This paper investigates the hardware implementation of arithmetical operations (multiplication and inversion) in symmetric and alternating groups, as well as in binary permutation groups (permutation groups of order 2). Various fast and space-efficient hardware architectures will be presented. High speed is achieved by employing switching networks, which effect multiplication in one clock cycle...

The (k,n)-perfect shuffle, a generalisation of the 2-way perfect shuffle, cuts a deck of kn cards into k equal size decks and interleaves them perfectly with the first card of the last deck at the top, the first card of the second-to-last deck as the second card, and so on. It is formally defined to be the permutation ρk,n : i→ ki (mod kn + 1), i ∈ {1,2, . . . ,kn}. We uncover the cycle structu...

Let G be a graph whose vertices are labeled 1, . . . , n, and π be a permutation on [n] := {1, 2, . . . , n}. A pebble pi that is initially placed at the vertex i has destination π(i) for each i ∈ [n]. At each step, we choose a matching and swap the two pebbles on each of the edges. Let rt(G, π), the routing number for π, be the minimum number of steps necessary for the pebbles to reach their d...

A square array whose all rows and columns are different permutations of the same length over symbol set is known as a Latin square. may or not be symmetric. For classification enumeration purposes, symmetric, non-symmetric, conjugate totally symmetric squares play vital roles. This article discusses Eigenproblem non-symmetric in well max-plus algebra. By defining certain vector corresponding to...

A permutation graph is a cubic graph admitting a 1-factor M whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if e is an edge of M such that every 4-cycle containing an edge of M contains e, then e is contained in a subdivision of the Petersen graph of a special type. In particular, if the graph is cyclically 5-edge-conn...

We consider the triangle-free process: Given an integer n, start by taking a uniformly random permutation of the edges of the complete n-vertex graph Kn. Then, traverse the edges of Kn according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving graph unless its addition creates a triangle. We study the evolving graph at around the time where Θ(...

We consider the triangle-free process: Given an integer n, start by taking a uniformly random permutation of the edges of the complete n-vertex graph Kn. Then, traverse the edges of Kn according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving graph unless its addition creates a triangle. We study the evolving graph at around the time where Θ(...

We say that a Euclidean lattice in Rn is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group Sn, i.e., if the lattice is closed under the action of some non-identity elements of Sn. Given a fixed element τ ∈ Sn, we study properties of the set of all lattices closed under the action of τ : we call such lattices τ -invariant. These lattices natura...

The set of synchronizing words a given n-state automaton forms regular language recognizable by an with \(2^n - n\) states. size recognizing for the is linked to computational problems related synchronization and length words. Hence, it natural investigate automata extremal this property, i.e., such that minimal deterministic has sync-maximal permutation groups have been introduced in [S. Hoffm...