On a conjecture concerning the fraction of invertible m-times persymmetric matrices over F_2

TL;DR

This paper proves that the proportion of invertible m-times persymmetric matrices over the finite field F_2 equals the product of (1 - 2^{-j}) for j from 1 to m, establishing a precise formula.

Contribution

The paper provides a new exact formula for the fraction of invertible m-times persymmetric matrices over F_2, confirming a conjecture.

Findings

Fraction of invertible m-times persymmetric matrices over F_2 equals prod_{j=1}^{m}(1-2^{-j})

Validates a conjecture about matrix invertibility over F_2

Provides a closed-form expression for a class of structured matrices

Abstract

In this paper we state that the fraction of invertible m-times persymmetric matrices over F_2 is equal to \prod_{j=1}^{m}(1-2^{-j})