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# Annales de l’Institut Henri Poincaré D

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**Volume 8, Issue 1, 2021, pp. 119–158**

**DOI: 10.4171/AIHPD/104**

Published online: 2021-03-08

Constellations and $\tau$-functions for rationally weighted Hurwitz numbers

John Harnad^{[1]}and Boris Runov

^{[2]}(1) Université de Montréal and Concordia University, Montreal, Canada

(2) St. Petersburg State University, Russia

Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the 2D Toda $\tau$-functions of hypergeometric type serving as generating functions for weighted Hurwitz numbers in the case of polynomial weight generating functions. The product over all vertex and edge weights of a given weighted constellation, summed over all configurations, reproduces the $\tau$-function. In the present work, this is generalized to constellations in which the weighting parameters are determined by a rational weight generating function. The associated $\tau$-function may be expressed as a sum over the weights of doubly labelled weighted constellations, with two types of weighting parameters associated to each equivalence class of branched coverings. The double labelling of branch points, referred to as “colour” and “flavour” indices, is required by the fact that, in the Taylor expansion of the weight generating function, a particular colour from amongst the denominator parameters may appear multiply, and the flavour labels indicate this multiplicity.

*Keywords: *$\tau$ functions, weighted constellations, weighted Hurwitz numbers, generating functions, graphical enumeration

Harnad John, Runov Boris: Constellations and $\tau$-functions for rationally weighted Hurwitz numbers. *Ann. Inst. Henri Poincaré Comb. Phys. Interact.* 8 (2021), 119-158. doi: 10.4171/AIHPD/104