Search

We enumerate and characterize some classes of alternating and reverse alternating involutions avoiding a single pattern of length three or four. If on one hand the case of patterns of length three is trivial, on the other hand, the length four case is more challenging and involves sequences of combinatorial interest, such as Motzkin and Fibonacci numbers.

We define a map between the set of permutations that avoid either the four patterns $3214,3241,4213,4231$ or $3124,3142,4123,4132$, and the set of Dyck prefixes. This map, when restricted to either of the two classes, turns out to be a bijection that allows us to determine some notable features of these permutations, such as the distribution of the statistics "number of ascents", "number of left-to-right maxima", "first element", and "position of the maximum element", Comment: 26 pages, 3 figures

We define a map $\nu$ between the symmetric group $S_n$ and the set of pairs of Dyck paths of semilength $n$. We show that the map $\nu$ is injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map., Comment: 16 pages, 8 figures

We study the descent distribution over the set of centrosymmetric permutations that avoid the pattern of length 3. Our main tool in the most puzzling case, namely, $\tau=123$ and $n$ even, is a bijection that associates a Dyck prefix of length $2n$ to every centrosymmetric permutation in $S_{2n}$ that avoids 123., Comment: 17 pages, 6 figures

The paper was withdrawn because of its significant overlap with a paper appeared recently., withdrawn paper

We present an extensive study of the Eulerian distribution on the set of self evacuated involutions, namely, involutions corresponding to standard Young tableaux that are fixed under the Sch$\ddot{\textrm{u}}$tzenberger map. We find some combinatorial properties for the generating polynomial of such distribution, together with an explicit formula for its coefficients. Afterwards, we carry out an analogous study for the subset of self evacuated involutions without fixed points., 18 pages

Centrosymmetric involutions in the symmetric group ${\mathcal S}_{2n}$ are permutations $\pi$ such that $\pi=\pi^{-1}$ and $\pi(i)+\pi(2n+1-i)=2n+1$ for all $i$ , and they are in bijection with involutions of the hyperoctahedral group. We describe the distribution of some natural descent statistics on $321$ -avoiding centrosymmetric involutions , including the number of descents in the first half of the involution, and the sum of the positions of these descents. Our results are based on two new bijections , one between centrosymmetric involutions in $ {\mathcal S} _{2n}$ and subsets of $\{1,\dots,n\}$ , and another one showing that certain statistics on Young diagrams that fit inside a rectangle are equidistributed . We also use the latter bijection to refine a known result stating that the distribution of the major index on $321$ -avoiding involutions is given by the $q$ -analogue of the central binomial coefficients.