Combinatorial properties of the numbers of tableaux of bounded height

We introduce an infinite family of lower triangular matrices ¡(s), where °s n;i counts the standard Young tableaux on n cells and with at most s columns on a suitable subset of shapes. We show that the entries of these matrices satisfy a three-term row recurrence and we deduce recursive and asymptotic properties for the total number ¿s(n) of tableaux on n cells and with at most s columns

Two permutation classes enumerated by the central binomial coefficients

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 figure

The Eulerian numbers on restricted centrosymmetric permutations

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 figure

Descent sets on 321-avoiding involutions and hook decompositions of partitions

We show that the distribution of the major index over the set of involutions in S_n that avoid the pattern 321 is given by the q-analogue of the n-th central binomial coefficient. The proof consists of a composition of three non-trivial bijections, one being the Robinson-Schensted correspondence, ultimately mapping those involutions with major index m into partitions of m whose Young diagram fits inside an n/2 by n/2 box. We also obtain a refinement that keeps track of the descent set, and we deduce an analogous result for the comajor index of 123-avoiding involutions

Restricted involutions and Motzkin paths

AbstractWe show how a bijection due to Biane between involutions and labelled Motzkin paths yields bijections between Motzkin paths and two families of restricted involutions that are counted by Motzkin numbers, namely, involutions avoiding 4321 and 3412. As a consequence, we derive characterizations of Motzkin paths corresponding to involutions avoiding either 4321 or 3412 together with any pattern of length 3. Furthermore, we exploit the described bijection to study some notable subsets of the set of restricted involutions, namely, fixed point free and centrosymmetric restricted involutions

and

Abstract. We study the descent distribution over the set of centrosymmetric permutations that avoid a pattern of length 3. In the most puzzling case, namely, τ = 123 and n even, our main tool is a bijection that associates a Dyck pre x of length 2n to every centrosymmetric permutation in S2n that avoids 123. Mathematics Subject Classi cation(2000). 05A05, 05A15, 05A19

Pattern avoiding alternating involutions

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

Pattern avoiding meandric permutations

We study and characterize meandric permutations avoiding one or more patterns of length three, and nd explicit formulae for the cardinality of each of these sets. We determine the distribution of the descent statistic for the set of meandric permutations avoiding the pattern 231. The sets of meandric permutations avoiding any other pattern of length three can be either trivially determined, or deduced from the 231 case via the symmetries of the square. In the 231 case we provide a bijection with a set of Motzkin paths that maps the statistic \number of descents of a permutation" to the statistic \number of non-horizontal steps of a path"