A362559 Number of integer partitions of n whose weighted sum is divisible by n.

1, 1, 2, 1, 2, 3, 3, 3, 5, 4, 5, 7, 8, 11, 14, 14, 18, 25, 28, 26, 42, 47, 52, 73, 77, 100, 118, 122, 158, 188, 219, 266, 313, 367, 412, 489, 578, 698, 809, 914, 1094, 1268, 1472, 1677, 1948, 2305, 2656, 3072, 3527, 4081, 4665, 5342, 6225, 7119, 8150, 9408

Offset

1,3

Comments

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.

Also the number of n-multisets of positive integers that (1) have integer mean, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.

Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.

Links

Table of n, a(n) for n=1..56.

Example

The weighted sum of y = (4,2,2,1) is 1*4+2*2+3*2+4*1 = 18, which is a multiple of 9, so y is counted under a(9).
The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)     (7)        (8)       (9)
            (111)       (11111)  (222)   (3211)     (3311)    (333)
                                 (3111)  (1111111)  (221111)  (4221)
                                                              (222111)
                                                              (111111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], Divisible[Total[Accumulate[Reverse[#]]], n]&]], {n, 30}]

Crossrefs

For median instead of mean we have A362558.

The complement is counted by A362560.

A000041 counts integer partitions, strict A000009.

A008284/A058398/A327482 count partitions by mean.

A264034 counts partitions by weighted sum.

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

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

Cf. A001227, A051293, A067538, A067539, A240219, A261079, A322439, A326622, A359893, A360068, A360069, A362051.

Sequence in context: A029165 A035431 A008681 * A097242 A183202 A306878

Adjacent sequences: A362556 A362557 A362558 * A362560 A362561 A362562

Keyword

nonn

Author

Gus Wiseman, Apr 24 2023

Status

approved