Derive path graph multidegree directly from the main combinatorial formula

Derive, using the combinatorial multidegree formula C(G) = sum over S in the set of vertex subsets with minimal value of |S| − c(V \ S) of (t1 t2)^{|S|} times the product over the sizes n of the connected components of G[V \ S] of (t1^n − t2^n)/(t1 − t2), a closed-form expression for the multidegree C(P_n) of the binomial edge ideal J_{P_n} of the path graph P_n, without resorting to alternative methods.

Background

The paper’s main result (Theorem thm:multidegree) gives a combinatorial formula for the multidegree of J_G in terms of a sum over a specific collection of vertex subsets and the sizes of connected components of induced subgraphs. While this formula is effective for several graph families (stars, cycles, wheels, friendship graphs, etc.), the authors note difficulty applying it directly to path graphs.

Instead, they compute the multidegree of path graphs via an alternative approach that uses the fact that J_{P_n} is generated by a regular sequence, obtaining C(P_n) = (t1 + t2)^{n−1}. The open issue is whether one can derive a comparable closed form directly from the main combinatorial formula, thereby illuminating how the summation over minimal-height prime components behaves for path graphs.