Metadata
bibkey: sun2022arithmeticpermanents
authors: Zhi-Wei Sun
year: 2022
title: "Arithmetic properties of some permanents"
doi: 10.48550/arXiv.2108.07723
url: https://arxiv.org/abs/2108.07723v7
claim: "Conjecture 4.2(i), equations (4.3) and (4.4): the two absolute-difference permanent congruences for every odd prime."
strata_touched:
- D5/S3/Combinatorics/Permanental/SunAbsoluteDifferenceCongruences
license: citation-only
triage: anchor
Verified locator
DOI: 10.48550/arXiv.2108.07723
Source: https://arxiv.org/abs/2108.07723v7
Source statements
Page 25, Conjecture 4.2(i), verbatim TeX (the source macros are retained):
\begin{conjecture} {\rm (i)} For any odd prime $p$, we have
\begin{equation}\label{|j-k|}\per[|j-k|]_{1\ls j,k\ls p}\eq-\f12\pmod p
\end{equation}
and
\begin{equation}\label{|j-k+1|}\per[|j-k+1|]_{1\ls j,k\ls p}\eq\f12\pmod p.
\end{equation}
Here \per denotes the permanent, \ls means ≤, \eq means ≡,
\f12 means 1/2, and \pmod p means (mod p).
Page 1 defines the permanent, verbatim TeX:
The permanent of a matrix $A=[a_{j,k}]_{1\ls j,k\ls n}$ over a field is defined by
$$\per(A)=\sum_{\tau\in S_n}\prod_{j=1}^na_{j,\tau(j)},$$
where $S_n$ is the symmetric group consisting all permutations of $\{1,\ldots,n\}$.
Integer encoding and proof
For j k : Fin p, the integer entry is
|((j.val : ℤ) + 1) - ((k.val : ℤ) + 1) + s|, with s = 0 or s = 1.
Both casts precede subtraction and absolute value. Source indices are j.val + 1
and k.val + 1. Mathlib’s Matrix.permanent sums ∏ i, M (σ i) i;
transposing the product indexing, or replacing the permutation by its inverse,
gives the source’s identical row-indexed sum. For an odd prime, clearing the
invertible denominator 2 gives the conjunction
(p : ℤ) ∣ 2 * Matrix.permanent (A p 0) + 1 and
(p : ℤ) ∣ 2 * Matrix.permanent (A p 1) - 1.
Alternating row and column signs turn these reductions into circulants. The prime-order fixed-point count on each permutation-weight fiber removes the nonfixed weights in characteristic p. Fixed permutations are translations; Fermat’s identity and a telescoping alternating sum evaluate their contribution. The prime circulant ingredient is the known Brualdi–Newman congruence (Theorem 3.2, 1965), as identified in issue #22003; it is not claimed as new mathematics. The entry reduction needs only odd order, while the counting step uses primality. Part (ii) is outside this statement.
Exact experiment
The experiment is in the-omega-institute/trureturing-experiments, commit
8478b9fc028dfcf192a82eff0aa7b6b25fe0a111, at
docs/reports/sun-2108-conj42i-permanents/check.py; its SHA256 is
a549c0f6573ad0e1452dbe7719d51f6d6f7ed11d83e48917650986aed9cc4678.
The recorded command is python3 -u docs/reports/sun-2108-conj42i-permanents/check.py,
exit 0, ending in ALL_CHECKS_OK. These finite integer computations support the
route and do not prove the universal assertion.
Literature boundary
Issue #22003 records the original v7 text and a bounded literature screen.
Jiang–Sun’s reciprocal-sum result concerns part (ii), rather than this pair.
Tao–Guo’s publisher full text was unread; some search services were inaccessible.
Literature completeness and worldwide priority remain ASSUMED-UNVERIFIED.