bibkey: koprowski2026enumeration authors: Brandon Koprowski, Joel Brewster Lewis year: 2026 title: “Enumeration of Nondegenerate 2 x (k+1) x k Hypermatrices” doi: 10.48550/arXiv.2602.22129 url: https://arxiv.org/abs/2602.22129v1 claim: “Section 4 supplies the orbit, Bruhat, triangular-invariance and augmented-cell count bridges; Section 5.1 identifies the actual eligible-permutation sum weighted by inv(sigma)+inv(pi)-h and asks for its general factorization.” strata_touched: [] license: citation-only triage: anchor

Nondegenerate hypermatrices and the published permutation-weight bridge

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DOI: 10.48550/arXiv.2602.22129

Immutable primary version: https://arxiv.org/abs/2602.22129v1. Its mathematical text is also available at https://arxiv.org/html/2602.22129v1.

The journal version is Enumerative Combinatorics and Applications 7:1, Article S2R3, DOI 10.54550/ECA2027V7S1R3, available at https://ecajournal.kms-ks.org/Volume2027/ECA2027_S2A3.pdf. The article bears publication date August 21, 2026, although the volume is labelled 2027. It is released under CC BY-ND 4.0. This note supplies citations and mathematical scope, rather than a source-text or code port.

The following arXiv-version numbers are the numbers displayed in the primary HTML text; the journal numbers are given separately because its theorem environments use separate counters.

Conjecture 3.1 is the original general count for both antitone masks over every finite field and every positive k. The two mask lengths satisfy $0\le {\mu }_j\le {\lambda }_j\le k-j$ and ${\mu }_j<k-j$ in zero-based column indices. The nondegeneracy condition is the nonvanishing of the boundary-format Cayley hyperdeterminant. Its integral coefficient determinant specification is described in the tensor-complex source.

• Definition 4.4 (wc) gives $w=\sigma \bar{\pi }$ and $c={\bar{\pi }}^{-1}(12\cdots k+1)\bar{\pi }$; hence $wc(i)=\sigma (\pi (i)+1)$ for $i\le k$, and $wc(k+1)=\sigma (1)$.
• Theorem 3.3 (thm:aitken) supplies the free transitive action of the quotient by the scalar subgroup on nondegenerate tensors. The full matrix group consequently represents each such tensor $q-1$ times.
• Theorem 4.13 (thm:bruhat) and Remark 4.14 (rem:equivalent bruhat) give the unique Bruhat decompositions used in the two factors. Their journal counterparts are Theorem 4.3 and Remark 4.2.
• Definition 4.16 (def:augHM) uses $A\in {\sigma }_{\ast }$ and $A^{\prime }\in {}^{\ast }({\pi }^{-1})$. The matrix multiplied on the right of each face is $(A^{\prime }{)}^{\mathrm{T}}\in {}^{\ast }\pi$. These transpose and inverse conventions are part of the supplier, not interchangeable choices of notation.
• Proposition 4.17 (prop:pull off U) gives the $q-1$ triangular/augmented representations of each nondegenerate tensor; Proposition 4.18 (prop:upper triangular) states preservation of the two zero regions in both directions. Their journal counterparts are Propositions 4.2 and 4.3.
• Corollary 4.19 (cor:count; journal Corollary 4.2) relates the actual tensor count to the augmented-cell sum, with multiplier $q^{k^2}(q-1{)}^{2k}$. The two triangular group orders multiply to $q^{k^2}(q-1{)}^{2k+1}$, and the scalar fiber has size $q-1$.
• Propositions 4.24 and 4.25 (prop:potentially bad entries in front face and prop:potentially bad entries in back face; journal Propositions 4.5 and 4.6) give the forbidden-entry criteria. The back criterion explicitly includes the distinguished target $j=k+1$; the front criterion excludes it.
• Proposition 4.28 (prop:acyclic) and Theorem 4.29 (thm:power of q; journal Proposition 4.7 and Theorem 4.4) supply the acyclic elimination of distinct coefficient-one final variables and the resulting cell counts.
• Section 5.1, “Proving the main conjecture,” displays the exact weighted sum over eligible pairs, with exponent $|\mathrm{Inv}(\sigma )|+|\mathrm{Inv}(\pi )|-h(\sigma ,\pi ;P)$. Here $h$ counts generically nonzero forbidden tensor entries, not nonzero values after a particular finite-field specialization.

Hypotheses and use boundary

The objects in the tensor-count supplier are pairs of $(k+1)$-by-$k$ matrices over a finite field, with both plane-partition zero regions, and Cayley’s second hyperdeterminant nonzero. Lemma 2.5 (lem:faceSum) states the algebraic-closure criterion: every linear combination of the two faces with coefficients in the algebraic closure, not both zero, has full column rank. It is not a condition restricted to rational combinations over the base finite field. The statement preceding that lemma restricts its three-dimensional boundary formats to positive $k_1,k_2$; its direct specialization with $k_1=1,k_2=k-1$ therefore requires $k\ge 2$. The $2\times 2\times 1$ case needs its own justification if this particular lemma is used to connect a rank-defined count to a hyperdeterminant-defined count. A size-zero permutation tail asserts no tensor-count statement.

The new coupled low-row digit construction uses the published row convention and forbidden-entry criteria to identify $h=H_f+H_b$, then proves an explicit bijection and factors the actual weighted permutation sum. The orbit, Bruhat, triangular-invariance and field-count results above are the published suppliers connecting that sum to the original tensor objects.

Published general and special statements

Both the arXiv v1 text and the journal article’s abstract and Section 5.1 retain the general factorization as a conjectural task; Section 5.1 says that grouping the terms into the desired product had been unsuccessful. Their proved special families and the unweighted hyperrook count are published results, not new contributions of the digit construction.