math.CO — Combinatorics

An Intrinsic Formula for the Ray–West Correction Term

Contributed by Donald Poindexter

For a nonempty permutation $\beta$ of length $m$, let $g_2(\beta)$ denote the number of distinct permutations of length $m+2$ containing $\beta$. Ray and West showed that $$ g_2(\beta)=\frac{m^4+2m^3+m^2+4m+4-2j(\beta)}{2}, \qquad 0\le j(\beta)\le m-1, $$ but their correction term was not expressed as a statistic of $\beta$. This was posed explicitly as Problem 3.7 by Vatter. We give a polynomial-time formula defined entirely from $\beta$: $j(\beta)=L(\beta)+R(\beta)$, where $L$ counts licensed gaps and $R$ counts anchor-free two-deletion classes containing a collision whose canonical alignment support is a permutation interval. We also derive the Ray–West bound intrinsically and characterize its upper-bound equality cases as precisely the layered or reverse-layered permutations. The proof is combinatorial. Exact computation is used only for independent corroboration; the accompanying proof-and-replay archive contains the full proof supplement and bounded verification. This preliminary technical report was first deposited on Zenodo on August 9, 2026: https://doi.org/10.5281/zenodo.21857882.