cp's OEIS Frontend

The rank of a partition is the largest summand minus the number of summands.

Also number of partitions of n with negative rank. - Omar E. Pol, Mar 05 2012

Column 1 of A208478. - Omar E. Pol, Mar 11 2012

Number of partitions p of n such that max(max(p), number of parts of p) is not a part of p. - Clark Kimberling, Feb 28 2014

The sequence enumerates the semigroup of partitions of positive rank for each number n. The semigroup is a subsemigroup of the monoid of partitions of nonnegative rank under the binary operation "*": Let A be the positive rank partition (a1,...,ak) where ak > k, and let B=(b1,...bj) with bj > j. Then let A*B be the partition (a1b1,...,a1bj,...,akb1,...,akbj), which has akbj > kj, thus having positive rank. For example, the partition (2,3,4) of 9 has rank 1, and its product with itself is (4,6,6,8,8,9,12,12,16) of 81, which has rank 7. A similar situation holds for partitions of negative rank--they are a subsemigroup of the monoid of nonpositive rank partitions. - Richard Locke Peterson, Jul 15 2018

The Dyson's rank of a partition is the largest part minus the number of parts.

For the definition of the k-th rank see A208478.

It appears that the sum of the k-th ranks of all partitions of n is equal to zero.

It appears that reversed rows converge to A000070, the same as A208478. - Omar E. Pol, Mar 10 2012

For the definition of k-th rank see A208478.

For the definition of k-th rank see A208478.

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 rank of a partition is the largest part minus the number of parts.

For more about this ordering, see A330370.

First differs from A105805 at a(49) = T(7,5).

a(n) is the sum of all parts of all partitions of n whose Ferrers diagrams are symmetric.

The k-th part of a partition equals the number of parts >= k of its conjugate partition. Hence, the k-th part of a self-conjugate partition equals the number of parts >= k.

The k-th rank of a partition is the k-th part minus the number of parts >= k. Thus all ranks of a conjugate-partitions are zero. Therefore, a(n) is also the sum of all parts of all partitions of n whose n ranks are zero, n >= 1. For more information about the k-th ranks see A208478.

Row n lists the partitions of n whose Ferrers diagrams are symmetrics.

The k-th part of a partition equals the number of parts >= k of its conjugate partition. Hence, the k-th part of a self-conjugate partition equals the number of parts >= k.

The k-th rank of a partition is the k-th part minus the number of parts >= k. Thus all ranks of a conjugate-partitions are zero. Therefore row n lists the partitions of n whose n ranks are zero, n >= 1. For more information about the k-th ranks see A208478.

In this triangle the partitions of n are ordered by their n-th rank. The partitions that have the same n-th rank appears ordered by their (n-1)-st rank. The partitions that have the same n-th rank and the same (n-1)-st rank appears ordered by their (n-2)-nd 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.

For further information see A330370.

First differs from A036037, A181317, A330370 and A334439 at a(48).

First differs from A080577 at a(56).