bibkey: wiseman2023a362559 authors: Gus Wiseman year: 2023 title: “OEIS A362559, Number of integer partitions of n whose weighted sum is divisible by n” doi: null url: https://oeis.org/A362559 claim: “Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.” strata_touched:

• D5/S3/Combinatorics/PartitionWeightedSumReversal license: citation-only triage: anchor

OEIS A362559

The entry counts the integer partitions of n whose one-based weighted sum is a multiple of n. Its terms begin 1, 1, 2, 1, 2, 3, 3, 3, 5, 4, 5, 7, 8, 11, 14, ….

Verified locator

• URL: https://oeis.org/A362559
• Locator: 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.” and, on the following comment line, “Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted sum divisible by n.”
• Revision read: #18, Apr 29 2023. The comment stands and carries no answer.
• The identical conjecture line stands on https://oeis.org/A362560, “Number of integer partitions of n whose weighted sum is not divisible by n”, at revision #6 of the same date. Both entries are by Gus Wiseman, Apr 24 2023.

Reading of the statement

Positions are counted from one, so the first part carries weight one. The entry’s own worked example fixes the reading: “The weighted sum of y = (4,2,2,1) is 14+22+32+41 = 18, which is a multiple of 9, so y is counted under a(9).”

The reverse of a partition presented in weakly decreasing order is the same multiset read in weakly increasing order. For y = (4,2,2,1) the reverse is (1,2,2,4), whose weighted sum is 1*1+2*2+3*2+4*4 = 27, also a multiple of 9.

Scope of the recorded answer

The comment holds for every n and every partition, and one identity is the whole reason.

For a finite list y of length k, write W y for its one-based weighted sum. Then

W y + W y.reverse = (k + 1) * (sum of y).

Reindexing the reversed sum by j = k + 1 - i turns W y.reverse into Sum_{j=1..k} (k+1-j) * y_j; adding it termwise to W y = Sum_j j * y_j leaves Sum_j (k+1) * y_j. On a partition of n the right-hand side is (k+1) * n, a multiple of n, so n divides one of the two weighted sums exactly when it divides the other.

The example above is this identity at k = 4 and n = 9: 18 + 27 = 45 = 5 * 9.

Neither the weak decrease of the parts nor their positivity enters the argument; the identity holds for every finite list of naturals. The formal statement carries the partition hypothesis so that it corresponds to the sentence on the entry, and that hypothesis does no work.

Bounded prior-resolution evidence

Both entries were read in full at the revisions above and record no proof and no reference to one; each still carries the word Conjecture. The cross-references A362558, A264034, A304818 and A318283 carry no proof of it either. A web search for a published proof returned only unrelated literature on divisibility of weighted k-regular partitions. Citation indices were not exhaustively reachable, so this is a bounded negative finding and no worldwide priority claim is made.