On Polynomials Counting Essentially Irreducible Maps

We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work Norbury that, if one disallows vertices degree one, the enumeration such related to counting lattice point in moduli space curves labeled points and given by a symmetric polynomial $N_{g,n}(\ell_1,\ldots,\ell_n)$ face degrees $2\ell_1, \ldots, 2\ell_n$. generalize this restricting that are essentially $2b$-irreducible for $b\geq 0$, which loosely speaking means they not allowed possess contractible cycles length less than $2b$ each cycle required bound $2b$. The shown be again $\hat{N}_{g,n}^{(b)}(\ell_1,\ldots,\ell_n)$ dependence $b$. These polynomials satisfy (generalized) string dilaton equations, $g\leq 1$ uniquely determine them. proofs rely heavily substitution approach Bouttier Guitter planar surfaces.