cp's OEIS Frontend

Row n lists the partial sums of row n of triangle A138785.

This triangle is a condensed version of the more irregular triangle A340031.

For further information about the correspondence divisor/part see A338156.

For further information about the correspondence divisor/part see A338156.

This triangle is a condensed version of the more irregular triangle A338156 which is the main sequence with further information about the correspondence divisor/part.

Row n lists the partial sums of row n of triangle A066633.

It appears that T(n,k) is also the total number of parts >= k in all overpartitions of n.

It appears that the first differences of row n together with 2 give row n of triangle A235798.

The equivalent sequence for partitions is A181187.

Theorem: the k-th part of a partition in nonincreasing order of a positive integer equals the number of parts >= k of its conjugate partition.

Example: for n = 9 consider the partition [5, 3, 1]. The first part is 5, so the conjugate partition [3, 2, 2, 1, 1] has five parts >= 1. The second part is 3, so the conjugate partition has three parts >= 2. The third part is 1, so the conjugate partition has only one part >= 3. And vice versa, consider now the partition [3, 2, 2, 1, 1]. The first part is 3, so the conjugate partition [5, 3, 1] has three parts >= 1. The second part is 2, so the conjugate partition has two parts >= 2. The third part is 2, so the conjugate partition has two parts >= 3. The fourth part is 1, so the conjugate partition has only one part >= 4. The fifth part is 1, so the conjugate partition has only one part >= 5.

Corollary: the difference between the k-th part and the (k+1)-st part of a partition in nonincreasing order of a positive integer equals the number of k's in its conjugate partition.

Example: consider the partition [5, 3, 1]. The difference between the first and the second parts is 5 - 3 = 2, which equals the number of 1's in its conjugate partition [3, 2, 2, 1, 1]. The difference between the second and third parts is 3 - 1 = 2, which equals the number of 2's in its conjugate partition. The difference between the third part and the fourth (virtual) part is 1 - 0 = 1, which equals the number of 3's in its conjugate partition. And vice versa, consider the partition [3, 2, 2, 1, 1]. The difference between the first and second parts is 3 - 2 = 1, which equals the number of 1's in its conjugate partition [5, 3, 1]. The difference between the second and third parts is 2 - 2 = 0, which equals the number of 2's in its conjugate partition. The difference between the third and fourth parts is 2 - 1 = 1, which equals the number of 3's in its conjugate partition, and so on.

Self-conjugate partitions are included in all the above comments.

A proof without words is as shown below:

.

+------------------------+

| +--------------------+ |

| | +----------------+ | |

| | | | | |

v v v P2 FD k | | |

| | |

+--------> * * * 3 1 1 --+ | |

| +------> * * 2 0 2 | |

| +------> * * 2 1 3 ----+ |

| | +----> * 1 0 4 |

| | +----> * 1 1 5 ------+

| | |

| | | P1 5 3 1

| | |

| | | FD 2 2 1

| | |

| | | k 1 2 3

| | |

| | | | | |

| | +-------+ | |

| +-----------+ |

+---------------+

.

Every partition of n has n ranks.

The k-th rank of a partition is the k-th part minus the number of parts >= k.

In accordance with the above theorem, the k-th rank of a partition is also the number of parts >= k of its conjugate partition minus the number of parts >= k of the partition.

All ranks of a partition are zeros if and only if the partition is a self-conjugate partition.

The list of ranks of a partition of n equals the list of ranks multipled by -1 of its conjugate partition.

For example, the nine ranks of the partition [5, 3, 1] are [2, 1, -1, -1, -1, -1, 0, 0, 0], and the nine ranks of its conjugate partition [3, 2, 2, 1, 1] are [-2, -1, 1, 1, 1, 1, 0, 0, 0].

Note that the first rank coincides with the Dyson's rank because the first part of a partition is also the largest part, and the number of parts >= 1 is also the total number of parts.

In this triangle the partitions of n appears ordered by their first rank. The partitions that have the same first rank appears ordered by their second rank. The partitions that have the same first rank and the same second rank appears ordered by their third rank, and so on. The partitions that have all k-ranks equal zero appears ordered by their largest parts, then by their second largest parts, then by their third largest parts, and so on.

Note that a partition and its conjugate partition both are equidistants from the center of the list of partitions of n.

The first ranks of the partitions of this triangle give A330368.

For more information about the k-th ranks see A208478.

First differs from A080577 at a(48), and from A036037 at a(56), and from A181317 at a(105).

In row n, the sum of all odd-indexed terms minus the sum of all even-indexed terms is equal to A194714(n).

Reversed rows converge to A014153. - Alois P. Heinz, Feb 13 2012

Note that for n >= 2 the tail of the last section of n starts at the second column and the second column contains only one part of size 1, thus both the first and the second columns contain the same number of parts. For more information see A135010 and A182703.

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.