Let \( n \) be a positive integer and let \( \mathcal{A} \) be a collection of subsets of \( [n] = \{1, \dots, n\} \) such that \( \emptyset \in \mathcal{A} \). Consider the following multivariate polynomial:
\(P_{\mathcal{A}}(x_1, \dots, x_n) = \sum\limits_{S \in \mathcal{A}} \prod_{i \in S} x_i\)
Our goal is to study the formal inverse of this polynomial, given by \(P_{\mathcal{A}}(x_1, \dots, x_n)^{-1} = \sum\limits_{\textbf{m}} a_{\mathcal{A}}(\textbf{m}) x^{\textbf{m}} \)
In this talk, we will explore the natural question: what combinatorial meaning do the coefficients \( a_{\mathcal{A}}(\textbf{m})\) have in terms of \(\mathcal{A}\)?
In particular, we will focus on some particular instances when \(\mathcal{A}\) comes as the independent subsets of hypergraphs.
