Maximizing a combinatorial expression arising from crowd estimation

TL;DR

This paper precisely determines the value of N that maximizes a specific combinatorial sum related to set union probabilities, providing insights into crowd estimation models and conjectures about rational functions.

Contribution

It introduces an exact method to find the maximum N for a combinatorial sum relevant to crowd estimation, connecting to conjectures on rational functions.

Findings

Exact value of N for maximum sum within 1

Insights into union size probabilities in crowd estimation

Conjectures on properties of related rational functions

Abstract

We determine, within 1, the value of N for which sum (s1 choose i)(s2 choose N)(s1 choose N-i)(N choose i) achieves its maximum value. Here s1 and s2 are fixed integers. This problem arises in studying the most likely value for the size of the union of A, B, and C if A and C are disjoint sets of size s1, and B is a set of size s2. Attempting to remove the 1 unit of indeterminacy leads to interesting conjectures about a family of rational functions.