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We find the generating function for the contributions of $n$-cylindersquare-tiled surfaces to the Masur-Veech volume of $\mathcal{H}(2g-2)$. It is abivariate generalization of the generating function for the total volumesobtained by Sauvaget via intersection theory. Our approach is, however, purelycombinatorial. It relies on the study of counting functions for certainfamilies of metric ribbon graphs. Their top-degree terms are polynomials, whose(normalized) coefficients are cardinalities of certain families of metric planetrees. These polynomials are analogues of Kontsevich polynomials that appear aspart of his proof of Witten's conjecture.

Contribution of $n$-cylinder square-tiled surfaces to Masur-Veech volume of $\mathcal{H}(2g-2)$