Related papers: Counting friezes in type D_n
In this article we prove that equation $\phi(x)=n$, for a fixed $n$, admits a finite number of solutions, we find the general form of these solutions, and we show that: if $x_0$ is a unique solution of this equation then $x_0$ is a product…
In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.
The real type of a finite family of univariate polynomials characterizes the combined sign behavior of the polynomials over the real line. We derive an explicit formula for the number of real types subject to given degree bounds. For the…
The fusion rules in $\mathrm{Rep}_f D(G)$ for a finite group $G$ can be computed in terms of character inner products. Using an explicit formula for these fusion rules, we show that $\mathrm{Rep}_f D(G)$ is multiplicity free for two…
Let D be the cluster category of Dynkin type A_{\infty}. This paper provides a bijection between torsion theories in D and certain configurations of arcs connecting non-neighbouring integers.
A classic result of Conway and Coxeter on frieze patterns has been generalized to a bijection between $p$-angulations of regular polygons and frieze patterns of type $\Lambda_p$. One of the features of Conway-Coxeter theory is a…
We provide two alternative ways to determine the number of (bi-)twisted conjugacy classes in a finite group: one by counting certain irreducible characters and one by counting certain twisted conjugacy classes of other endomorphisms. In…
We briefly review the construction of N=2 WZW models in terms of Manin triples. We analyse the restrictions which should be imposed on the gluing conditions of the affine currents in order to preserve half of the bulk supersymmetry. In…
We count subrings of small index of $\mathbb{Z}^n$, where the addition and multiplication are defined componentwise. Let $f_n(k)$ denote the number of subrings of index $k$. For any $n$, we give a formula for this quantity for all integers…
Knecht considers the enumeration of coronas. This is a counting problem for two specific types of lozenge tilings. Their exact closed formulas are conjectured in [A380346] and [A380416] on the OEIS. We prove this conjecture by using the…
A packing of subsets $\mathcal S_1,..., \mathcal S_n$ in a group $G$ is a sequence $(g_1,...,g_n)$ such that $g_1\mathcal S_1,...,g_n\mathcal S_n$ are disjoint subsets of $G$. We give a formula for the number of packings if the group $G$ is…
We prove that for all $n$, simultaneously, we can choose prime filtrations of $R/I^n$ such that the set of primes appearing in these filtrations is finite.
The aim of this article is to obtain a formula giving, for a positive integer $n$, the number of roots of the $n^{th}$ continuant polynomial over a finite local ring. In particular, we will give counting formulae for the roots of the…
In this paper, we give the enumeration of z-classes in finite Coxeter groups.
We speculate on whether a certain p-adic stability phenomenon, observed by David Robbins empirically for Dodgson condensation, appears in other nonlinear recurrence relations that "unexpectedly" produce integer or nearly-integer sequences.…
We discuss the problem of counting vertices in Gelfand-Zetlin polytopes. Namely, we deduce a partial differential equation with constant coefficients on the exponential generating function for these numbers. For some particular classes of…
We show that for a Frobenius split variety, there are only finitely many closed subvarieties which are compatibly-split.
D-finite functions and P-recursive sequences are defined in terms of linear differential and recurrence equations with polynomial coefficients. In this paper, we introduce a class of numbers closely related to D-finite functions and…
We establish some upper and lower bounds for the number of rational points of Prym varieties over finite fields.
We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all d >= 4 and n sufficiently large, depending on d. As a…