Related papers: Combinatorics and asymptotic behavior for double H…
In this paper we revisit several recent results on monotone and strictly monotone Hurwitz numbers, providing new proofs. In particular, we use various versions of these numbers to discuss methods of derivation of quantum spectral curves…
The harmonic numbers and higher-order harmonic numbers appear frequently in several areas which are related to combinatorial identities, many expressions involving special functions in analytic number theory, and analysis of algorithms. The…
In this paper, we study a certain type of Hurwitz numbers which count branched covers over the Riemann sphere admitting several branch points with fixed ramification types, one branch point with a fixed number of preimages, and one branch…
In general, Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. In this paper, we initiate the study of…
Using a combinatorial description of the Bernstein operator and its action on Schur functions, we describe the formal power series solutions to a family of partial differential equations known as the 2-Toda hierarchy. We also characterize…
We prove two explicit formulae for one-part double Hurwitz numbers with completed 3-cycles. We define "combinatorial Hodge integrals" from these numbers in the spirit of the celebrated ELSV formula. The obtained results imply some explicit…
Single Hurwitz numbers enumerate branched covers of the Riemann sphere with specified genus, prescribed ramification over infinity, and simple branching elsewhere. They exhibit a remarkably rich structure. In particular, they arise as…
We consider ramified coverings of P^1 with arbitrary ramification type over 0 and infinity and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings is a tau-function for the Toda lattice…
We consider weighted double Hurwitz numbers, with the weight given by arbitrary rational function times an exponent of the completed cycles. Both special singularities are arbitrary, with the lengths of cycles controlled by formal…
We study a generalization of the Harish-Chandra - Itzykson - Zuber integral to tensors and its expansion over trace-invariants of the two external tensors. This gives rise to natural generalizations of monotone double Hurwitz numbers, which…
The asymptotic study of class numbers of binary quadratic forms is a foundational problem in arithmetic statistics. Here, we investigate finer statistics of class numbers by studying their self-correlations under additive shifts.…
We present the geometry lying behind counting twin prime polynomials in $\mathbb{F}_q[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties…
We generalize the determinant representation of the KP $\tau$ functions to the case of the 2D Toda $\tau$ functions. The generating functions for the weighted Hurwitz numbers are a parametric family of 2D Toda $\tau$ functions; for which we…
In this paper, we study combinatorial and asymptotic properties of some interesting rational numbers called the Br\'ezin--Gross--Witten (BGW) numbers, which can be represented as the intersection numbers of psi and Theta classes on the…
This manuscript studies a special case of the Hurwitz enumeration problem: for branched covers from genus g compact Riemann surface to the Riemann sphere, with three branch points, and require the branching data at one of the branch points…
In this paper we present an example of a derivation of an ELSV-type formula using the methods of topological recursion. Namely, for orbifold Hurwitz numbers we give a new proof of the spectral curve topological recursion, in the sense of…
Classical Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. Monotone Hurwitz numbers restrict the…
This paper aims at extending the criterion that the quasi-stability of a polynomial is equivalent to the total nonnegativity of its Hurwitz matrix. We give a complete description of functions generating doubly infinite series with totally…
In this paper, we investigate the parity of three class of Hurwitz-type cyclotomic Euler sums using the methods of contour integration and residue computation, and derive explicit parity formulas for linear, quadratic, and some higher-order…
Hurwitz numbers with completed cycles are standard Hurwitz numbers with simple branch points replaced by completed cycles. In fact, simple branch points correspond to completed $2$-cycles. Okounkov and Pandharipande have established the…