Related papers: s-Overlap Cycles for Permutations

In this paper, we study discrepancy questions for spanning subgraphs of $k$-uniform hypergraphs. Our main result is that, for any integers $k \ge 3$ and $r \ge 2$, any $r$-colouring of the edges of a $k$-uniform $n$-vertex hypergraph $G$…

In 1973 Bermond, Germa, Heydemann and Sotteau conjectured that if $n$ divides $\binom{n}{k}$, then the complete $k$-uniform hypergraph on $n$ vertices has a decomposition into Hamilton Berge cycles. Here a Berge cycle consists of an…

S. Baker (2019), B. B\'ar\'any and A. K\"{a}enm\"{a}ki (2019) independently showed that there exist iterated function systems without exact overlaps and there are super-exponentially close cylinders at all small levels. We adapt the method…

It is well known that Universal Cycles of $k$-letter words on an $n$-letter alphabet exist for all $k$ and $n$. In this paper, we prove that Universal Cycles exist for restricted classes of words, including: non-bijections, equitable words…

A $k$-path is a hypergraph P_k = e_1,e_2,...,e_k such that |e_i \cap e_j| = 1 if |j - i| = 1 and e_i \cap e_j is empty otherwise. A k-cycle is a hypergraph C_k = e_1,e_2,.. ,e_k obtained from a (k-1)-path e_1,e_2,...,e_{k-1} by adding an…

The ``overlapping-cycles shuffle'' mixes a deck of $n$ cards by moving either the $n$th card or the $(n-k)$th card to the top of the deck, with probability half each. We determine the spectral gap for the location of a single card, which,…

We define generalized de Bruijn words as those words having a Burrows-Wheeler transform that is a concatenation of permutations of the alphabet. We show that generalized de Bruijn words are in 1-to-1 correspondence with Hamiltonian cycles…

In 1981, Erd\H{o}s and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let $\mathcal{C}(G)$ be the set of cycle lengths in a graph $G$ and let…

Until recently, it was an important open problem in Fractal Geometry to determine whether there exists an iterated function system acting on $\mathbb{R}$ with no exact overlaps for which cylinders are super-exponentially close at all small…

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 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…

In order to study certain questions concerning the distribution of the overlap in Sherrington--Kirkpatrick type models, such as the chaos and ultrametricity problems, it seems natural to study the free energy of multiple systems with…

We show that for any square-free natural number $n$ and any global field $K$ with $(\text{char}(K), n)=1$ containing the $n$th roots of unity, the pairs $(x,y)\in K^*\times K^*$ such that $x$ is not a norm of $K(\sqrt[n]{y})/K$ form a…

The Permutation Pattern Matching problem, asking whether a pattern permutation $\pi$ is contained in a permutation $\tau$, is known to be NP-complete. In this paper we present two polynomial time algorithms for special cases. The first…

Consider the process of random transpositions on the complete graph. We use representation theory to give an exact, simple formula for the expected number of cycles of size k at time t, in terms of an incomplete Beta function. Using this we…

In earlier versions of the community discovering problem, the overlap between communities was restricted by a simple count upper-bound [17,5,11,8]. In this paper, we introduce the $\Pi$-Packing with $\alpha()$-Overlap problem to allow for…

Each positive increasing integer sequence $\{a_n\}_{n\geq 0}$ can serve as a numeration system to represent each non-negative integer by means of suitable coefficient strings. We analyse the case of $k$-generalized Fibonacci sequences…

Common intervals of K permutations over the same set of n elements were firstly investigated by T. Uno and M.Yagiura (Algorithmica, 26:290:309, 2000), who proposed an efficient algorithm to find common intervals when K=2. Several particular…

We introduce the "Median Inverse Problem" for metric spaces. In particular, having a permutation $\pi$ in the symmetric group $S_n$ (endowed with the breakpoint distance), we study the set of all $k$-subsets $\{x_1,...,x_k\}\subset S_n$ for…

Consider the random walk on the permutation group obtained when the step distribution is uniform on a given conjugacy class. It is shown that there is a critical time at which two phase transitions occur simultaneously. On the one hand, the…