Problem 993: add OEIS A000055 (free trees); retain possible flag - #375
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Problem 993 (unimodality of independence polynomials of trees) quantifies over free trees on n vertices, enumerated by A000055. Exhaustive verifications of the conjecture enumerate exactly this sequence: the per-order tree counts from independent verification runs through n = 32 (109,972,410,221 trees at n = 32 alone) match A000055 term for term. The possible flag is retained since sequences derived from the independence polynomials themselves could be linked later. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
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Problem 993 (the 1987 Alavi-Malde-Schwenk-Erdős conjecture that the independence polynomial of every tree is unimodal) quantifies over free trees on n vertices, which are enumerated by A000055 (number of trees with n unlabeled nodes). This is the sequence that every exhaustive verification of the conjecture walks through, so it seems the natural first OEIS link for this problem.
Verification of the association (per the CONTRIBUTING guidance on computing values): the conjecture has been exhaustively verified for all free trees on up to 32 vertices, and the per-order tree counts produced by these runs match A000055 term for term at every order — including a(32) = 109,972,410,221. Two independent implementations agree: BrettRey/erdos-problem-993 (n ≤ 29; see also arXiv:2604.18824) and Tyorden/erdos-993-trees-n31 (independent reproduction of n ≤ 29 and extension to n ≤ 32, gentreeg + exact integer subtree DP), with identical counts on every common order.
The
possibleflag is retained: sequences derived from the independence polynomials themselves (e.g., coefficient statistics for notable tree families) could be linked in the future.AI disclosure (per the CONTRIBUTING policy): my verification pipeline was written and run with Claude (Anthropic) assistance; its outputs are independently validated by the exact term-for-term match with A000055's published values and by agreement with the independent implementation above. No new sequence is being submitted to the OEIS here — this PR only links an existing, long-established sequence.
🤖 Generated with Claude Code