bibkey: oeis2021a344611
authors: Gus Wiseman
year: 2021
title: OEIS A344611 — reverse-alternating sums of integer partitions
doi: null
url: https://oeis.org/A344611
claim: The number of partitions of 2n with nonnegative reverse-alternating sum is at most the number of partitions of 2n+1 into an odd number of parts.
strata_touched: []
license: citation-only
triage: anchor

OEIS A344611

Verified locator

Source: https://oeis.org/A344611

OEIS A344611, revision 27, May 17, 2026, 17:23:31, FORMULA, NAME, COMMENTS and OFFSET; the electronic entry has no page numbers. The author is Gus Wiseman; the entry was created May 30, 2021. There is no source arXiv identifier or DOI.

Source statements

NAME: “Number of integer partitions of 2*n with reverse-alternating sum >= 0.”

COMMENTS: “The reverse-alternating sum of a partition (y_1,…,y_k) is Sum_i (-1)^(k-i) y_i.”

FORMULA: “Conjecture: a(n) <= A160786(n). The difference is 0, 0, 0, 0, 1, 2, 4, 9, 16, 28, 48, 79, …”

OFFSET: “0,2”. The example includes the empty partition at n=0.

A160786, revision 25, FORMULA: “a(n) = A027193(2n+1).” A027193, revision 78, NAME: “Number of partitions of n into an odd number of parts.”

Mathematical interpretation

Mathlib’s Nat.Partition m carries positive multiset parts of sum m. Sort those parts in decreasing order. The existing integer recurrence TrimmedAlternatingPartitions.trimmedSums starts from zero and replaces z by a-z for each part a; its last endpoint, defaulting to zero on the empty list, is the source’s reverse-alternating sum. Nat.card counts finite subtypes of partitions. The comparison includes every natural n, including zero.

The repository proof increases the first part of an odd-length source and appends 1 to an even-length source. A two-step endpoint bound forces the first two parts of every nonempty even-length source to agree. The images therefore have respectively unique and repeated largest parts; the empty source maps to [1]. This gives an injection for every source weight m into odd-length partitions of weight m+1.

Evidence and limits

The source and partial literature screen are specified in preregistration #23757. Literature completeness is ASSUMED-UNVERIFIED: the screen includes explicit rate limits and unread classical references. No exhaustive novelty claim is made.

The separate unbounded strict inequality for n≥4 is not formalized here. The adjacent A344743 comparison has a different target count and is not identified with A344611.

The experiment is docs/reports/wiseman-a344611/check.py in trureturing-experiments, commit 6c424c1a68a8d81ed46f83a441a14a179d1708c4, SHA-256 99253e529398497e64c2e444787a2a1c97a40cc99d29219d1432c547fcc396ba. Preregistration reports python3 docs/reports/wiseman-a344611/check.py, exit 0, final line ALL_CHECKS_OK, over n=0..20 and 61,515 qualifying source partitions. These finite computations are distinct from the unbounded Lean proof.