A209918 -id:A209918 - cp's OEIS frontend

A209655 Tetrahedron in which the n-th slice is also one of the three views of the shell model of partitions of A207380 with n shells.

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 5, 4, 1, 2, 2, 1, 1, 2, 1, 1, 7, 6, 1, 4, 2, 1, 2, 3, 1, 1, 1, 2, 2, 1, 1

Offset: 1

Omar E. Pol, Mar 25 2012

Each slice of the tetrahedron is a triangle, thus the number of elements in the n-th slice is A000217(n). The slices are perpendicular to the slices of A026792. Each element of the n-th slice equals the volume of a column of the shell model of partitions with n shells. The sum of each row of the n-th slice is A000041(n). The sum of all elements of the n-th slice is A066186(n).
It appears that the triangle formed by the last row of each slice gives A008284 and A058398.
It appears that the triangle formed by the first column of each slice gives A058399.
Also consider a vertical rectangle on the infinite square grid with shorter side = n and longer side = p(n) = A000041(n). Each row of rectangle represents a partition of n. Each part of each partition of n is a horizontal rectangle with shorter side = 1 and longer side = k, where k is the size of the part. It appears that T(n,k,j) is also the number of k-th parts of all partitions of n in the j-th column of rectangle.

			--------------------------------------------------------
Illustration of first five
slices of the tetrahedron                       Row sum
--------------------------------------------------------
. 1,                                               1
.    2,                                            2
.    1, 1,                                         2
.          3,                                      3
.          2, 1,                                   3
.          1, 1, 1,                                3
.                   5,                             5
.                   4, 1,                          5
.                   2, 2, 1,                       5
.                   1, 2, 1, 1,                    5
.                               7,                 7
.                               6, 1,              7
.                               4, 2, 1,           7
.                               2, 3, 1, 1,        7
.                               1, 2, 2, 1, 1,     7
--------------------------------------------------------
. 1, 3, 1, 6, 2, 1,12, 5, 2, 1,20, 8, 4, 2, 1,
.
Written as a triangle begins:
1;
2, 1, 1;
3, 2, 1, 1, 1, 1;
5, 4, 1, 2, 2, 1, 1, 2, 1, 1;
7, 6, 1, 4, 2, 1, 2, 3, 1, 1, 1, 2, 2, 1, 1;
In which row sums give A066186.

Column sums give A181187. Main diagonal gives A210765. Another version is A209918.

Cf. A000041, A000217, A002260, A004736, A008284, A026792, A058398, A058399, A066186, A135010, A182703, A182715, A207380.

A210763 Tetrahedron T(j,n,k) in which the slice j is a finite triangle read by rows T(n,k) which lists the sums of the columns of the shell model of partitions with n shells.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 2, 2, 3, 1, 1, 2, 5, 1, 1, 2, 2, 2, 3, 2, 2, 3, 5, 1, 1, 1, 2, 7, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 3, 3, 4, 4, 7, 1, 1, 1, 2, 4, 11, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 4, 4, 5, 4, 7, 3, 3, 3, 5, 6, 11, 1, 1, 1, 1, 2, 4, 15

Offset: 1

Omar E. Pol, Apr 24 2012

			--------------------------------------------------------
Illustration of first five                      A210952
slices of the tetrahedron                       Row sum
--------------------------------------------------------
. 1,                                               1
.    1,                                            1
.    1, 2,                                         3
.          1,                                      1
.          1, 2,                                   3
.          1, 1, 3,                                5
.                   1,                             1
.                   1, 2,                          3
.                   2, 2, 3,                       7
.                   1, 1, 2, 5,                    9
.                               1,                 1
.                               1, 2,              3
.                               2, 2, 3,           7
.                               2, 2, 3, 5,       12
.                               1, 1, 1, 2, 7,    12
--------------------------------------------------------
. 1, 2, 2, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 7,
Each column sum in the slice j is equal to A000041(j).
.
Also this sequence can be written as a triangle read by rows in which each row is a flattened triangle. The sequence begins:
1;
1,1,2;
1,1,2,1,1,3;
1,1,2,2,2,3,1,1,2,5;
1,1,2,2,2,3,2,2,3,5,1,1,1,2,7;
1,1,2,2,2,3,3,3,3,5,3,3,4,4,7,1,1,1,2,4,11;
1,1,2,2,2,3,3,3,3,5,4,4,5,4,7,3,3,3,5,6,11,1,1,1,1,2,4,15;
Row n has length A000217(n). Row sums give A066186. Right border gives A000041(n), n >= 1.

