For an array of length n (n ≥ 1) with arbitrary integer values, what is the maximum possible number of distinct frequency counts among its distinct values? Derive a tight expression in terms of n and prove optimality. Discuss constructive examples that achieve the bound.

The question evaluates combinatorial reasoning, discrete mathematics, and constructive extremal analysis by asking for the maximum number of distinct frequency counts in an array and a matching construction.

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Derive max distinct frequencies for n items

Maximum Number of Distinct Frequency Counts in an Array

Context

You are given an array of length n ≥ 1 whose elements are arbitrary integers (values may repeat). For each distinct value, compute its frequency (number of occurrences). Among these frequencies, some values may coincide. We ask:

What is the maximum possible number of distinct frequency values that can appear?
Give a tight expression in terms of n, prove it is optimal, and present constructions that achieve the bound.

Task

Define the function m_max(n): the maximum number of distinct frequency counts achievable by any length-n array.
Derive a closed-form expression for m_max(n) in terms of n.
Prove optimality (upper bound and matching construction).
Provide small illustrative examples.

Maximum Number of Distinct Frequency Counts in an Array

Context

What is the maximum possible number of distinct frequency values that can appear?
Give a tight expression in terms of n, prove it is optimal, and present constructions that achieve the bound.

Task

Define the function m_max(n): the maximum number of distinct frequency counts achievable by any length-n array.
Derive a closed-form expression for m_max(n) in terms of n.
Prove optimality (upper bound and matching construction).
Provide small illustrative examples.

Derive max distinct frequencies for n items

Quick Overview

Maximum Number of Distinct Frequency Counts in an Array

Context

Task

Solution

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Derive max distinct frequencies for n items

Quick Overview

Maximum Number of Distinct Frequency Counts in an Array

Context

Task

Solution

Comments (0)