A194436 -id:A194436 - cp's OEIS frontend

A194448 Number of parts > 1 in the n-th region of the shell model of partitions.

0, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 3, 1, 1, 14, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 1, 21, 1, 2, 1, 4, 1, 2

Offset: 1

Omar E. Pol, Dec 10 2011

Also triangle read by rows: T(n,k) = number of parts > 1 in the k-th region of the last section of the set of partitions of n.

			Written as a triangle:
0;
1;
1;
1,2;
1,2;
1,2,1,4;
1,2,1,4;
1,2,1,4,1,1,7;
1,2,1,4,1,2,1,8;
1,2,1,4,1,1,7,1,2,1,1,12;
1,2,1,4,1,2,1,8,1,1,3,1,1,14;
1,2,1,4,1,1,7,1,2,1,1,12,1,2,1,4,1,2,1,1,21;

Row sums give A138135. Row n has length A187219(n).

Cf. A000041, A002865, A135010, A138121, A138137, A138879, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A194447, A194449.

A210942 Triangle read by rows in which row n lists the parts > 1 of the n-th region of the shell model of partitions, with a(1) = 1.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 18 2012

For the definition of "region of n" see A206437. See also A186114. Row n lists the largest part and the parts > 1 of the n-th region of the shell model of partitions. Also 1 together with the numbers > 1 of A206437.

			Written as a triangle begins:
1;
2;
3;
2;
4,2;
3;
5,2,
2;
4,2;
3;
6,3,2,2;
3;
5,2;
4;
7,3,2,2;

Omar E. Pol, Illustration of the seven regions of 5
Omar E. Pol, Illustration of the partitions and regions of n, for n = 1..12

Column 1 is A141285. Records give A000027. The n-th record is T(A000041(n),1).

Cf. A135010, A138121, A182699, A182709, A183152, A186114, A187219, A194436-A194439, A194447-A194448, A196025, A198381, A206437, A210941.

A182377 Total sum of positive ranks of all regions in the last shell of n.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 8, 14, 21, 32, 45, 67, 91

Offset: 1

Omar E. Pol, Apr 29 2012

The rank of a region of n is the largest part minus the number of parts. For the definition of "region of n" see A206437. For the definition of "last shell of n" see A135010.

a(n) is also the sum of positive integers in row n of triangle A194447. First differs from A000094 at a(12).

			For n = 7 the last shell of 7 contains four regions: [3],[5,2],[4],[7,3,2,2,1,1,1,1,1,1,1,1,1,1,1] so we have:
----------------------------------------------------------
.        Largest    Number
Region     part    of parts    Rank
----------------------------------------------------------
.  1        3         1          2
.  2        5         2          3
.  3        4         1          3
.  4        7        15         -8
.
The sum of positive ranks is a(7) = 2 + 3 + 3 = 8.

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

Cf. A000094, A135010, A138121, A182699, A182709, A183152, A186114, A187219, A194436-A194439, A194447-A194448, A196025, A206437, A209616.

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).

Original entry on oeis.org

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).

A194799 Triangle read by rows: T(n,k) = number of partitions of n that are formed by k sections, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 4, 2, 2, 1, 1, 1, 4, 4, 2, 2, 1, 1, 1, 7, 4, 4, 2, 2, 1, 1, 1, 8, 7, 4, 4, 2, 2, 1, 1, 1, 12, 8, 7, 4, 4, 2, 2, 1, 1, 1, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 1, 21, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 1, 24, 21, 14

Offset: 1

Omar E. Pol, Jan 30 2012

Also triangle read by rows in which row n lists the first k terms of A187219 in reverse order. For more information see A135010.

			Triangle begins:
1,
1,1,
1,1,1,
2,1,1,1,
2,2,1,1,1,
4,2,2,1,1,1,
4,4,2,2,1,1,1,
7,4,4,2,2,1,1,1,
8,7,4,4,2,2,1,1,1,
12,8,7,4,4,2,2,1,1,1,
14,12,8,7,4,4,2,2,1,1,1,
21,14,12,8,7,4,4,2,2,1,1,1

Columns are A187219. Row sums give A000041, n >= 1.

Cf. A002865, A116598, A135010, A138121, A182748, A186114, A193870, A194436-A194438, A194446.

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.

Original entry on oeis.org

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.

cp's OEIS Frontend

A194448 Number of parts > 1 in the n-th region of the shell model of partitions.

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A210942 Triangle read by rows in which row n lists the parts > 1 of the n-th region of the shell model of partitions, with a(1) = 1.

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A182377 Total sum of positive ranks of all regions in the last shell of n.

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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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Formula

A194799 Triangle read by rows: T(n,k) = number of partitions of n that are formed by k sections, k >= 1.

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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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