A194447 -id:A194447 - cp's OEIS frontend

A206441 Triangle read by rows. T(n,k) = number of distinct parts in the k-th region of the last section of the set of partitions of n.

1, 2, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 3, 1, 1, 5, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 1, 6

Offset: 1

Omar E. Pol, Feb 13 2012

a(n) is also the number of distinct parts in the n-th region of the shell model of partitions (see A135010 and A206437).

			The first region in the last section of the set of partitions of 6 looks like this:
.        **
There is only one part, so T(6,1) = 1.
The second region in the last section of the set of partitions of 6 looks like this:
.        ****
.          **
There are two distinct parts, so T(6,2) = 2.
The third region in the last section of the set of partitions of 6 looks like this:
.        ***
There is only one part, so T(6,3) = 1.
The 4th region in the last section of the set of partitions of 6 looks like this:
.        ******
.           ***
.            **
.            **
.             *
.             *
.             *
.             *
.             *
.             *
.             *
There are four distinct parts, so T(6,4) = 4.
Written as a triangle:
1;
2;
2;
1, 3;
1, 3;
1, 2, 1, 4;
1, 2, 1, 4;
1, 2, 1, 3, 1, 1, 5;
1, 2, 1, 3, 1, 2, 1, 5;
1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 1, 6;

Row n has length A187219(n).

Cf. A135010, A138121, A141285, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A194447.

A210451 n minus the number of parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j), with a(0) = 0.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 5, 0, 7, 7, 9, 0, 11, 11, 13, 0, 15, 15, 17, 15, 19, 20, 0, 22, 22, 24, 22, 26, 26, 28, 0, 30, 30, 32, 30, 34, 35, 30, 37, 37, 39, 40, 0, 42, 42, 44, 42, 46, 46, 48, 42, 50, 51, 50, 53, 54, 0, 56, 56, 58, 56, 60, 61, 56, 63, 63, 65, 66, 56, 68, 68, 70, 68, 72, 72, 74, 75, 0

Offset: 0

Omar E. Pol, Jan 27 2013

nonn

a(0) = 0 iff n is a partition number A000041, n >= 1.
a(n) is also the number of zeros in the n-th row of triangle A186114 and also of triangle A193870, n >= 1.
a(n) is also the value of the index "i" mentioned in the definition of "regions of the set of partitions" in A206437.

Cf. A135010, A141285, A194446, A186114, A193870, A194447, A206437.

a(n) = n - A194446(n) = n - A141285(n) + A194447(n).

a(n+1) = a(n+2) = n, if a(n) = 0 and n >= 7.

A211004 Number of distinct regions in the set of partitions of n.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51

Offset: 1

Omar E. Pol, Oct 22 2012

The number of regions in the set of partitions of n equals the number of partitions of n. The sequence counts only the distinct regions. For the definition of "regions of the set of partitions of n" (or more simply "regions of n") see A206437.

Is this the same as A001840 for all positive integers? If not, where is the first place these sequences differ?

			For n = 6 the 11 regions of 6 are [1], [2,1], [3,1,1], [2], [4,2,1,1,1], [3], [5,2,1,1,1,1,1], [2], [4,2], [3], [6,3,2,2,1,1,1,1,1,1,1]. These number are the first A006128(6) terms of triangle A206437 in which the first A000041(6) rows are the 11 regions of 6. We can see that the 8th region is equal to the 4th region: [2] = [2]. Also the 10th region is equal to the 6th region: [3] = [3]. There are two repeated regions, therefore a(6) = A000041(6) - 2 = 11 - 2 = 9.

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

Cf. A000041, A006128, A135010, A141285, A182244, A186114, A186412, A187219, A193870, A194446, A194447, A206437, A211009.

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

A206441 Triangle read by rows. T(n,k) = number of distinct parts in the k-th region of the last section of the set of partitions of n.

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A210451 n minus the number of parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j), with a(0) = 0.

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A211004 Number of distinct regions in the set of partitions of n.

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

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