On The Morse Ensemble Polynomial Of Simplicial Complexes

We introduce the \emph{Morse ensemble polynomial} $\ME_K(z_0,\ldots,z_d)$ of a finite simplicial complex $K$, defined as the generating function $\ME_K = \sum_M \prod_i z_i^{c_i(M)}$ over all acyclic matchings $M$ on the face poset of $K$, where $c_i(M)$ counts critical $i$-simplices. This polynomial records the complete distribution of Morse vectors across all discrete Morse functions on $K$, and is an isomorphism invariant of simplicial complexes. Our main results are the following. \textbf{(I) The Laplacian Formula}: for any connected graph $G$, $\ME_G = z_1^{m-n}\det(z_0z_1\,I_n + L_G)$, identifying $\ME_G$ as a complete Laplacian spectral invariant and showing $\ME_G$ to be incomparable with the Tutte polynomial. \textbf{(II) The Top-Face Recursion}: adding a $d$-simplex $\sigma$ (with $\partial\sigma\subset K$) to a complex $K$ gives a recursion $\ME_{K\cup\{\sigma\}} = z_d\cdot\ME_K + \sum_{\tau\prec\sigma}(\ME_{P(K')\setminus\{\sigma,\tau\}}-F(K,\sigma,\tau))$. The correction term is controlled by the top incidence graph: an incidence-separation criterion detects exactly when $F=0$, and the incidence distance gives the leading obstruction term. As a topological application, this recursion gives exact coefficient recursions for perfect and optimal discrete Morse vectors. \textbf{(III) The independence ME polynomial} $\Phi(G) := \ME_{\mathrm{Ind}(G)}$ is a fine graph invariant which strictly refines the graph-level Morse ensemble $\ME_G$, separates examples not distinguished by $T_G$ and $I(G;t)$, and records collapse-level information of $\mathrm{Ind}(G)$ through coefficients such as $[z_0]\Phi(G)$.