We examine the enumeration of certain Motzkin objects according to numbers crossings and nestings. With respect continued fractions, we compute express distributions statistics nestings over three sets, namely set $4321$-avoiding involutions, $3412$-avoiding $(321,3\bar{1}42)$-avoiding permutations. To get our results, exploit bijection Biane restricted sets $4321$- involutions which was charac...

LetM2(K) be the matrix algebra of order two over an in nite eld K of characteristic p 6= 2. If K is algebraically closed then, up to isomorphism, there are two involutions of rst kind on M2(K), namely the transpose and the symplectic. Even ifK is not algebraically closed, studying -identities it is su cient to consider only these two involutions. We describe bases of the polynomial identities w...

In this article we study a VOA with two Miyamoto involutions generating S3. In [M3], Miyamoto showed that a VOA generated by two conformal vectors whose Miyamoto involutions generate an automorphism group isomorphic to S3 is isomorphic to one of the four candidates he listed. We construct one of them and prove that our VOA is actually the same as VA(e, f) studied by Miyamoto. We also show that ...

Suppose that G is a group, X a subset of G and π a set of natural numbers. The π-product graph Pπ(G,X) has X as its vertex set and distinct vertices are joined by an edge if the order of their product is in π. If X is a set of involutions, then Pπ(G,X) is called a π-product involution graph. In this paper we study the connectivity and diameters of Pπ(G,X) when G is a finite symmetric group and ...

Bent functions are maximally nonlinear Boolean functions. They are important functions introduced by Rothaus and studied firstly by Dillon and next by many researchers for four decades. Since the complete classification of bent functions seems elusive, many researchers turn to design constructions of bent functions. In this note, we show that linear involutions (which are an important class of ...

An element of a group is reversible if it is conjugate to its own inverse, and it is strongly reversible if it is conjugate to its inverse by an involution. A group element is strongly reversible if and only if it can be expressed as a composite of two involutions. In this paper the reversible maps, the strongly reversible maps, and those maps that can be expressed as a composite of involutions...

1. Preliminaries. We are concerned with involutions without fixed points, together with equivariant maps connecting such involutions. An involution T is a homeomorphism of period 2 of a Hausdorff space X onto itself; that is, T(x) = x for all x £ X . There is associated with an involution T on X the orbit space X/T, obtained by identifying x with T(x) for all x G Z . Denote by v\ X—+X/T the dec...

The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of i...

In [6], the compatibility of Baer involutions and elations was studied. Further compatibility results for translation planes of dimension 2 were obtained in [10]. For example [6, Theorem (2.2)], we have the following. Let n be a derivable translation plane of even order q, q # 2, with derivable net N. Let EL denote an elation group in the translation complement which fixes the component L of N ...

In 1969, Hirsch posed the following problem: given a diffeomorphism f : N → N , and a compact invariant hyperbolic set Λ of f , describe the topology of Λ and the dynamics of f restricted to Λ. We solve the problem where Λ = M is a closed 3-manifold: if M is orientable, then it is a connected sum of tori and handles; otherwise it is a connected sum of tori and handles quotiented by involutions....