We discuss some problems and permutation statistics involving two different types of random permutations. Under the usual model of random permutations, we prove that the shifted coverage of the elements of {1, 2, . . . , k} of a random permutation over {1, 2, . . . , n}; that is, the size of the union of the cycles containing these elements, excluding these elements themselves, follows a negati...

Let G be a transitive permutation group acting on a finite set Ω. A cycle c is involved in a permutation g if and only if gc−1 fixes all points of Supp(c). We define a function Cyc(G) which takes the permutation group G to the group generated by the cycles involved in its elements. Let Cyc1(G) = Cyc(G) and Cyci+1(G) = Cyc(Cyci(G))). It is known that Cyc3(G) = Cyc4(G) for all such groups. We inv...

A permutation with no fixed points is called a derangement.The subset $mathcal{D}$ of a permutation group is derangement if all elements of $mathcal{D}$ are derangement.Let $G$ be a permutation group, a derangementgraph is one with vertex set $G$ and derangement set $mathcal{D}$ as connecting set. In this paper, we determine the spectrum of derangement graphs of order a product of three primes.

2. Permutation groups 1. Find the orbits of the permutation group on {1, . . . , 12} generated by the two permutations (1, 2)(3, 4)(5, 7)(6, 8)(9, 11)(10, 12) and (1, 2, 3)(4, 5, 6)(8, 9, 10). (We use the convention that points omitted from the cycle notation are fixed, that is, in cycles of length 1.) Where does this group fall in the hierarchy of properties defined in this section of the note...

We give a theoretical answer to a natural question arising from a few years of computational experiments on the problem of sorting a permutation by the minimum number of reversals, which has relevant applications in computational molecular biology. The experiments carried out on the problem showed that the so-called alternating-cycle lower bound is equal to the optimal solution value in almost ...

A number of fields, including the study of genome rearrangements and the design of interconnection networks, deal with the connected problems of sorting permutations in “as few moves as possible”, using a given set of allowed operations, or computing the number of moves the sorting process requires, often referred to as the distance of the permutation. These operations often act on just one or ...

We introduce and study k-commuting permutations. One of our main results is a characterization of permutations that k-commute with a given permutation. Using this characterization, we obtain formulas for the number of permutations that k-commute with a permutation [Formula: see text], for some cycle types of [Formula: see text]. Our enumerative results are related with integer sequences in "The...

We investigate the reduction to finite fields of polynomial automorphisms of the plane, which lead to invertible dynamics (permutations) of a finite space. We provide evidence that, if the map of the plane is non-integrable, then the presence or absence of a type of time-reversal symmetry called R-reversibility produces a clear signature in the cycle statistics of the associated permutation. If...

The single cycle T-function is a particular permutation function with complex algebraic structures, maximum period and efficient implementation in software and hardware. In this paper, on the basis of existing methods, we present a new construction using a class of single cycle T-functions meeting certain conditions to construct a family of new single cycle T-functions, and we also give the num...

There are many combinatorial expressions for evaluating characters of the Hecke algebra of type A. However, with rare exceptions, they give simple results only for permutations that have minimal length in their conjugacy class. For other permutations, a recursive formula has to be applied. Consequently, quantum immanants are complicated objects when expressed in the standard basis of the quantu...