# Upper Hessenberg and Toeplitz Bohemians

## TL;DR

This paper investigates special classes of Bohemian matrices, focusing on upper Hessenberg Toeplitz forms with specific entries, analyzing their characteristic polynomials, and establishing bounds and asymptotic behaviors.

## Contribution

It characterizes the characteristic polynomials of upper Hessenberg Bohemian matrices with maximal height and provides bounds and conjectures on their asymptotic behavior.

## Key findings

- Maximal characteristic height grows exponentially with matrix size.
- The height of the matrices remains constant despite polynomial height growth.
- Theorems on the enumeration of normal and stable matrices in these families.

## Abstract

We look at Bohemians, specifically those with population $\{-1, 0, {+1}\}$ and sometimes $\{0,1,i,-1,-i\}$. More, we specialize the matrices to be upper Hessenberg Bohemian. From there, focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give useful bounds on their height, and conjecture an accurate asymptotic formula. The lower bound for the maximal characteristic height is exponential in the order of the matrix; in contrast, the height of the matrices remains constant. We give theorems about the numbers of normal matrices and the numbers of stable matrices in these families.