Tutte polynomials and h-vectors of convex polytopes

Question

Revised question:

Aluffi and Marcolli give a recursion relation for the Tutte polynomials of altered graphs (as described in Machacek's post below) on page 42 of "Feynman graphs and deletion-contraction relations" and the e.g.f.s $f$ and $g$ of the coefficients of this recursion. The general e.g.f.s are related to the h-vectors of convex polytopes:

$f= T $, $g=C/S $, and $f/g = P $,

where $T,C,S,$ and $P$ are the bivariate e.g.f.s of the h-vectors of the $(n-1)-$simplices (the complete homogeneous symmetric polynomials in two variables, A000012 read as a lower triangular array), hypercubes (A007318, Pascal's triangle), stellahedra (A046802), and permutahedra (A008292, the Eulerian numbers), respectively (cf. pg. 60 of Toric Topology by Buchstaber and Panov).

Is there some heuristic or intuition that "explains" the appearance of these e.g.f.s in the recursion relation?

---

Prior question:

Aluffi and Marcolli state on page 42 of "Feynman graphs and deletion-contraction relations" that tanh is related to a Tutte polynomial. Can someone characterize the associated family of graphs? (Edit per S.H.'s remark.)

Tanh, of course, is related to soliton solutions of the KdV equation, up-down permutations (cf. OEIS A000182), normalized Bernoulli numbers, and the magnetization of the spin 1/2 paramagnet.

Answer 1

They are taking a graph $\Gamma$ and letting $\Gamma^{(m)}$ denote $\Gamma$ with $m$ paralel edges inserted between two chosen vertices. Then one of the invariants $U$ they study (i.e. the Tutte polynomial) will satisfy $$U(\Gamma^{(m)}) = f_{m+1}U(\Gamma) + g_{m+1}U(\Gamma \setminus e) + h_{m+1}U(\Gamma / e)$$ for some sequences $f_m, g_m, h_m$. They want to find the associated exponential generating functions for the sequences (i.e. $f(s) = \sum f_m(s^m/m!)$, etc.). The paper then claims $f(s) = \sinh(s)$ and $g(s) = \cosh(s)$ is then solution when $U$ is the Tutte polynomial (they make a choice of 'representation' of the Tutte polynomial in (6.3) and choice of initial conditions in (6.1) to arrive at these particular functions which lead to these exponential generating functions for the Tutte polynomial). Hence, then ratio gives $\tanh(s)$.