Non-separable graphs meet Ledoux's polynomials

In the pathbreaking article \cite{LED16}, an integral representation of the derivatives of entropy along the heat flow of a probability measure was established under suitable moment conditions. These integral representations have found significant applications in diverse domains - notably in information theory (e.g., entropy power inequalities, monotonicity of Fisher information) and in estimation theory (through the link between entropy derivatives and the minimum mean square error, MMSE, in Gaussian channels). The representations involve multivariate polynomials $(R_n)_n$, arising from a Lie algebra framework on multilinear operators. Despite their central role, the combinatorial structure of these polynomials remains only partially understood. In this note, we prove that the number of monomials in $R_n$ coincides with the number of degree sequences with degree sum $2n$ having a non-separable graph realization, thereby resolving a conjecture from \cite{MPS24}, and drawing an interesting link between these two domains.