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

Author

Gus Wiseman, Apr 24 2023

Keywords

• nonn

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.

Examples

			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)
		

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.

Programs

Mathematica

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