Here is a 3-paragraph summary of the blog post for mathopen.com:
Binomial coefficients, the numbers that count combinations and appear throughout mathematics, occasionally satisfy surprising equations where two seemingly different binomial coefficients turn out to be equal. These unexpected equalities are called "coincidences" because they fall outside the predictable patterns mathematicians have already identified and understood. Researchers have catalogued four known infinite families of such equalities, meaning structured, rule-based relationships that generate infinitely many examples.
Beyond those families, mathematicians have found exactly seven additional equations that cannot be explained by any of the four known patterns. These are the true coincidences, isolated numerical accidents that appear to arise from sheer numerical luck rather than any deeper algebraic structure. The Dutch mathematician Benjamin de Weger studied these cases extensively in a 1997 paper in the Journal of Number Theory, laying out careful elementary arguments about when and why such equalities can occur.
De Weger went further by conjecturing that the list of seven coincidences is complete, meaning no additional examples exist outside the four infinite families. At the time of his work, he and his collaborators had computationally verified this conjecture up to a certain range of values. The blog post revisits this open problem, highlighting it as a fascinating intersection of combinatorics, number theory, and computational mathematics where a simple question about counting has surprisingly deep and unresolved implications.