In the symmetric group on a set of size 2n, let P_{2n} denote the conjugacy class of involutions with no fixed points (equivalently, we refer to these as ``pairings'', since each disjoint cycle has length 2). Harer and Zagier explicitly determined the distribution of the number of disjoint cycles in the product of a fixed cycle of length 2n and the elements of P_{2n}. Their famous result has been reproved many times, primarily because it can be interpreted as the genus distribution for 2-cell embeddings in an orientable surface,of a graph with a single vertex attached to n loops. In this paper we give a new formula for the cycle distribution when a fixed permutation with two cycles (say the lengths are p,q, where p+q=2n) is multiplied by the elements of P_{2n}. It can be interpreted as the genus distribution for 2-cell embeddings in an orientable surface, of a graph with two vertices, of degrees p and q. In terms of these graphs, the formula involves a parameter that allows us to specify, separately, the number of edges between the two vertices and the number of loops at each of the vertices. The proof is combinatorial, and uses a new algorithm that we introduce to create all rooted forests containing a given rooted forest.
We show an instance of the cyclic sieving phenomenon on annular noncrossing permutations with given cycle types. We define annular $q$-Kreweras numbers, annular $q$-Narayana numbers, and annular $q$-Catalan number, all of which are polynomials in $q$. We then show that these polynomials exhibit the cyclic sieving phenomenon on annular noncrossing permutations. We also show that a sum of annular $q$-Kreweras numbers becomes an annular $q$-Narayana number and a sum of $q$-Narayana numbers becomes an annular $q$-Catalan number.
We give two combinatorial proofs of Goulden and Jackson's formula for the number of minimal transitive factorizations of a permutation when the permutation has two cycles. We use the recent result of Goulden, Nica, and Oancea on the number of maximal chains of annular noncrossing partitions of type $B$.
We present the Möbius functions of several posets of annular noncrossing objects, namely a self-dual extension of the annular noncrossing permutations, minimal length annular partitioned permutations, and annular noncrossing partitons.
We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a four-colored noncrossing partition. This representation may be selected in a unique way for permutations containing no trivial cycles. The conclusion follows from a general generating function formula that holds for any class of permutations that is closed under the removal and reinsertion of trivial cycles. Ou…
Recently, Gross et al. posed the LLC conjecture for the locally log-concavity of the genus distribution of every graph, and provided an equivalent combinatorial version, the CLLC conjecture, on the log-concavity of the generating function counting cycles of some permutation compositions. In this paper, we confirm the CLLC conjecture for cyclic permutations, with the aid of Hultman numbers and by applying the Hermite--Biehler theorem on the generating function of Stirling numbers of the first kind. This leads to a further conjecture that every local genus polynomial is real-rooted.
Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface with any genus and number of boundary components. If only bigons are allowed, then it becomes an arc diagram. The count of arc diagrams is known to have a rich structure. We show that the count of polygon diagrams exhibits the same interesting behaviours, in particular it is almost polynomial in the number of points on the boundary components, and the leadi…
The central object of study is a formal power series that we call the content series, a symmetric function involving an arbitrary underlying formal power series $f$ in the contents of the cells in a partition. In previous work we have shown that the content series satisfies the KP equations. The main result of this paper is a new partial differential equation for which the content series is the unique solution, subject to a simple initial condition. This equation is expressed in terms of families of operators that we call $\mathcal{U}$ and $\mathcal{D}$ operators, whose action on the Schur sym…
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 the branched covers counted by the Hurwitz numbers, 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-cu…
In this paper, we introduce plane permutations, i.e. pairs $\mathfrak{p}=(s,π)$ where $s$ is an $n$-cycle and $π$ is an arbitrary permutation, represented as a two-row array. Accordingly a plane permutation gives rise to three distinct permutations: the permutation induced by the upper horizontal ($s$), the vertical $π$) and the diagonal ($D_{\mathfrak{p}}$) of the array. The latter can also be viewed as the three permutations of a hypermap. In particular, a map corresponds to a plane permutation, in which the diagonal is a fixed point-free involution. We study the transposition action on plan…
We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting transitive factorisations of permutations into transpositions.) These results prove a conjecture of Goulden and Jackson, and extend results of Hurwitz and many others.
