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Formal Reversion, Diamond Avoidance, and Binary Support Laws

Year: 2026
Type: preprint
Abstract: We study a two-parameter family of integer formal reversion series associated with rational factors $(1-x)^a/(1+x)^b$. For the member corresponding to $a=b=4$, we solve the published recursion for $231$-avoiding labelings of four-vertex diamond posets and obtain an all-coefficient generating-function identity that was stated as a conjecture in OEIS A365847. For two other parameter subfamilies, we derive exact binary support tests: a carry-free condition for negative powers of two, and a unique radix-digit description when $a-b=2^r-1$. These descriptions yield exact Fibonacci-product counts in dyadic windows and polynomial $(k+1)^r$ counts in radix windows, respectively. Eight specified inverse series are treated through the same formal reversion framework. The published combinatorial recursions, Lagrange inversion, Lucas's theorem, and earlier Fuss--Catalan and Schröder path identities are credited to their sources. Three exact coefficient implementations agree with all 1,805 entries in the eight supplied OEIS b-files; an additional computation from the original four-component diamond recursion checks the poset connection through degree 100; these finite checks are not part of the infinite proofs. Licensing note: The manuscript is licensed under CC BY 4.0. The OEIS b-file snapshots included in the supplementary archive retain their original CC BY-SA 4.0 license and attribution. The independently authored verification code is made available under CC BY 4.0.
DOI: 10.5281/zenodo.24082557
Language: English
Open Access status: green
OpenAlex ID: W7225648420
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