Related papers: On the decycling of powers and products of cycles

In this note, we obtain a formula which leads to a practical and efficient method to calculate the number of partitions of n into parts not divisible by m for given natural numbers n and m.

We discuss and prove a number of results for calculating characteristic cycles, or graded, enriched characteristic cycles. We concentrate particularly on results related to hypersurfaces.

Based on the earlier work of Li (European J. Combin. 1997) and Dobson (Discrete Math. 2008), in this paper we complete the classification of cyclic $m$-DCI-groups and $m$-CI-groups. For a positive integer $m$ such that $m \ge 3$, we show…

We prove a recursive formula for the exterior and symmetric powers of modules for a cyclic 2-group. This makes computation straightforward. Previously, a complete description was only known for cyclic groups of prime order.

We deduce a simple representation and the invariant factor decompositions of the subgroups of the group $\Bbb{Z}_m \times \Bbb{Z}_n$, where $m$ and $n$ are arbitrary positive integers. We obtain formulas for the total number of subgroups…

Two algorithms for computing $P(n,m)$, the number of integer partitions of $n$ into exactly $m$ parts, are described, and using a combination of these two algorithms, the resulting algorithm is $O(n^{3/2})$. The second algorithm uses a list…

Let $m, n$ be positive integers such that $m>1$ divides $n$. In this paper, we introduce a special class of piecewise-affine permutations of the finite set $[1, n]:=\{1, \ldots, n\}$ with the property that the reduction $\pmod m$ of $m$…

We give a new expression for the number of factorizations of a full cycle into an ordered product of permutations of specified cycle types. This is done through purely algebraic means, extending work of Biane. We deduce from our result a…

We call $n$ a cyclic number if every group of order $n$ is cyclic. It is implicit in work of Dickson, and explicit in work of Szele, that $n$ is cyclic precisely when $\gcd(n,\phi(n))=1$. With $C(x)$ denoting the count of cyclic $n\le x$,…

We study the mixing properties of permutations obtained as a product of two uniformly random permutations of fixed cycle types. For instance, we give an exact formula for the probability that elements $1,2,...,k$ are in distinct cycles of…

We study the one-dimensional wave equation with an inverse power potential that equals $const.x^{-m}$ for large $|x|$ where $m$ is any positive integer greater than or equal to 3. We show that the solution decays pointwise like $t^{-m}$ for…

If $n$ is even, the $n$-dimensional hypercube can be decomposed into edge-disjoint cycles of length $2^i$ for every value of $i$ from $2$ to $n$.

Recently, there has been intensive research on the weight distributions of cyclic codes. In this paper, we compute the weight distributions of three classes of cyclic codes with Niho exponents. More specifically, we obtain two classes of…

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Cyclic codes have been studied for many years, but their weight…

We obtain sharp bounds for the number of n-cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. En route, we prove some sharp estimates on power…

Strong decay probabilities are calculated using the Lorentz contracted wave functions of decay products, determined in the arbitrary dynamical scheme with the instantaneous interaction. It is shown that the decay width obtains an additional…

Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let $\alpha $ be a generator of…

In this paper we consider combinatorial numbers $C_{m, k}$ for $m\ge 1$ and $k\ge 0$ which unifies the entries of the Catalan triangles $ B_{n, k}$ and $ A_{n, k}$ for appropriate values of parameters $m$ and $k$, i.e., $B_{n,…

In this paper we study the cycle descent statistic on permutations. Several involutions on permutations and derangements are constructed. Moreover, we construct a bijection between negative cycle descent permutations and Callan perfect…

We define a generalization of the winding number of a piecewise $C^1$ cycle in the complex plane which has a geometric meaning also for points which lie on the cycle. The computation of this winding number relies on the Cauchy principal…