We define an excedance number for the multi-colored permutation group i.e. the wreath product Zr1 × · · ·×Zrk ≀Sn and calculate its multi-distribution with some natural parameters. We also compute the multi-distribution of the parameters exc(π) and fix(π) over the sets of involutions in the multi-colored permutation group. Using this, we count the number of involutions in this group having a fi...

Dickson polynomials which are permutations are interesting combinatorial objects and well studied. In this paper, we describe Dickson polynomials of the first kind in F2[x] that are involutions over finite fields of characteristic 2. Such description is obtained using modular arithmetic’s tools. We give results related to the cardinality and the number of fixed points (in the context of cryptog...

Interesting properties of the partitions of a finite field Fq induced by the combination of involutions and trace maps are studied. The special features of involutions of the form u z , u being a fixed element of Fq, are exploited to generate pseudorandom numbers, the randomness resting on the uniform distribution of the images of zero-trace elements among the sets of non-zero trace elements of...

We show that the principal order ideal below an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motz...

Using the ideas and techniques developed by Bayer-Fluckiger, Shapiro and Tignol about hyperbolic involutions of central simple algebras, criteria for the hyperbolicity of involutions of the form σ⊗ τ and σ⊗ ρ, where σ is an involution of a central simple algebra A, τ is the nontrivial automorphism of a quadratic extension of the center of A and ρ is an involution of a quaternion algebra are obt...

In this article we study VOAs 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.

Let In,k (respectively, Jn,k) be the number of involutions (respectively, fixed-point free involutions) of {1, . . . , n} with k descents. Motivated by Brenti’s conjecture which states that the sequence In,0, In,1, . . . , In,n−1 is log-concave, we prove that the two sequences In,k and J2n,k are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers an,k such t...

The automorphism group AutfSy of the full 2-shift is conjectured to be generated by the shift and involutions. We approach this problem by studying a certain family of automorphisms whose order was unknown, but which we show to be finite and for which we find factorizations as products of involutions. The result of this investigation is the explicit construction of a subgroup %Z of Aut(Z2) ; %Z...