A138137 -id:A138137 - cp's OEIS frontend

A194449 Largest part minus the number of parts > 1 in the n-th region of the set of partitions of j, if 1 <= n <= A000041(j).

1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 2, 2, 3, 3, 3, 1, 2, 2, 2, 4, 3, 1, 2, 3, 3, 3, 2, 4, 4, 1, 1, 2, 2, 2, 4, 3, 1, 3, 5, 5, 4, -2, 2, 3, 3, 3, 2, 4, 4, 1, 4, 3, 5, 6, 5, -3, 1, 2, 2, 2, 4, 3, 1, 3, 5, 5, 4, -2, 2, 4, 4, 5, 3, 6, 6, 5, -9

Offset: 1

Omar E. Pol, Dec 10 2011

Also triangle read by rows: T(j,k) = largest part minus the numbers of parts > 1 in the k-th region of the last section of the set of partitions of j. It appears that the sum of row j is equal to A000041(j-1). For the definition of "region" of the set of partitions of j see A206437. See also A135010.

			The 7th region of the shell model of partitions is [5, 2, 1, 1, 1, 1, 1]. The largest part is 5 and the number of parts > 1 is 2, so a(7) = 5 - 2 = 3 (see an illustration in the link section).
Written as an irregular triangle T(j,k) begins:
1;
1;
2;
1,2;
2,3;
1,2,2,2;
2,3,3,3;
1,2,2,2,4,3,1;
2,3,3,3,2,4,4,1;
1,2,2,2,4,3,1,3,5,5,4,-2;
2,3,3,3,2,4,4,1,4,3,5,6,5,-3;
1,2,2,2,4,3,1,3,5,5,4,-2,2,4,4,5,3,6,6,5,-9;

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

Row j has length A187219(j).

Cf. A000041, A135010, A138121, A138137, A138879, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A194447, A206437.

a(n) = A141285(n) - A194448(n).

A195821 Total number of parts that are not the smallest part in all partitions of n that do not contain 1 as a part.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 5, 7, 12, 19, 25, 37, 56, 72, 102, 138, 187, 246, 330, 422, 563, 721, 931, 1177, 1523, 1903, 2421, 3020, 3797, 4700, 5875, 7218, 8956, 10954, 13474, 16401, 20083, 24316, 29576, 35685, 43179, 51870, 62490, 74757, 89666, 106927, 127687

Offset: 1

Omar E. Pol, Oct 19 2011

nonn

Total number of parts that are not the smallest part in all partitions of the head of the last section of the set of partitions of n. For more information see A195820.

			For n = 8 the seven partitions of 8 that do not contain 1 as a part are:
.   8
.   4  +  4
.  (5) +  3
.  (6) +  2
.  (3) + (3) +  2
.  (4) +  2  +  2
.   2  +  2  +  2  +  2
Note that in every partition the parts that are not the smallest part are shown between parentheses. The total number of parts that are not the smallest part is 0+0+1+1+2+1+0 = 5, so a(8) = 5.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000041, A000070, A002865, A092269, A135010, A138121, A138135, A138137, A182984, A195820.

a(n) = A138135(n) - A195820(n) = A138137(n) - A195820(n) - A000041(n-1).

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

Original entry on oeis.org

1, 2, 4, 5, 6, 9, 10, 11, 12, 14, 16, 20, 21, 22, 23, 24, 25, 27, 30, 35, 36, 37, 38, 39, 40, 41, 42, 44, 46, 48, 50, 54, 57, 60, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 81, 84, 86, 91, 94, 98, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114

Offset: 1

Omar E. Pol, Apr 25 2012

This sequence shows the growth of the shell model of A135010 step by step. At stage n one part of size A135010(n) is added to the structure.

			Written as a triangle begins:
1;
2,  4;
5,  6, 9;
10,11,12,14,16,20;
21,22,23,24,25,27,30,35;
36,37,38,39,40,41,42,44,46,48,50,54,57,60,66;
67,68,69,70,71,72,73,74,75,76,77,79,81,84,86,91,94,98,105;

Partial sums of A135010. Row j has length A138137(j). Right border give A066186.

