cp's OEIS Frontend

First differs from the revlex (instead of colex) version for partitions of 12.

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

First differs from A036036 for partitions of 6.

First differs from A334442 for partitions of 6.

Also reversed partitions in reverse-colexicographic order.

The ordering of reversed partitions is first by sum, then by length, and finally lexicographically. The version for non-reversed partitions is A335123.

Differs from A339195 in having 77 before 66.

A proper definition is needed for this sequence.

Are the row sums A074139(n) and the row lengths A000041(n)? - R. J. Mathar, May 08 2019 [Not exactly: see below. - M. F. Hasler, Jan 07 2024]

From M. F. Hasler, Jan 06 2024: (Start)

I get this triangle as T(n,k) = # { v in S(p_n), |v| = k }, where p_n is the n-th partition as listed in A036036 or A036037 (which has a nice table of the p's), and S(p) = {0, ..., p[1]} x ... x {0, ..., p[#p]}, the set of vectors v with 0 <= v[i] <= p[i] for all indices i from 1 to #p = number of parts in p.

Then the row sums are indeed the total number of elements in S(p_n) which is equal to the product (p[1]+1)*...*(p[#p]+1) which is also the number of divisors of the Heinz number of p (cf. A185974).

The row lengths are 1 + |p| = 1 + sum of all parts of p (corresponding to the possible values of |v| ranging from 0 to |p|), repeated A000041(|p|) times: A000041(0) = 1 row of length 0+1 for the partition () of 0, A000041(1) = 1 row of length 1+1 for partition (1) of 1; A000041(2) = 2 rows of length 2+1 for the two partitions (2) and (1,1) of 2; A000041(3) = 3 rows of length 3+1 for the 3 partitions {(3), (2,1), (1,1,1)} of 3; etc. (End)

Each partition is listed in nonincreasing order.

The partitions in each row are listed in decreasing lexicographic order.

Also the numbers of vertices of the connected components of the 2-regular simple graphs on n vertices.

Values represented by more than one set of indices are listed once per set; otherwise A176212 results.

Each term is a permanent of a quadratic symmetric (0,1) matrix with 1's on the main diagonal and exactly three 1's in each row and column.

For fixed Sum m_i=n with m_i >= 3, Product A000211(m_i) >= 6(4/3)^(n-3) and max(Product A000211(m_i)) = 6^((n-h)/3)*floor((3/2)^h), where h is the remainder of n (mod 3).

The parts of each partition are listed in decreasing order.

Differs from A080577 at a(48) and from A036037 at a(56). A181087 uses the same order for all partitions.

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).

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.