Cf. A135010, A138121, A209655, A209918, A210952, A210960, A210961.

A210765 Triangle read by rows in which row n lists the number of partitions of n together with n-1 ones.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 22, 1, 1, 1, 1, 1, 1, 1, 30, 1, 1, 1, 1, 1, 1, 1, 1, 42, 1, 1, 1, 1, 1, 1, 1, 1, 1, 56, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 77, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 101, 1, 1, 1

Offset: 1

Omar E. Pol, Mar 26 2012

The sum of row n is S_n = n - 1 + A000041(n) = A133041(n) - 1.

Also consider a vertical rectangle on the infinite square grid with shorter side = n and longer side = p(n) = A000041(n). Each row of rectangle represents a partition of n. Each part of each partition of n is a horizontal rectangle with shorter side = 1 and longer side = k, where k is the size of the part. It appears that T(n,k) is also the number of k-th parts of all partitions of n in the k-th column of rectangle.

			Triangle begins:
1;
2,  1;
3,  1, 1;
5,  1, 1, 1;
7,  1, 1, 1, 1;
11, 1, 1, 1, 1, 1;
15, 1, 1, 1, 1, 1, 1;
22, 1, 1, 1, 1, 1, 1, 1;
30, 1, 1, 1, 1, 1, 1, 1, 1;
42, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Main diagonal of A209655 and of A209918.

Cf. A000041, A008284, A058399, A133041, A135010, A141285, A209656, A207380.

A210960 Tetrahedron T(j,n,k) in which the slice j is a finite triangle read by rows T(n,k) which list the number of parts in the columns of the shell model of partitions with n shells mentioned in A210970.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 3, 3, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 2, 1, 1, 3, 4, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Apr 22 2012

			--------------------------------------------------------
Illustration of first five
slices of the tetrahedron                       Row sum
--------------------------------------------------------
. 1,                                               1
.    1,                                            1
.    1, 1,                                         2
.          1,                                      1
.          1, 1,                                   2
.          1, 1, 1,                                3
.                   1,                             1
.                   1, 1,                          2
.                   2, 1, 1,                       4
.                   1, 2, 1, 1,                    5
.                               1,                 1
.                               1, 1,              2
.                               2, 1, 1,           4
.                               2, 2, 1, 1,        6
.                               1, 2, 2, 1, 1,     7
--------------------------------------------------------
. 1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 7, 6, 4, 2, 1,
.
It appears that column sums give A058399.
Also, written as a triangle read by rows in which each row is a flattened triangle, begins:
1;
1,1,1,
1,1,1,1,1,1;
1,1,1,2,1,1,1,2,1,1;
1,1,1,2,1,1,2,2,1,1,1,2,2,1,1;
1,1,1,2,1,1,3,2,1,1,3,3,2,1,1,1,3,3,2,1,1;
1,1,1,2,1,1,3,2,1,1,4,3,2,1,1,3,4,3,2,1,1,1,3,4,3,2,1,1;
In which row sums give A006128.

Cf. A058399, A135010, A138121, A182703, A209655, A209918, A210763, A210961.

cp's OEIS Frontend

A209655 Tetrahedron in which the n-th slice is also one of the three views of the shell model of partitions of A207380 with n shells.

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A210763 Tetrahedron T(j,n,k) in which the slice j is a finite triangle read by rows T(n,k) which lists the sums of the columns of the shell model of partitions with n shells.

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A210765 Triangle read by rows in which row n lists the number of partitions of n together with n-1 ones.

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Examples

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A210960 Tetrahedron T(j,n,k) in which the slice j is a finite triangle read by rows T(n,k) which list the number of parts in the columns of the shell model of partitions with n shells mentioned in A210970.

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