An Intrinsic Formula for the Ray–West Correction Term

Abstract

For a nonempty permutation β of length m, Ray and West expressed the number of length-(m+2) permutations containing β using a correction term j(β), but did not give j as a statistic intrinsic to β. This is Problem 3.7 of Vatter’s 2026 survey. This preliminary technical report proves the intrinsic formula j(β) = L(β) + R(β). Here L counts licensed gaps of β, while R counts anchor-free equal-two-deletion classes containing a collision whose canonical alignment support is an interval in both position and value. The proof consists of a fibre-forest defect identity, a classwise boundary theorem, and a strong-interval-tree charging theorem. It also derives 0 ≤ j(β) ≤ m−1 intrinsically and characterizes equality as occurring precisely for layered and reverse-layered permutations. The companion proof-and-replay archive contains the complete human-proof supplement, LaTeX source, deterministic standard-library verification code, mutation tests, integrity tooling, and an exhaustive cleanup check over all 362,880 permutations of length 9. Computation is used as corroboration only: the theorem is presented as a human proof and has not yet been externally refereed or fully machine-checked.