Related papers: Factorizations of cycles and multi-noded rooted tr…
We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that…
We construct a family of root-finding algorithms which exploit the branched covering structure of a polynomial of degree $d$ with a path-lifting algorithm for finding individual roots. In particular, the family includes an algorithm that…
We build on recent work of Yeats, Courtiel, and others involving connected chord diagrams. We first derive from a Hopf-algebraic foundation a class of tree-like functional equations and prove that they are solved by weighted generating…
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on…
We present a nice result on the probability of a cycle occurring in a randomly generated graph. We then provide some extensions and applications, including the proof of the famous Cayley formula, which states that the number of labeled…
Combinatorial classes T that are recursively defined using combinations of the standard multiset, sequence, directed cycle and cycle constructions, and their restrictions, have generating series T(z) with a positive radius of convergence;…
We evaluate combinatorially certain connection coefficients of the symmetric group that count the number of factorizations of a long cycle as a product of three permutations. Such factorizations admit an important topological interpretation…
Cayley's formula states that the number of labelled trees on $n$ vertices is $n^{n-2}$, and many of the current proofs involve complex structures or rigorous computation. We present a bijective proof of the formula by providing an…
Parking functions of length $n$ are well known to be in correspondence with both labelled trees on $n+1$ vertices and factorizations of the full cycle $\sigma_n=(0\,1\,\cdots\,n)$ into $n$ transpositions. In fact, these correspondences can…
We consider a new type of Hurwitz number, the number of ordered transitive factorizations of an arbitrary permutation into d-cycles. In this paper, we focus on the special case d = 3. The minimal number of transitive factorizations of any…
We present a coordinate space version of the factorization formula for the connected tree part of the chronological products. We consider a general framework and then we apply it for the QCD case.
Phylogenetic networks are leaf-labelled directed acyclic graphs that are used to describe non-treelike evolutionary histories and are thus a generalization of phylogenetic trees. The hybridization number of a phylogenetic network is the sum…
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…
The problem of counting ramified covers of a Riemann surface up to homeomorphism was proposed by Hurwitz in the late 1800's. This problem translates combinatorially into factoring a permutation of specified cycle type, with certain…
We give a short proof, based on symmetric function theory, of a formula due to Goupil and Schaeffer, counting the number of factorizations of a cycle of maximal length in the symmetric group, into the product of two permutations of given…
We prove a generalization of Orlov's theorem for matrix factorizations with $n$ steps. Let $X$ be a regular scheme, $W\colon X\to \mathbb{A}^1$ a flat morphism and $D:=W^{-1}(0)$ its central fiber. We construct an appropriate triangulated…
We prove factorization of the generating functional of connected tree diagrams by exploring that it is the Legendre transform of the action. This theorem is then applied to the example of a massive real scalar field theory in 2D. In the…
We investigate random minimal factorizations of the $n$-cycle, that is, factorizations of the permutation $(1 \, 2 \cdots n)$ into a product of cycles $\tau_1, \ldots, \tau_k$ whose lengths $\ell(\tau_1), \ldots, \ell(\tau_k)$ verify the…
Two factorizations of a permutation into products of cycles are equivalent if one can be obtained from the other by repeatedly interchanging adjacent disjoint factors. This paper studies the enumeration of equivalence classes under this…
We consider the determination of the number $c_k(\alpha)$ of ordered factorisations of an arbitrary permutation on n symbols, with cycle distribution $\alpha$, into k-cycles such that the factorisations have minimal length and such that the…