Sun’s absolute-difference permanent congruences

Abstract

Both absolute-difference permanent congruences in Sun Conjecture 4.2(i).

Definition 1.1 (A).

$$\forall p\in \mathbb{N},\forall s\in \mathbb{Z},\forall j\in \mathrm{Fin}\left(p\right),\forall k\in \mathrm{Fin}\left(p\right),A\left(p,s,j,k\right)=|(\mathrm{val}\left(j\right):\mathbb{Z})+1-\left((\mathrm{val}\left(k\right):\mathbb{Z})+1\right)+s|$$

Formalization. D5/S3/Combinatorics/Permanental/SunAbsoluteDifferenceCongruences.A (✓ std3).

Citation. Zhi-Wei Sun (2022). Arithmetic properties of some permanents. DOI: 10.48550/arXiv.2108.07723. URL: https://arxiv.org/abs/2108.07723v7.

Commentary.

The matrices in (4.3) and (4.4), p. 25, are “per[|j − k|]” and “per[|j − k + 1|]”, with 1 ⩽ j,k ⩽ p. The integer shift s unifies their entries. Source indices are val(j)+1 and val(k)+1; each value is cast to ℤ before subtraction and absolute value.

Definition 1.2 (claim).

$$claim=(\forall p\in \mathbb{N},Nat.Prime\left(p\right)\Rightarrow \left(\mathrm{Odd}\left(p\right)\Rightarrow \left(((p:\mathbb{Z})\mid (2\cdot Matrix.permanent\left(A\left(p,0\right)\right)+1))\wedge ((p:\mathbb{Z})\mid (2\cdot Matrix.permanent\left(A\left(p,1\right)\right)-1))\right)\right))$$

Formalization. D5/S3/Combinatorics/Permanental/SunAbsoluteDifferenceCongruences.claim (✓ std3).

Citation. Zhi-Wei Sun (2022). Arithmetic properties of some permanents. DOI: 10.48550/arXiv.2108.07723. URL: https://arxiv.org/abs/2108.07723v7.

Commentary.

Conjecture 4.2(i), p. 25: “(i) For any odd prime p, we have per[|j − k|]₁⩽ⱼ,ₖ⩽ₚ ≡ −½ (mod p) and per[|j − k + 1|]₁⩽ⱼ,ₖ⩽ₚ ≡ ½ (mod p).” The displays are (4.3) and (4.4), respectively.

Nat.Prime and Odd express exactly an odd prime. Matrix.permanent uses the sum of column-indexed permutation products, equal to the source’s row-indexed sum by inversion of the permutation. Since 2 is invertible modulo an odd prime, the two integer divisibilities are precisely (4.3) and (4.4). The same p serves both conjuncts.

Theorem 1.3 (alternating_sum).

$$\forall m\in \mathbb{N},\forall s\in \mathbb{Z},\sum _{r<2\cdot m+1}(-1{)}^r\cdot ((r:\mathbb{Z})+s)=(m:\mathbb{Z})+s$$

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Permanental/SunAbsoluteDifferenceCongruences.alternating_sum (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every nonnegative m and integer shift s, the alternating affine sum through 2m is m+s. This telescoping identity supplies the two parity cases in the permanent evaluation.

Theorem 1.4 (result).

$$\forall p\in \mathbb{N},Nat.Prime\left(p\right)\Rightarrow \left(\mathrm{Odd}\left(p\right)\Rightarrow \left(((p:\mathbb{Z})\mid (2\cdot Matrix.permanent\left(A\left(p,0\right)\right)+1))\wedge ((p:\mathbb{Z})\mid (2\cdot Matrix.permanent\left(A\left(p,1\right)\right)-1))\right)\right)$$

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

Resolves. Problems/sun210842 (proved) by D5/S3/Combinatorics/Permanental/SunAbsoluteDifferenceCongruences.result.

Source. Repository-derived.

Acknowledgement. Zhi-Wei Sun (2022). Arithmetic properties of some permanents. DOI: 10.48550/arXiv.2108.07723. URL: https://arxiv.org/abs/2108.07723v7.

Commentary.

Alternating row and column signs give circulants without changing their permanents. A prime-order fixed-point count on the permutation-weight fibers leaves only translations in characteristic p. Fermat’s identity and a telescoping alternating sum then evaluate both permanents. The prime circulant ingredient is the Brualdi–Newman congruence; the two absolute-difference reductions yield the stated divisibilities.

References

• Truth anchor: D5/S3/Combinatorics/Permanental/SunAbsoluteDifferenceCongruences.A
• Truth anchor: D5/S3/Combinatorics/Permanental/SunAbsoluteDifferenceCongruences.alternating_sum
• Truth anchor: D5/S3/Combinatorics/Permanental/SunAbsoluteDifferenceCongruences.claim
• Truth anchor: D5/S3/Combinatorics/Permanental/SunAbsoluteDifferenceCongruences.result