A363526 - OEIS

A363526

Number of integer partitions of n with reverse-weighted sum 3*n.

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 4, 3, 2, 4, 4, 4, 5, 5, 4, 7, 7, 5, 8, 7, 6, 11, 9, 8, 11, 10, 10, 13, 12, 11, 15, 15, 12, 17, 16, 14, 20, 18, 16, 22, 20, 19, 24, 22, 20, 27, 26, 23, 29, 27, 25, 33, 30, 28, 35, 33, 31, 38, 36, 33, 41, 40

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OFFSET

0,11

COMMENTS

Are the partitions counted all of length 4 or 5?

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. The reverse-weighted sum is the weighted sum of the reverse, also the sum of partial sums. For example, the weighted sum of (4,2,2,1) is 1*4 + 2*2 + 3*2 + 4*1 = 18 and the reverse-weighted sum is 4*4 + 3*2 + 2*2 + 1*1 = 27.

LINKS

Table of n, a(n) for n=0..70.

EXAMPLE

The partition (6,4,4,1) has sum 15 and reverse-weighted sum 45 so is counted under a(15).

The a(n) partitions for n = {5, 10, 15, 16, 21, 24}:

(1,1,1,1,1) (4,3,2,1) (6,4,4,1) (6,5,4,1) (8,6,6,1) (9,7,7,1)

(2,2,2,2,2) (6,5,2,2) (6,6,2,2) (8,7,4,2) (9,8,5,2)

(7,3,3,2) (7,4,3,2) (9,5,5,2) (9,9,3,3)

(3,3,3,3,3) (9,6,3,3) (10,6,6,2)

(10,4,4,3) (10,7,4,3)

(11,5,5,3)

(12,4,4,4)

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], Total[Accumulate[#]]==3n&]], {n, 0, 30}]

CROSSREFS

Positions of terms with omega > 4 appear to be A079998.

The version for compositions is A231429.

The non-reverse version is A363527.

These partitions have ranks A363530, reverse A363531.

A000041 counts integer partitions, strict A000009.

A053632 counts compositions by weighted sum, rank statistic A029931/A359042.

A264034 counts partitions by weighted sum, reverse A358194.

A304818 gives weighted sum of prime indices, row-sums of A359361.

A318283 gives weighted sum of reversed prime indices, row-sums of A358136.

A320387 counts multisets by weighted sum, zero-based A359678.

Cf. A000016, A008284, A067538, A222855, A222970, A359755, A360672, A360675, A362559, A362560, A363525, A363528.

Sequence in context: A291708 A163822 A029203 * A159805 A208216 A208217

Adjacent sequences: A363523 A363524 A363525 * A363527 A363528 A363529

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 10 2023

STATUS

approved