On reduced unicellular hypermonopoles

Robert Cori; Gábor Hetyei

Home/Archive/arXiv

arXiv · 2403.19569

On reduced unicellular hypermonopoles

Robert Cori·Gábor Hetyei

View original source ↗Read source document ↗Export metadata ↓

Abstract

The problem of counting unicellular hypermonopoles by the number of their hyperedges is equivalent to describing the cycle length distribution of a product of two circular permutations, first solved by Zagier. The solution of this problem has also been used in the study of the cycle graph model of Bafna and Pevzner and of related models in mathematical biology. In this paper we develop a method to compute the finite number of reduced unicellular hypermonopoles of a given genus. The problem of representing any hypermap as a drawing is known to be simplifiable to solving the same problem for reduced unicellular hypermonopoles. We also outline a correspondence between our hypermap model, the cycle graph model of Bafna and Pevzner, and the polygon gluing model of Alexeev and Zograf. Reduced unicellular hypermonopoles correspond to reduced objects in the other models as well, and the notion of genus is the same.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Robert Cori, Gábor Hetyei. 2024-04-24. On reduced unicellular hypermonopoles. https://arxiv.org/abs/2403.19569

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Small doublings in abelian groups of prime power torsion

Let $A$ be a subset of $G$, where $G$ is a finite abelian group of torsion $r$. It was conjectured by Ruzsa that if $|A+A|\leq K|A|$, then $A$ is contained in a coset of $G$ of size at most $r^{CK}|A|$ for some constant $C$. The case $r=2$ received considerable attention in a sequence of papers, and was resolved by Green and Tao. Recently, Even-Zohar and Lovett settled the case when $r$ is a prime. In this paper, we confirm the conjecture when $r$ is a power of prime. In particular, the bound we obtain is tight.

math.CO↗

arXiv2026

Graph Polynomial for Colored Embedded Graphs: A Topological Approach

We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in physics. We also analyze a variant of these polynomials for colored embedded graphs. This is used to describe the change in the polynomial under basic graph theoretic operations. We conclude with several applications of this polynomial including detection of certain classes of graphs and the connection of this polynomial with topological entanglement entropy.

math.CO↗

arXiv2026

Schur positivity of the spiders $S(a,2,1)$ and $S(a,4,1)$ via noncommutative symmetric functions

We prove that the spider graphs $S(a,2,1)$ and $S(a,4,1)$ are Schur positive for all integers $a\ge1$. Together with the known $e$-positivity results, this completes the $e$- and Schur-positivity classification of both families. Our approach uses noncommutative symmetric functions, including a particularly simple ribbon expansion for the path lift with coefficients given by powers of two. We give a new proof of the Shareshian--Wachs path formula at $t=1$ and construct corresponding lifts for spiders. The Littlewood--Richardson rule converts their ribbon expansions into a general Schur-coefficient formula in terms of weighted Yamanouchi words. Mass-preserving multi-injections and a reduction to finitely many inequalities in degree $10$ then prove the required positivity; exact computer verification of these inequalities completes the proof in full generality.

math.CO↗