Enumeration of Nondegenerate $2 \times (k+1) \times k$ Hypermatrices

TL;DR

This paper investigates the enumeration of specific 3D hypermatrices with nonzero hyperdeterminant over finite fields, proposing a conjectural product formula and establishing polynomial enumeration properties.

Contribution

It introduces a conjecture for counting hypermatrices with nonzero hyperdeterminant and proves it in several cases, linking enumeration to rook placements.

Findings

Proposes a product formula conjecture for enumeration.

Shows enumeration is a polynomial in q with nonnegative coefficients.

Connects enumeration at q=1 to 3D rook placements.

Abstract

We consider the problem of enumerating hypermatrices of format $2 \times (k + 1) \times k$ over a finite field that have nonzero hyperdeterminant and whose nonzero entries are restricted to a plane partition. We conjecture an attractive product formula for the enumeration, and prove it in many cases. In general, we show that the enumeration is given (up to a power of $q - 1$) by a polynomial in $q$ with nonnegative integer coefficients, whose value at $q = 1$ enumerates a natural family of three-dimensional rook placements.