1, 2, 3, 5;
1, 1, 2, 2, 3, 4, 6;
1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 7;
1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 6, 8;
1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 7, 9;
...

Paolo Xausa, Table of n, a(n) for n = 1..11732 (rows 1..27 of triangle, flattened)

Mirror of A336811.
Row n has length A000041(n-1).
Row sums give A000070.
Right border gives A000027.
Cf. A000070, A000041, A002865, A027750, A028310, A133735, A135010, A138121, A138137, A176206, A182703, A187219, A207378, A237593, A336812, A338156, A339278, A340061.

A341049[rowmax_]:=Table[Flatten[Table[ConstantArray[n-m,PartitionsP[m]-PartitionsP[m-1]],{m,n-1,0,-1}]],{n,rowmax}];
A341049[10] (* Generates 10 rows *) (* Paolo Xausa, Feb 17 2023 *)

A341049(rowmax)=vector(rowmax,n,concat(vector(n,m,vector(numbpart(n-m)-numbpart(n-m-1),i,m))));
A341049(10) \\ Generates 10 rows - Paolo Xausa, Feb 17 2023

A185370 Triangle read by rows: T(n,k) is the number of occurrences of k in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 0, 1, 3, 1, 0, 1, 0, 0, 1, 5, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 7, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 11, 2, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 15, 4, 1, 1, 0, 0, 0, 1

Offset: 1

Omar E. Pol, Jan 25 2013

For the definition of "region of the set of partitions of j" see A206437.
T(n,k) is the number of occurrences of k in the n-th region of the shell model of partitions (see A135010).
T(n,k) is also the number of occurrences of k in the n-th row of triangles A186114, A193870, A206437 (and possibly more).
If the length of row n is a record then the length of row n is j and also A000041(j) = n.
If A000041(j) = n then the sum of the last A187219(j) elements of column k is A182703(j,k) and also the sum of all elements of column k is A066633(j,k).

			First seven regions of any integer >= 5 are
[1], [2,1], [3,1,1], [2], [4,2,1,1,1], [3], [5,2,1,1,1,1,1] (see illustrations, see also A206437). The 7th region contains five 1's, only one 2 and only one 5. There are no 3's. There are no 4's, so row 7 is [5, 1, 0, 0, 1].
-----------------------------------------
n    j  m    k : 1  2  3  4  5  6  7  8
-----------------------------------------
1    1  1        1;
2    2  1        1, 1;
3    3  1        2, 0, 1;
4    4  1        0, 1;
5    4  2        3, 1, 0, 1;
6    5  1        0, 0, 1;
7    5  2        5, 1, 0, 0, 1;
8    6  1        0, 1;
9    6  2        0, 1, 0, 1;
10   6  3        0, 0, 1;
11   6  4        7, 2, 1, 0, 0, 1;
12   7  1        0, 0, 1;
13   7  2        0, 1, 0, 0, 1;
14   7  3        0, 0, 0, 1;
15   7  4       11, 2, 1, 0, 0, 0, 1;
16   8  1        0, 1;
17   8  2        0, 1, 0, 1;
18   8  3        0, 0, 1;
19   8  4        0, 2, 1, 0, 0, 1;
20   8  5        0, 0, 0, 0, 1;
21   8  6        0, 0, 0, 1;
22   8  7       15, 4, 1, 1, 0, 0, 0, 1;

Omar E. Pol, Illustration of the seven regions of 5

Row n has length A141285(n). Row sums give A194446. Positive terms of column 1 give A000041.

Cf. A006128, A066633, A135010, A182703, A186114, A187219, A193870, A206437.

A210946 Triangle read by rows: T(n,k) = sum of parts in the k-th column of the mirror of the last section of the set of partitions of n with its parts aligned to the right margin.

Original entry on oeis.org

1, 3, 5, 9, 2, 12, 3, 20, 9, 2, 25, 11, 3, 38, 22, 9, 2, 49, 28, 14, 3, 69, 44, 26, 9, 2, 87, 55, 37, 14, 3, 123, 83, 62, 29, 9, 2, 152

Offset: 1

Omar E. Pol, Apr 21 2012

Row n lists the positive terms of the n-th row of triangle A210953 in decreasing order.

