17 research outputs found
The signed Eulerian numbers on involutions
We define an analogue of signed Eulerian numbers for involutions of
the symmetric group and derive some combinatorial properties of this sequence.
In particular, we exhibit both an explicit formula and a recurrence for
arising from the properties of its generating function.Comment: 10 page
Permutations and Pairs of Dyck Paths
We define a map v between the symmetric group Sn and the set of pairs of Dyck paths of semilength n. We show that the map v is injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map
Consecutive patterns in restricted permutations and involutions
It is well-known that the set of involutions of the symmetric
group corresponds bijectively - by the Foata map - to the set
of -permutations that avoid the two vincular patterns
We consider a bijection from the set
to the set of histoires de Laguerre, namely, bicolored Motzkin paths with
labelled steps, and study its properties when restricted to In particular, we show that the set
of permutations that avoids the
consecutive pattern and the classical pattern
corresponds via to the set of Motzkin paths, while its image under
is the set of restricted involutions We exploit these
results to determine the joint distribution of the statistics des and inv over
and over
Moreover, we determine the distribution in these two sets of every
consecutive pattern of length three. To this aim, we use a modified version of
the well-known Goulden-Jacson cluster method.Comment: 24 page
An algorithmic approach to maximal unions of chains in a partially ordered set
We exhibit a recursive procedure that enables us to construct a maximal union of k chains in a finite partially ordered set P for every positive integer k. As a consequence, we obtain an algorithmic proof of Greene's Duality Theorem on the relations between the cardinalities of maximal unions of chains and antichains in a finite poset
Young tableaux and k-matchings in finite posets
We present a recursive procedure that directly associates a Young tableau to an arbitrary finite poset with a linear extension, where no a priori information about its Ferrers shape is required
The joint distribution of consecutive patterns and descents in permutations avoiding 3-1-2
We exploit Krattenthaler’s bijection between the set Sn(3-1-2) of permutations in Sn avoiding the classical pattern 3-1-2 and Dyck n-paths to study the joint distribution over the set Sn(3-1-2) of a given consecutive pattern of length 3 and of descents. We utilize a involution on Dyck paths due to E. Deutsch to show that these consecutive patterns split into 3 equidistribution classes. In addition, we state equidistribution theorems concerning quadruplets of statistics relative to occurrences of consecutive patterns of length 3 and of descents in a permutation