Let $M(z)$ be the generating function of edge-rooted connected quadrangulations, with $z$ marking the number of edges. I derived the following Lagrangean equations for $M(z)$:
$$M(z) = \psi(L(z)),~\psi(t) = \frac{1-4t}{(1-3t)^2}~,$$
$$L(z) = z \phi(L(z)),~\phi(t)=\left(\frac{t}{1-3t}\right)^{1/2}~.$$
Here is how I derived them. First note that, for any $n\in\mathbb{N}$, there is a one-to-one correspondence between the set of edge-rooted connected planar maps with $n$ edges and the set of edge-rooted connected quadrangulations with $2n$ edges. Also note that the Lagrangean equations for the generating function $m(z)$ of edge-rooted connected planar maps are:
$$m(z) = \Psi(l(z)),~\Psi(t) = t(2-t)/3~,$$
$$l(z) = z \Phi(l(z))~, \Phi(t) = 3(1+t)^2~,$$
where $z$ marks the number of edges. Tutte used the quadratic method to derive the above equations. What I did was to change the variable $z$ to $z^2$ in the intermediate step of the quadratic method. Then that gave rise to the Lagrangean equations for $M(z)$ described at the beginning.
My questions are:
- The function $\phi(t)$ that I got doesn't seem to satisfy one of the conditions (page 402 (35) in Analytic Combinatorics) for the singular inversion theorem (page 404 Theorem VI.6 in Analytic Combinatorics). What is the reason for imposing that quoted condition in the book? Is there a more general version of the singular inversion theorem which doesn't require that condition?
2. Is there another way to derive Lagrangean equations for quadrangulations?
Please let me know if further clarification is needed. Thanks!
