Derive max distinct frequencies for n items

Q: Derive max distinct frequencies for n items

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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Q: What difficulty level is this interview question?

This is a medium difficulty Statistics & Math question, commonly asked during Technical Screen rounds at Meta.

Q: What role is this question designed for?

This question is commonly asked for Software Engineer candidates at Meta during technical interviews.

Question

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.

Derive max distinct frequencies for n items

Quick Overview

Derive max distinct frequencies for n items

Maximum Number of Distinct Frequency Counts in an Array

Context

Task

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