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A Counterexample to the Literal Adjacent Difference

Abstract

The literal adjacent-column identity with the zero-based index n-4 fails at n=5: the actual cardinality difference is twenty and the absolute sequence value is six.

Definition 1.1 (The zero-based auxiliary sequence).

Formalization. D5/S3/Combinatorics/Derangements/AdjacentDifferenceRefutation.b (✓ std3).

Citation. Wolfdieter Lang (2014). OEIS A238474: a signed factorial quotient. URL: https://oeis.org/A238474.

Commentary.

For every natural m, b(m) is (-1)^m times (m+3)! divided by 2(m+1), computed in the rationals. In the formulas, castQ denotes the natural-number embedding into the rationals. The factorial and m+1 are first computed in the natural numbers and then cast. The denominator is nonzero for every m >= 0. This is the formula of A238474 with offset zero: b(0)=3 and b(1)=-6. The exponent uses the rational number minus one, rather than natural truncated subtraction.

Definition 1.2 (The literal adjacent-column conjecture).

Formalization. D5/S3/Combinatorics/Derangements/AdjacentDifferenceRefutation.claim (✓ std3).

Citation. Peter Kagey (2024). OEIS A378495: derangements with no k-cycles. URL: https://oeis.org/A378495.

Commentary.

The proposition claim asserts that for every natural n with 4 <= n, T(n,n-1)-T(n,n-2) equals the absolute value of b(n-4). Here T is the cardinality of all permutations of Fin n with no fixed point and no cycle factor of the forbidden length. Each cardinality is cast to the rationals before ordinary subtraction; the absolute value is rational. Every index subtraction is natural subtraction, and the hypothesis ensures that n-4 has its ordinary nonnegative meaning. The shift is exactly n-4, with no change of the zero offset of b and no upper bound on n.

Theorem 1.3 (Five labels contradict the asserted equality).

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Derangements/AdjacentDifferenceRefutation.result (✓ std3). ∎

Resolves. Problems/oeis-a378495-adjacent-difference (refuted) by D5/S3/Combinatorics/Derangements/AdjacentDifferenceRefutation.result.

Source. Repository-derived.

Acknowledgement. Peter Kagey (2024). OEIS A378495: derangements with no k-cycles. URL: https://oeis.org/A378495.

Commentary.

The proposition claim is false. At n=5, a derangement cannot contain a 4-cycle: the remaining label would be fixed. Thus excluded(5,4) is empty, whereas the cycle type (3,2) contributes exactly 5!/(2 times 3)=20 excluded permutations at length three. Applying the complementary-count identities to the same derangement set gives T(5,4)=T(5,3)+20 and hence the rational difference twenty. The source sequence has b(5-4)=b(1)=-6, whose absolute value is six. Since 20 differs from 6 and five satisfies the stated lower bound, this one exact counterexample negates the whole universal proposition. It does not assert a replacement formula or negate the separate terminal-column identity.

References