Related papers: Long mutation cycles

We characterise and enumerate permutations that are sortable by n-4 passes through a stack. We conjecture the number of permutations sortable by n-5 passes, and also the form of a formula for the general case n-k, which involves a…

A Hamiltonian cycle of a graph is a closed path that visits every vertex once and only once. It has been difficult to count the number of Hamiltonian cycles on regular lattices with periodic boundary conditions, e.g. lattices on a torus,…

A class of one-dimensional convolutional codes will be presented. They are all MDS codes, i. e., have the largest distance among all one-dimensional codes of the same length n and overall constraint length delta. Furthermore, their extended…

Identifying and characterizing mutational paths is an important issue in evolutionary biology and in bioengineering. We here introduce a generic description of mutational paths in terms of the goodness of sequences and of the mutational…

A biased graph is a graph $G$, together with a distinguished subset $\mathcal{B}$ of its cycles so that no Theta-subgraph of $G$ contains precisely two cycles in $\mathcal{B}$. A large number of biased graphs can be constructed by choosing…

We classify rotary (orientably-regular) maps whose underlying graphs are multicycles. For the multicycle $\mathrm{C}_n^{(\lambda)}$ of length $n$ and edge-multiplicity $\lambda$, we determine all rotary embeddings for $n\geqslant 3$ and…

We consider the distribution of cycles in two models of random permutations, that are related to one another. In the first model, cycles receive a weight that depends on their length. The second model deals with permutations of points in…

We give a generating function for the number of unimodal permutations with a given cycle structure.

Signature is an infinite graded sequence of statistics known to characterize geometric rough paths, which includes the paths with bounded variation. This object has been studied successfully for machine learning with mostly applications in…

The study of pattern containment and avoidance for linear permutations is a well-established area of enumerative combinatorics. A cyclic permutation is the set of all rotations of a linear permutation. Callan initiated the study of…

For a positive constant $\alpha$ a graph $G$ on $n$ vertices is called an $\alpha$-expander if every vertex set $U$ of size at most $n/2$ has an external neighborhood whose size is at least $\alpha\left|U\right|$. We study cycle lengths in…

Let $G$ be a finite connected simple graph with $n$ vertices and $m$ edges. We show that, when $G$ is not bipartite, the number of $4$-cycles contained in $G$ is at most $\binom{m-n+1}{2}$. We further provide a short combinatorial proof of…

We enumerate permutations in the two permutation classes $\text{Av}_n(312, 4321)$ and $\text{Av}_n(321, 4123)$ by the number of cycles each permutation admits. We also refine this enumeration with respect to several statistics.

We solve four similar problems: For every fixed $s$ and large $n$, we describe all values of $n_1,\ldots,n_s$ such that for every $2$-edge-coloring of the complete $s$-partite graph $K_{n_1,\ldots,n_s}$ there exists a monochromatic (i)…

Mutational signatures describe the pattern of mutations over the different mutation types. Each mutation type is determined by a base substitution and the flanking nucleotides to the left and right of that base substitution. Due to the…

Li, Nikiforov and Schelp conjectured that a 2-edge coloured graph G with order n and minimal degree strictly greater than 3n/4 contains a monochromatic cycle of length l, for all l at least four and at most n/2. We prove this conjecture for…

A pattern class is a set of permutations closed under the formation of subpermutations. Such classes can be characterised as those permutations not involving a particular set of forbidden permutations. A simple collection of necessary and…

We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which…

We study the evolution of graphs densifying by adding edges: Two vertices are chosen randomly, and an edge is (i) established if each vertex belongs to a tree; (ii) established with probability $p$ if only one vertex belongs to a tree;…

Let $G$ be an $n$-vertex graph obtained by adding chords to a cycle of length $n$. Markstr\"{o}m asked for the maximum number of edges in $G$ if there are no two cycles in $G$ with the same length. A simple counting argument shows that such…