Matteo Silimbani

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.

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.

Semistandard tableaux associated with generalized labellings of posets

We describe a correspondence between a family of labelled partially ordered sets and semistandard Young tableaux. Moreover, we define some operations among labelled posets which naturally correspond to operations among the associated semistandard Young tableaux.