			For n = 7 the illustration shows two arrangements of the last section of the set of partitions of 7:
.
.       (7)        (7)
.     (4+3)        (3+4)
.     (5+2)        (2+5)
.   (3+2+2)        (2+2+3)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.                 ---------
.                  25,11,3
.
The left hand picture shows the last section of 7 with its parts aligned to the right margin. In the right hand picture (the mirror) we can see that the sum of all parts of the columns 1..3 are 25, 11, 3 therefore row 7 lists 25, 11, 3.
Written as a triangle begins:
1;
3;
5;
9,    2;
12,   3;
20,   9,  2;
25,  11,  3;
38,  22,  9,  2;
49,  28, 14,  3;
69,  44, 26,  9,  2;
87,  55, 37, 14,  3,
123, 83, 62, 29,  9,  2;

Column 1 is A046746, n >= 1. Row sums give A138879.

Cf. A135010, A138121, A182703, A194714, A196807, A206437, A207031, A207034, A207035, A210945, A210952, A210953.

A330242 Sum of largest emergent parts of the partitions of n.

Original entry on oeis.org

0, 0, 0, 2, 3, 9, 12, 24, 33, 54, 72, 112, 144, 210, 273, 379, 485, 661, 835, 1112, 1401, 1825, 2284, 2944, 3652, 4645, 5745, 7223, 8879, 11080, 13541, 16760, 20406, 25062, 30379, 37102, 44761, 54351, 65347, 78919, 94517, 113645, 135603, 162331, 193088, 230182, 272916, 324195, 383169, 453571

Offset: 1

Omar E. Pol, Dec 06 2019

nonn

In other words: a(n) is the sum of the largest parts of all partitions of n that contain emergent parts.
The partitions of n that contain emergent parts are the partitions that contain neither 1 nor n as a part. All parts of these partitions are emergent parts except the last part of every partition.
For the definition of emergent part see A182699.

			For n = 9 the diagram of
the partitions of 9 that
do not contain 1 as a part
is as shown below:           Partitions
.
    |_ _ _|   |   |   |      [3, 2, 2, 2]
    |_ _ _ _ _|   |   |      [5, 2, 2]
    |_ _ _ _|     |   |      [4, 3, 2]
    |_ _ _ _ _ _ _|   |      [7, 2]
    |_ _ _|     |     |      [3, 3, 3]
    |_ _ _ _ _ _|     |      [6, 3]
    |_ _ _ _ _|       |      [5, 4]
    |_ _ _ _ _ _ _ _ _|      [9]
.
Note that the above diagram is also the "head" of the last section of the set of partitions of 9, where the "tail" is formed by A000041(9-1)= 22 1's.
The diagram of the
emergent parts is as
shown below:                 Emergent parts
.
    |_ _ _|   |   |          [3, 2, 2]
    |_ _ _ _ _|   |          [5, 2]
    |_ _ _ _|     |          [4, 3]
    |_ _ _ _ _ _ _|          [7]
    |_ _ _|     |            [3, 3]
    |_ _ _ _ _ _|            [6]
    |_ _ _ _ _|              [5]
.
The sum of the largest emergent parts is 3 + 5 + 4 + 7 + 3 + 6 + 5 = 33, so a(9) = 33.

Cf. A000041, A002865, A006128, A135010, A138135, A138137, A141285, A182699, A182703, A182709, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A207031, A299474, A299475.

a(n) = A138137(n) - n.

a(n) = A207031(n,1) - n.

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cp's OEIS Frontend

A194711 Triangle read by rows: T(n,k) = number of partitions or zones in the last section of the set of partitions of n that contains k as a part.

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A210692 Number of parts that are visible in one of the three views of the shell model of partitions with n regions mentioned in A210991.

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A211030 Sum of all parts in the structure of the shell model of partitions of A135010 after n-th stage.

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A211990 Triangle read by rows: T(n,k) = total number of regions in the last k shells of n.

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A228524 Triangle read by rows: T(n,k) = total number of occurrences of parts k in the n-th section of the set of compositions (ordered partitions) of any integer >= n.

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A341049 Irregular triangle read by rows T(n,k) in which row n lists the terms of n-th row of A336811 in nondecreasing order.

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A185370 Triangle read by rows: T(n,k) is the number of occurrences of k in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

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A210946 Triangle read by rows: T(n,k) = sum of parts in the k-th column of the mirror of the last section of the set of partitions of n with its parts aligned to the right margin.

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A330242 Sum of largest emergent parts of the partitions of n.

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