Flavio Bonetti

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 a ? n 2 ? × ? 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.

Some permutations on Dyck words

We examine three permutations on Dyck words. The first one, α, is related to the Baker and Norine theorem on graphs, the second one, β, is the symmetry, and the third one is the composition of these two. The first two permutations are involutions and it is not difficult to compute the number of their fixed points, while the third one has cycles of different lengths. We show that the lengths of these cycles are odd numbers. This result allows us to give some information about the interplay between α and β, and a characterization of the fixed points of α ź β .

Bijections and recurrences for integer partitions into a bounded number of parts

Abstract Let Π s ( n ) denote the set of partitions of the integer n into exactly s parts, and Π s ( 2 ) ( n ) the subset of Π s ( n ) containing all partitions whose two largest parts coincide. We present a bijection between Π s ( 2 ) ( n ) and Π s − 1 ( m ) for a suitable m n in the cases s = 3 , 4 . Such bijections yield recurrence formulas for the numbers P 3 ( n ) and P 4 ( n ) of partitions of n into 3 and 4 parts. Furthermore, we show that the present approach can be extended to the case s = 5 , 6 .

Permutations and Pairs of Dyck Paths

We define a map between the symmetric group and the set of pairs of Dyck paths of semilength . We show that the map is injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map.