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Lemma 3.1: Kendall's tau of an arbitrary signed shuffle

S records the horizontal and vertical tilings, with widths allowed to be zero. π specifies their relative order. A true sign means +1 and a false sign means -1. The paper's pair sum is written with the indices renamed j < i; because π is injective, its sign is +1 exactly when π j < π i.

theorem Papers.OrendayLaresRockel2026TauFootruleBeta.shuffle_tau {n : ℕ} (S : Verification.PositiveShuffle n) (ε : Fin n → Bool) (π : Equiv.Perm (Fin n)) (hx : ∀ (i j : Fin n), i < j → (S.strip i).x + (S.strip i).width ≤ (S.strip j).x) (hy : ∀ (i j : Fin n), π i < π j → (S.strip i).y + (S.strip i).width ≤ (S.strip j).y) : (S.signedCopula ε).kendallTau = ∑ i : Fin n, (if ε i = true then 1 else -1) * (S.strip i).width ^ 2 + 2 * ∑ i : Fin n, ∑ j : Fin n, if j < i then (if π j < π i then 1 else -1) * (S.strip j).width * (S.strip i).width else 0