Cf. A211025, A182703, A206437.

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

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 02 2013

For the definition of "region" see A206437.

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

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

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

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

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

A182289 Triangle read by rows. Let p be one of the parts of size A135010(n,k) in one of the partitions of n and S(n,k) = sum of all preceding parts to p in the mentioned partition of n. So T(n,k) = 2*S(n,k) + A135010(n,k).

Original entry on oeis.org

1, 3, 2, 5, 5, 3, 7, 7, 7, 6, 2, 4, 9, 9, 9, 9, 9, 8, 3, 5, 11, 11, 11, 11, 11, 11, 11, 10, 6, 2, 10, 4, 9, 3, 6, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 12, 8, 3, 12, 5, 11, 4, 7, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 14, 10, 6

Offset: 1

Omar E. Pol, Aug 14 2012

Consider a physical model of the partitions of n in which each part p of size A135010(n,j) is represented by a right circular cylinder with radius j and height 2. T(n,k) is also the distance (or coordinate X) from the axis Y to the center of the base of cylinder of the part p in the structure of A135010.

			Written as an irregular triangle the sequence begins:
1;
3,2;
5,5,3;
7,7,7,6,2,4;
9,9,9,9,9,8,3,5;
11,11,11,11,11,11,11,10,6,2,10,4,9,3,6;
13,13,13,13,13,13,13,13,13,13,13,12,8,3,12,5,11,4,7;
15,15,15,15,15,15,15,15,15,15,15,15,15,15,15,14,10,6,2,14,10,4,14,9,3,14,6,13,5,10,4,8;

Row n starts with A000041(n-1) terms equal to A005408(n-1). Row n has length A138137(n). Right border gives A000027.

Cf. A135010.

A220487 Partial sums of triangle A206437.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 11, 15, 17, 18, 19, 20, 23, 28, 30, 31, 32, 33, 34, 35, 37, 41, 43, 46, 52, 55, 57, 59, 60, 61, 62, 63, 64, 65, 66, 69, 74, 76, 80, 87, 90, 92, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 107, 111, 113, 116, 122, 125, 127, 129, 134, 138

Offset: 1

Omar E. Pol, Jan 18 2013

			When written as an irregular triangle in which row j has length A194446(j) then the right border gives A182244. Also the records of row lengths give the partition numbers (A000041) of the positive integers as shown below:
1;
3, 4;
7, 8, 9;
11;
15,17,18,19,20;
23;
28,30,31,32,33,34,35;
37;
41,43;
46;
52,55,57,59,60,61,62,63,64,65,66;
69;
74,76;
80;
87,90,92,94,95,96,97,98,99,100,101,102,103,104,105;
...
Also when written as an irregular triangle in which row j has length A138137(j) then the right border gives A066186 as shown below:
1;
3, 4;
7, 8, 9;
11,15,17,18,19,20;
23,28,30,31,32,33,34,35;
37,41,43,46,52,55,57,59,60,61,62,63,64,65,66;
69,74,76,80,87,90,92,94,95,96,97,98,99,100,101,102,103,104,105;
...

Cf. A000041, A006128, A066186, A135010, A138137, A138879, A141285, A182181, A182244, A186412, A194446, A206437.

a(A182181(n)) = A182244(n), n >= 1.

a(A006128(n)) = A066186(n), n >= 1.

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.

cp's OEIS Frontend

A194449 Largest part minus the number of parts > 1 in the n-th region of the set of partitions of j, if 1 <= n <= A000041(j).

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A195821 Total number of parts that are not the smallest part in all partitions of n that do not contain 1 as a part.

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

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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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A182289 Triangle read by rows. Let p be one of the parts of size A135010(n,k) in one of the partitions of n and S(n,k) = sum of all preceding parts to p in the mentioned partition of n. So T(n,k) = 2*S(n,k) + A135010(n,k).

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A220487 Partial sums of triangle A206437.

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

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