We define a statistic, called weight, on involutions and consider two applications in which this statistic arises. Let I (n)denote the set of all involutions on [n](= {1, 2, . . . , n}) and let F(2n)denote the set of all fixed point free involutions on [2n]. For an involution δ, let |δ| denote the number of 2-cycles in δ. Let [n]q = 1+q+· · ·+qn−1 and let (k)q denote the q-binomial coefficient....

We study generating functions for the number of involutions in Sn avoiding (or containing once) 132, and avoiding (or containing once) an arbitrary permutation τ on k letters. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind. In particular, we establish that involutions avoiding both 132 and 12 . . . k have the ...

Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. In t...

In [GM] Guibert and Mansour studied involutions on n letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary pattern on k letters. They also established a bijection between 132-avoiding involutions and Dyck word prefixes of same length. Extending this bijection to bilateral words allows to determine more parameters; in particular, we consider the...

In this paper, a classification of free involutions on 3-dimensional homotopy complex projective spaces is given. By the Z2-equivariant Montgomery-Yang correspondence, we obtain all smooth involutions on S with fixed-point set an embedded S.

Let Ak be the set of permutations in the symmetric group Sk with prefix 12. This paper concerns the enumeration of involutions which avoid the set of patterns Ak. We present a bijection between symmetric Schröder paths of length 2n and involutions of length n + 1 avoiding A4. Statistics such as the number of right-to-left maxima and fixed points of the involution correspond to the number of ste...

In this note we propose a novel algorithm for locating in a text T every occurrence of a strings that can be obtained from a given pattern P by successively applying antimorphic involutions on some of its factors. When the factors on which these involutions are applied overlap, a linear time algorithm is obtained. When we apply the involutions to non-overlapping factors we obtain an algorithm r...

We study the orientation preserving involutions of the orientable 3-dimensional handlebody Hg, for any genus g. A complete classification of such involutions is given in terms of their fixed points.

We study generating functions for the number of involutions of length n avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary permutation τ of length k. In several interesting cases these generating functions depend only on k and can be expressed via Chebyshev polynomials of the second kind. In particular, we show that involutions of length n avoiding ...

This paper is concerned with a class of functional differential equations whose argument transforms are involutions. In contrast to the earlier works in this area, which have used only involutions with a fixed point, we also admit involutions without a fixed point. In the first case, an initial value problem for a differential equation with involution is reduced to an initial value problem for ...