bibkey: spezia2018a322844
authors: Stefano Spezia
year: 2018
title: OEIS A322844 and the serpentine matrix A317614
doi: null
url: https://oeis.org/A322844
claim: "The two COMMENTS conjectures assert the prescribed absolute second exterior trace and vanishing of every exterior trace above order two."
strata_touched:
  - D5/S3/Quantum/Matrix/SpeziaSerpentineSecondExteriorTrace
  - D5/S3/Quantum/Matrix/SpeziaSerpentineHigherExteriorTrace
license: citation-only
triage: anchor

OEIS A322844 and the serpentine matrix A317614

Verified locator

Source: https://oeis.org/A322844

Matrix source: https://oeis.org/A317614

A322844 revision 68, COMMENTS and FORMULA, is unpaginated. A317614 revision 194 supplies the same matrix entries. No DOI, arXiv identifier or manuscript is identified for this source.

Source statements

a(n) = (1/12)n^2(3*(1 + n^2) - 2*(2 + n^2)*(n mod 2)).

For n > 1, a(n) is the absolute value of the trace of the 2nd exterior power of an n X n square matrix M(n) defined as M[i,j,n] = j + n*(i-1) if i is odd and M[i,j,n] = n*i - j + 1 if i is even (see A317614). Equivalently, a(n) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of the matrix M(n), or the absolute value of the sum of all principal minors of M(n) of size 2.

For k > 2, the trace of the k-th exterior power of the matrix M(n) is equal to zero.

The entries have integer arithmetic and one-based source indices. The Lean matrix uses Fin n and adds one to each index. The exterior trace is encoded by the signed characteristic coefficient for k ≤ n, and by zero for k > n. This is the characteristic-coefficient formulation stated by the source; the encoding does not construct an exterior-power linear map. The sequence value is rational, with the integer absolute trace cast into the rationals.

Result and boundary

Over the rationals the matrix factors through a two-dimensional space. The rectangular characteristic-polynomial identity gives the higher-coefficient vanishing. Centered linear, quadratic and alternating moments evaluate the second coefficient as the negative of the displayed sequence value, which is nonnegative. A separate exact-rank theorem and a separately expanded full characteristic polynomial are not supplied.

Supporting computation

The exact experiment is in the-omega-institute/trureturing-experiments, commit 72e99409d60c1747efec95cc67de4f5dd612c133, directory docs/reports/spezia-a322844/. The command python3 check.py exited 0 with ALL_CHECKS_OK; check.py has SHA-256 2573b4928f9051382c4692a3d3d1680b30d477d8d5d8c898145398a8d08f5159. These finite computations are separate from the universal Lean results.

Literature boundary

Issues #21795 and #21796 record the source readings and literature screen. Literature completeness is ASSUMED-UNVERIFIED: the cited OpenAlex, Semantic Scholar and Google requests were unavailable. The source has no identified arXiv version.