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In this talk, I show that the kernel of $XB$ is generated by vertex-relabellings and a finite set of modular relations, in the same style as a recent analogous result by Penaguiao for the chromatic symmetric function $X$. Additionally, I give a structural characterization of all local modular relations of the chromatic and Tutte symmetric functions, and prove that there is no single local modification that preserves either function on simple graphs.
This is joint work with Sophie Spirkl.
A generalization of handshaking lemma for complexes is the following identity:
\begin{equation*}
\sum_{\sigma:~\sigma~is~a~k-face} val_{\mathcal{K}}(\sigma) = {k+1 \choose 1} f_{k}(\mathcal{K}), ~~~(k \geq 1).
\end{equation*}
This identitiy implies the well-known combinatorial interpretation of
the first derivative of face polynomials.
\Our main goal here is to give a generalization of
the combinatorial interpretation of the first derivative of
face polynomials to higher-order derivatives.
We also present some interesting research questions related to face polynomials and
their higher derivatives.