Monotone Hurwitz Numbers in Genus Zero

Abstract

Abstract. Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.

Monotone Orbifold Hurwitz Numbers

“…Theorem 9 (Cut-and-join recursion for monotone Hurwitz numbers [14]). The monotone Hurwitz numbers satisfy the recursion…”

Topological recursion and a quantum curve for monotone Hurwitz numbers

“…, (a m b m ) of the factorisation to satisfy the property of monotonicity -namely, that when the transpositions are written with the convention that a i < b i , then b 1 ≤ b 2 ≤ · · · ≤ b m . The monotone Hurwitz numbers first appeared in a series of papers by Goulden, Guay-Paquet and Novak, in which they arose as coefficients in the large N asymptotic expansion of the Harish-Chandra-Itzykson-Zuber (HCIZ) matrix integral over the unitary group U(N) [21,22,23]. The monotonicity condition is also natural from the standpoint of the Jucys-Murphy elements in the symmetric group algebra C[S |µ| ].…”

On the Convergence of Monotone Hurwitz Generating Functions

“…[10,18,19] and references therein, and it is known that they exhibit a wide array of subtle structural properties. In [7,8,9], we introduced and studied the monotone double Hurwitz numbers H g (α, β), which count the same walks as the double Hurwitz numbers H g (α, β), but with the monotonicity constraint imposed. In [7,8], we demonstrated that the monotone double Hurwitz numbers enjoy a high degree of structural similarity with the classical double Hurwitz numbers, and in [9] we applied this structure to the asymptotic analysis of unitary matrix integrals via (1.1).…”