Light-cone string diagrams have been used to reproduce the orbifold Euler characteristic of moduli spaces of punctured Riemann surfaces at low genus and with few punctures. Nakamura studied the meromorphic differential introduced by Giddings and Wolpert to characterise light-cone diagrams and introduced a class of graphs related to this differential. These Nakamura graphs were used to parametrise the cells in a light-cone cell decomposition of moduli space. We develop links between Nakamura graphs and realisations of the worldsheet as branched covers. This leads to a development of the combina…
Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, 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 previous work we gave an explicit formula for monotone Hurwitz numbers in genus zero. In this paper we consider monotone Hurwitz…
We consider the problem of counting and of listing topologically inequivalent "planar" {4-valent} maps with a single component and a given number n of vertices. This enables us to count and to tabulate immersions of a circle in a sphere (spherical curves), extending results by Arnold and followers. Different options where the circle and/or the sphere are/is oriented are considered in turn, following Arnold's classification of the different types of symmetries. We also consider the case of bicolourable and bicoloured maps or immersions, where faces are bicoloured. Our method extends to immersio…
We study the asymptotic behaviour of random factorizations of the $n$-cycle into transpositions of fixed genus $g>0$. They have a geometric interpretation as branched covers of the sphere and their enumeration as Hurwitz numbers was extensively studied in algebraic combinatorics and enumerative geometry. On the probabilistic side, several models and properties of permutation factorizations were studied in previous works, in particular minimal factorizations of cycles into transpositions (which corresponds to the case $g=0$ of this work). Using the representation of factorizations via unicellul…
Starting with a collection of $n$ oriented polygonal discs, with an even number $N$ of sides in total, we generate a random oriented surface by randomly matching the sides of discs and properly gluing them together. Encoding the surface in a random permutation $γ$ of $[N]$, we use the Fourier transform on $S_N$ to show that $γ$ is asymptotic to the permutation distributed uniformly on the alternating group $A_N$ ($A_N^c$ resp.) if $N-n$ and $N/2$ are of the same (opposite resp.) parity. We use this to prove a local central limit theorem for the number of vertices on the surface, whence for its…
Let E be a finite set. Given permutations x and y of E that together generate a transitive subgroup, for which s is it true that x and the conjugate of y by s also generate a transitive subgroup? Such transitive permutation pairs encode dessins d'enfants, important graph-theoretic objects which are also known to have great arithmetic significance. The absolute Galois group acts on dessins d'enfants and permutes them in a very mysterious way. Two dessins d'enfants that share certain elementary combinatorial features are related by conjugations as above, and dessins d'enfants in the same Galois-…
The classical Hurwitz numbers count the fixed-length transitive transposition factorizations of a permutation, with a remarkable product formula for the case of minimum length (genus $0$). We study the analogue of these numbers for reflection groups with the following generalization of transitivity: say that a reflection factorization of an element in a reflection group $W$ is full if the factors generate the whole group $W$. We compute the generating function for full factorizations of arbitrary length for an arbitrary element in a group in the combinatorial family $G(m, p, n)$ of complex ref…
We present a procedure to sample uniformly from the set of combinatorial isomorphism types of balanced triangulations of surfaces - also known as graph-encoded surfaces. For a given number $n$, the sample is a weighted set of graph-encoded surfaces with $2n$ triangles. The sampling procedure relies on connections between graph-encoded surfaces and permutations, and basic properties of the symmetric group. We implement our method and present a number of experimental findings based on the analysis of $138$ million runs of our sampling procedure, producing graph-encoded surfaces with up to $280$ …
In this paper we continue the perturbative analysis of the quartic Kontsevich model. We investigate meromorphic functions $Ω^{(0)}_m$ with $m=1,2$, that obey blobbed topological recursion. We calculate their expansions and check their equivalence to sums of ribbon graph weights, which are obtained with common methods of perturbation theory in QFT, up to fifth order in the coupling using Mathematica. Furthermore, we provide a catalog of permutation pairs $(α,σ)$, which encode all 5660 vacuum ribbon graphs that contribute to the free energy $\mathcal{F}^{(g)}$ with genus $g\geq 0$ up to fifth or…
