Related papers: Necklaces over a group with identity product

Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every…

A circular word, or a necklace, is an equivalence class under conjugation of a word. A fundamental question concerning regularities in standard words is bounding the number of distinct squares in a word of length $n$. The famous conjecture…

A classic problem in enumerative combinatorics is to count the number of derangements, that is, permutations with no fixed point. Inspired by a recent generalization to facet derangements of the hypercube by Gordon and McMahon, we…

Let G be a Lie group, $g = Lie(G)$ - its Lie algebra, $g*$ - the dual vector space and $\widehat G$ - the set of equivalence classes of unitary irreducible representations of $G$. The orbit method [1] establishes a correspondence between…

We show several geometric and algebraic aspects of a necklace: a link composed with a core circle and a series of circles linked to this core. We first prove that the fundamental group of the configuration space of necklaces (that we will…

Fix a finite alphabet. A necklace is a circular word. For positive integers $n$ and~$k$, a necklace is $(n,k)$-perfect if all words of length $n$ occur $k$ times but at positions with different congruence modulo $k$, for any convention of…

Let $G$ be a finitely generated group. We prove that the $n$-fold tensor product $G^{\otimes n}$ is finite (resp. polycyclic) if and only $G$ is finite (resp. polycyclic). Further, assuming that $G$ is finitely presented, we show that…

Cochran defined the nth-order integral Alexander module of a knot in the three sphere as the first homology group of the knot's (n+1)th-iterated abelian cover. The case n=0 gives the classical Alexander module (and polynomial). After a…

Ezra Getzler notes in the proof of the main theorem of "The semi-classical approximation for modular operads" that "A proof of the theorem could no doubt be given using [a combinatorial interpretation in terms of a sum over necklaces];…

We consider the irreducible representations each of dimension 2 of the necklace braid group $\mathcal{NB}_n$ ($n=2,3,4$). We then consider the tensor product of the representations of $\mathcal{NB}_n$ ($n=2,3,4$) and determine necessary and…

We provide a new combinatorial approach to studying the collection of N-infinity-operads in G-equivariant homotopy theory for G a finite cyclic group. In particular, we show that for G the cyclic group of order p^n the natural order on the…

Let $G$ be a group and $G_0 \subseteq G$ be a subset. A sequence over $G_0$ means a finite sequence of terms from $G_0$, where the order of elements is disregarded and the repetition of elements is allowed. A product-one sequence is a…

The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and Ushakov. It is defined for each finitely generated group $G$ and takes as input group elements $g_1,\ldots,g_n,g\in G$ and asks whether there are $x_1,\ldots,x_n\ge…

A necklace can be considered as a cyclic list of $n$ red and $n$ blue beads in an arbitrary order, and the goal is to fold it into two and find a large cross-free matching of pairs of beads of different colors. We give a counterexample for…

We look into computational aspects of two classical knot invariants. We look for ways of simplifying the computation of the coloring invariant and of the Alexander module. We support our ideas with explicit computations on pretzel knots.

This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let $a_1G_1,...,a_kG_k$ be left cosets in a group $G$ such that ${a_iG_i}_{i=1}^k$ covers each element of $G$ at least $m$ times but none of its…

Let $n$ be a positive integer and $G(n)$ denote the number of non-isomorphic finite groups of order $n$. It is well-known that $G(n) = 1$ if and only if $(n,\phi(n)) = 1$, where $\phi(n)$ and $(a, b)$ denote the Euler's totient function and…

Let $G=< x,t\mid w>$ be a one-relator group, where $w$ is a word in $x,t$. If $w$ is a product of conjugates of $x$ then, associated with $w$, there is a polynomial $A_w(X)$ over the integers, which in the case when $G$ is a knot group, is…

M. B. Levin used Sobol-Faure low discrepancy sequences with Pascal matrices modulo $2$ to construct, for each integer $b$, a real number $x$ such that the first $N$ terms of the sequence $(b^n x \mod 1)_{n\geq 1}$ have discrepancy $O((\log…

Let $(G, 1_G)$ be a finite group and let $S=g_1\bdot \ldots\bdot g_{\ell}$ be a nonempty sequence over $G$. We say $S$ is a tiny product-one sequence if its terms can be ordered such that their product equals $1_G$ and…