bibkey: wiseman2025a391620
authors: Gus Wiseman
year: 2025
title: OEIS A391620, partitions that are not first sums of a composition with all parts > 1
doi: null
url: https://oeis.org/A391620
claim: "Conjecture: These are integer partitions of n whose least part is < 4. The first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...)."
strata_touched:
  - D5/S1/Words/Compositions/FirstSumsPartitionCharacterization
license: citation-only
triage: anchor

OEIS A391620

NAME “Number of integer partitions of n that are not the first sums of any composition with all parts > 1.”

Gus Wiseman’s entry of December 30, 2025 counts partitions of positive n that are not first sums of a composition with every part greater than one. The conjecture and definition above are quoted from the entry supplied by the orchestrator on September 8, 2026. The ordered-part reading writes the partition in weakly decreasing order and requires equality with the list of adjacent sums, as in the entry’s example (5,4) from (3,2,2).

The Lean module proves this reading by backward reconstruction, and proves equality of the corresponding counts on Mathlib partitions. Its list theorem also allows zero parts; its counting theorem includes the empty partition at n = 0, where both counts vanish. The proof is derived in this repository.

Verified locator

• URL: https://oeis.org/A391620