Nonsymmetric normal entry patterns with the maximum number of distinct indeterminates

TL;DR

This paper proves an upper bound on the number of distinct indeterminates in nonsymmetric normal entry patterns of order n and characterizes the unique pattern attaining this maximum, up to permutation similarity.

Contribution

It establishes a maximum count of distinct indeterminates in nonsymmetric normal entry patterns and explicitly describes the unique pattern achieving this maximum.

Findings

Maximum of n(n-3)/2+3 distinct indeterminates for nonsymmetric normal patterns

Uniqueness of the pattern attaining the maximum up to permutation similarity

Explicit description of the extremal pattern

Abstract

We prove that a nonsymmetric normal entry pattern of order $n$ ($n\ge 3$) has at most $n(n-3)/2+3$ distinct indeterminates and up to permutation similarity this number is attained by a unique pattern which is explicitly described.