login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Number of integer partitions of n with reverse-weighted sum 3*n.
6

%I #10 Jun 12 2023 08:44:13

%S 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,

%T 11,10,10,13,12,11,15,15,12,17,16,14,20,18,16,22,20,19,24,22,20,27,26,

%U 23,29,27,25,33,30,28,35,33,31,38,36,33,41,40

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

%C Are the partitions counted all of length 4 or 5?

%C 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.

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

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

%e (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)

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

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

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

%e (10,4,4,3) (10,7,4,3)

%e (11,5,5,3)

%e (12,4,4,4)

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

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

%Y The version for compositions is A231429.

%Y The non-reverse version is A363527.

%Y These partitions have ranks A363530, reverse A363531.

%Y A000041 counts integer partitions, strict A000009.

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

%Y A264034 counts partitions by weighted sum, reverse A358194.

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

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

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

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

%K nonn

%O 0,11

%A _Gus Wiseman_, Jun 10 2023